Formulas · Mathematics

Trigonometric functions

23 formulas, each with worked examples. Revision sheet · Practise these formulas

Class 11

Degrees to radians

R = \frac{π}{180} D

What each letter means
$R$	the angle in radians
$D$	the same angle in degrees (degrees)

A full turn is 360° or 2π radians, so 180° is π radians. To change degrees to radians, multiply by π/180; to change radians to degrees, multiply by 180/π.

Why it works

One radian is the angle at the centre of a circle made by an arc as long as the radius. The whole circumference is 2π radii long, so a full turn is 2π radians. The same full turn is 360°, so π radians = 180°, and every angle changes in the same ratio.

When to use it

Whenever a formula needs the angle in radians (the arc l = rθ, calculus with sin x), or when a book gives an angle like π/6 and you want it in degrees.

How to use it

Degrees to radians: multiply by π and divide by 180, then simplify the fraction (60° = 60π/180 = π/3).
Radians to degrees: multiply by 180 and divide by π (3π/4 = 3 × 180 ÷ 4 = 135°).
Keep π in the answer unless a decimal is asked for.

To remember: π radians is a half turn: 180°.

Other forms

Rearranged $D = \frac{180}{π} R$ Radians to degrees.
To remember it1 radian is about 57.3°, and 1° is about 0.01745 radian.

Worked examples

Story

Through how many radians does the minute hand of a clock turn in 20 minutes?

In 60 minutes it turns 360°, so in 20 minutes it turns 120°.
120 × π ÷ 180 = 2π/3.
$\frac{120 π}{180} = \frac{2 π}{3}$

Answer: 2π/3 radians

Picture

The angles of a triangle are in the ratio 1 : 2 : 3. Write them in radians.

The three angles add up to 180°, which is π radians.
Six equal shares of π: π/6, 2π/6 and 3π/6.
So the angles are π/6, π/3 and π/2.

Answer: π/6, π/3 and π/2

Direct

Change 150° to radians.

150 × π ÷ 180
$\frac{150 π}{180} = \frac{5 π}{6}$

Answer: 5π/6

Reverse

Change 7π/12 radians to degrees.

Try it first, then show the working

7π/12 × 180/π
7 × 15 = 105
$\frac{7 π}{12} \times \frac{180}{π} = 105$

Answer: 105°

Exam

A wheel makes 360 turns in one minute. Through how many radians does it turn in one second?

Try it first, then show the working

360 turns a minute is 6 turns a second.
One turn is 2π radians, so 6 turns is 12π radians.
$6 \times 2 π = 12 π$

Answer: 12π radians

Common mistake: Multiplying by 180/π when going to radians, the wrong way round. A quick check: for the same angle the radian number is the smaller, since 1 radian is about 57°.

Class 11

Arc length with the angle in radians

l = r θ

What each letter means
$l$	the length of the arc (length)
$r$	the radius of the circle (length)
$θ$	the angle the arc makes at the centre, in radians

When the angle at the centre is in radians, an arc is simply the radius times the angle: l = rθ.

Why it works

An angle of 1 radian cuts off an arc exactly one radius long; that is what a radian means. So an angle of θ radians cuts off θ radii of arc, a length of r × θ.

When to use it

Finding an arc length, a radius, or an angle in radians; lengths along curved paths such as a wheel's rim or a bend in a road.

How to use it

Make sure θ is in radians (change degrees with θ = D × π/180).
Multiply: l = r × θ.
For the angle, divide: θ = l ÷ r. For the radius, divide: r = l ÷ θ.

To remember: Radius times radians gives the rim.

Other forms

Rearranged $θ = \frac{l}{r}$ The angle in radians from an arc and its radius.
To remember itWith the angle D in degrees the same arc is πrD/180, the Class 10 form 2πr × D/360.

Worked examples

Story

A pendulum 75 cm long swings through an angle of π/10 radians. How long is the arc its tip travels?

l = rθ = 75 × π/10
= 7.5π, about 23.6 cm
$75 \times \frac{π}{10} = \frac{15 π}{2}$

Answer: 7.5π cm, about 23.6 cm

Picture

In a circle of radius 21 cm, an arc 22 cm long makes what angle at the centre, in radians and in degrees? (Use π = 22/7.)

θ = l ÷ r = 22/21 radian.
In degrees: 22/21 × 180 ÷ (22/7) = 22/21 × 180 × 7/22 = 60°.
$\frac{22}{21} \times 180 \times \frac{7}{22} = 60$

Answer: 22/21 radian, which is 60°

Direct

Find the length of an arc of a circle of radius 10 cm that makes an angle of 1.5 radians at the centre.

l = rθ
10 × 1.5 = 15
$10 \times 1.5 = 15$

Answer: 15

Reverse

An arc 44 cm long makes an angle of 2 radians at the centre. Find the radius.

Try it first, then show the working

r = l ÷ θ
44 ÷ 2 = 22
$\frac{44}{2} = 22$

Answer: 22

Exam

The minute hand of a clock is 1.5 cm long. How far does its tip move in 40 minutes? (Use π = 3.14.)

Try it first, then show the working

In 40 minutes the hand turns 40/60 of a full turn: 2/3 × 2π = 4π/3 radians.
l = 1.5 × 4π/3 = 2π.
2 × 3.14 = 6.28
$1.5 \times \frac{4 π}{3} = 2 π$

Answer: 6.28 cm

Common mistake: Putting the angle in degrees: for a 60° angle, l = r × 60 is far too big. Use π/3 instead.

Class 11

sin(A + B)

sin (A + B) = sin A cos B + cos A sin B

What each letter means
$A$	any angle
$B$	any other angle

The sine of a sum is not the sum of the sines. Instead sin(A + B) = sin A cos B + cos A sin B: each angle's sine times the other's cosine.

Why it works

Put the two angles one after the other at the centre of a unit circle. The height of the final point splits into two pieces, sin A cos B and cos A sin B. The book proves cos(x + y) first on the unit circle, then turns it into this rule with cos(π/2 − x) = sin x.

When to use it

Exact values of angles like 75° = 45° + 30°, expanding sin(x + a), and proving identities.

How to use it

Write the angle as a sum of two angles you know (75° = 45° + 30°).
Replace sin(A + B) by sin A cos B + cos A sin B.
Put in the known values and simplify.

To remember: Sine of a sum: sin cos plus cos sin.

Other forms

Special case $sin 2 A = 2 sin A cos A$ When B is the same angle as A.

Worked examples

Expand

Expand sin(x + 60°).

sin x cos 60° + cos x sin 60°
= ½ sin x + (√3/2) cos x
$sin (x + \frac{π}{3}) = \frac{1}{2} sin x + \frac{3}{2} cos x$

Answer: ½ sin x + (√3/2) cos x

Factorise

Write sin 3x cos x + cos 3x sin x as a single sine.

It has the pattern sin A cos B + cos A sin B with A = 3x and B = x.
So it is sin(3x + x) = sin 4x.

Answer: sin 4x

Simplify

Simplify sin(A + B) + sin(A − B).

(sin A cos B + cos A sin B) + (sin A cos B − cos A sin B)
The cos A sin B terms cancel: 2 sin A cos B.
$sin (A + B) + sin (A - B) = 2 sin A cos B$

Answer: 2 sin A cos B

Prove

Prove that sin(π/4 + x) = (sin x + cos x)/√2.

Try it first, then show the working

sin(π/4 + x) = sin(π/4) cos x + cos(π/4) sin x
= (1/√2) cos x + (1/√2) sin x = (sin x + cos x)/√2.
$sin (\frac{π}{4} + x) = \frac{s i n x + c o s x}{2}$

Answer: Both sides are (sin x + cos x)/√2

Quick calculation

Find the exact value of sin 75°.

Try it first, then show the working

sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
= (1/√2)(√3/2) + (1/√2)(1/2)
= (√3 + 1)/(2√2)
$\frac{1}{2} \times \frac{3}{2} + \frac{1}{2} \times \frac{1}{2} = \frac{3 + 1}{2 2}$

Answer: (√3 + 1)/(2√2)

Common mistake: Writing sin(A + B) = sin A + sin B. Test it with A = B = 30°: sin 60° is about 0.87, but sin 30° + sin 30° = 1.

Class 11

sin(A − B)

sin (A - B) = sin A cos B - cos A sin B

What each letter means
$A$	any angle
$B$	any other angle

The sine of a difference: sin(A − B) = sin A cos B − cos A sin B. It is the sum rule with a minus in the middle.

Why it works

Put −B for B in sin(A + B) = sin A cos B + cos A sin B. Since cos(−B) = cos B and sin(−B) = −sin B, the second term changes sign.

When to use it

Exact values like sin 15° = sin(45° − 30°), angles such as π − x, and identities.

How to use it

Write the angle as a difference of two angles you know.
Replace sin(A − B) by sin A cos B − cos A sin B; keep the order, since the minus belongs to cos A sin B.
Put in the known values and simplify.

To remember: Sine keeps the sign: plus for a sum, minus for a difference.

Other forms

Special case $sin (π - A) = sin A$ When the first angle is π (a half turn).

Worked examples

Expand

Expand sin(x − 30°).

sin x cos 30° − cos x sin 30°
= (√3/2) sin x − ½ cos x
$sin (x - \frac{π}{6}) = \frac{3}{2} sin x - \frac{1}{2} cos x$

Answer: (√3/2) sin x − ½ cos x

Factorise

Write sin 80° cos 20° − cos 80° sin 20° as one sine, and find its value.

It has the pattern sin A cos B − cos A sin B: sin(80° − 20°) = sin 60°.
sin 60° = √3/2.

Answer: sin 60° = √3/2

Simplify

Simplify sin(A + B) − sin(A − B).

(sin A cos B + cos A sin B) − (sin A cos B − cos A sin B)
= 2 cos A sin B
$sin (A + B) - sin (A - B) = 2 cos A sin B$

Answer: 2 cos A sin B

Prove

Prove that sin(π − x) = sin x.

Try it first, then show the working

sin(π − x) = sin π cos x − cos π sin x
= 0 × cos x − (−1) sin x = sin x
$sin (π - x) = sin x$

Answer: Both sides are sin x

Quick calculation

Find the exact value of sin 15°.

Try it first, then show the working

sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30°
= (1/√2)(√3/2) − (1/√2)(1/2)
= (√3 − 1)/(2√2)
$\frac{1}{2} \times \frac{3}{2} - \frac{1}{2} \times \frac{1}{2} = \frac{3 - 1}{2 2}$

Answer: (√3 − 1)/(2√2)

Common mistake: Swapping the order: sin(A − B) is not sin B cos A − cos B sin A (that is sin(B − A), the negative).

Class 11

cos(A + B)

cos (A + B) = cos A cos B - sin A sin B

What each letter means
$A$	any angle
$B$	any other angle

The cosine of a sum: cos(A + B) = cos A cos B − sin A sin B. The cosines go together, the sines go together, and the sign flips to minus.

Why it works

On a unit circle, the points at angles A + B and at 0 are the same distance apart as the points at A and at −B (the same arc between them). Writing both distances with the distance formula and comparing gives cos(A + B) = cos A cos B − sin A sin B; this is the book's first proof, and every other sum rule follows from it.

When to use it

Exact values like cos 75°, angles such as π/2 + x or π + x, and identities.

How to use it

Write the angle as a sum of two known angles.
Replace cos(A + B) by cos A cos B − sin A sin B.
Put in the values and simplify.

To remember: Cosine is contrary: a sum gets a minus.

Other forms

Special case $cos 2 A = cos^{2} A - sin^{2} A$ When B is the same angle as A.

Worked examples

Expand

Expand cos(x + 45°).

cos x cos 45° − sin x sin 45°
= (cos x − sin x)/√2
$cos (x + \frac{π}{4}) = \frac{c o s x - s i n x}{2}$

Answer: (cos x − sin x)/√2

Factorise

Write cos 2x cos x − sin 2x sin x as a single cosine.

It has the pattern cos A cos B − sin A sin B with A = 2x and B = x.
So it is cos(2x + x) = cos 3x.

Answer: cos 3x

Simplify

Simplify cos(A + B) + cos(A − B).

(cos A cos B − sin A sin B) + (cos A cos B + sin A sin B)
= 2 cos A cos B
$cos (A + B) + cos (A - B) = 2 cos A cos B$

Answer: 2 cos A cos B

Prove

Prove that cos(π/2 + x) = −sin x.

Try it first, then show the working

cos(π/2) cos x − sin(π/2) sin x
= 0 × cos x − 1 × sin x = −sin x
$cos (\frac{π}{2} + x) = - sin x$

Answer: Both sides are −sin x

Quick calculation

Find the exact value of cos 75°.

Try it first, then show the working

cos(45° + 30°) = cos 45° cos 30° − sin 45° sin 30°
= (1/√2)(√3/2) − (1/√2)(1/2)
= (√3 − 1)/(2√2)
$\frac{1}{2} \times \frac{3}{2} - \frac{1}{2} \times \frac{1}{2} = \frac{3 - 1}{2 2}$

Answer: (√3 − 1)/(2√2)

Common mistake: Using a plus sign: cos(A + B) has a minus. Test with A = B = 45°: cos 90° = 0, and cos 45° cos 45° − sin 45° sin 45° = ½ − ½ = 0.

Class 11

cos(A − B)

cos (A - B) = cos A cos B + sin A sin B

What each letter means
$A$	any angle
$B$	any other angle

The cosine of a difference: cos(A − B) = cos A cos B + sin A sin B. The sign flips again, so a difference gets a plus.

Why it works

Put −B for B in cos(A + B) = cos A cos B − sin A sin B. Since cos(−B) = cos B and sin(−B) = −sin B, the minus becomes a plus.

When to use it

Exact values like cos 15°, the rule cos(π/2 − x) = sin x, and identities.

How to use it

Write the angle as a difference of two known angles.
Replace cos(A − B) by cos A cos B + sin A sin B.
Put in the values and simplify.

To remember: Cosine is contrary: a difference gets a plus.

Other forms

Special case $cos (- B) = cos B$ When A is 0, which shows cosine does not change when the angle turns the other way.

Worked examples

Expand

Expand cos(x − 60°).

cos x cos 60° + sin x sin 60°
= ½ cos x + (√3/2) sin x
$cos (x - \frac{π}{3}) = \frac{1}{2} cos x + \frac{3}{2} sin x$

Answer: ½ cos x + (√3/2) sin x

Factorise

Write cos 70° cos 10° + sin 70° sin 10° as one cosine, and find its value.

It has the pattern cos A cos B + sin A sin B: cos(70° − 10°) = cos 60°.
cos 60° = ½.

Answer: cos 60° = ½

Simplify

Simplify cos(A − B) − cos(A + B).

(cos A cos B + sin A sin B) − (cos A cos B − sin A sin B)
= 2 sin A sin B
$cos (A - B) - cos (A + B) = 2 sin A sin B$

Answer: 2 sin A sin B

Prove

Prove that cos(π/2 − x) = sin x.

Try it first, then show the working

cos(π/2) cos x + sin(π/2) sin x
= 0 × cos x + 1 × sin x = sin x
$cos (\frac{π}{2} - x) = sin x$

Answer: Both sides are sin x

Quick calculation

Find the exact value of cos 15°.

Try it first, then show the working

cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30°
= (1/√2)(√3/2) + (1/√2)(1/2)
= (√3 + 1)/(2√2)
$\frac{1}{2} \times \frac{3}{2} + \frac{1}{2} \times \frac{1}{2} = \frac{3 + 1}{2 2}$

Answer: (√3 + 1)/(2√2)

Common mistake: Writing a minus: that is cos(A + B). For a difference, cosine has a plus.

Class 11

tan(A + B)

tan (A + B) = \frac{tan A + tan B}{1 - tan A tan B}

What each letter means
$A$	any angle (none of A, B or A + B an odd multiple of π/2)
$B$	any other angle

The tangent of a sum is the sum of the tangents divided by one minus their product.

Why it works

Divide sin(A + B) = sin A cos B + cos A sin B by cos(A + B) = cos A cos B − sin A sin B, then divide the top and the bottom by cos A cos B: every sine over cosine becomes a tangent.

When to use it

Exact values like tan 75°, the angle between two straight lines, and identities with tangents.

How to use it

Find tan A and tan B.
Top: add them. Bottom: 1 minus their product.
Divide and simplify.

To remember: Sum over one minus the product.

Other forms

Special case $tan 2 A = \frac{2 t a n A}{1 - t a n ^{2} A}$ When B is the same angle as A.

Worked examples

Expand

Expand tan(x + 45°).

(tan x + tan 45°) ÷ (1 − tan x tan 45°)
tan 45° = 1, so (tan x + 1)/(1 − tan x).
$tan (x + \frac{π}{4}) = \frac{1 + t a n x}{1 - t a n x}$

Answer: (1 + tan x)/(1 − tan x)

Factorise

Write (tan 20° + tan 25°)/(1 − tan 20° tan 25°) as one tangent, and find its value.

It has the pattern (tan A + tan B)/(1 − tan A tan B): tan(20° + 25°) = tan 45°.
tan 45° = 1.

Answer: tan 45° = 1

Simplify

Simplify (1 + tan x)/(1 − tan x) to a single tangent.

Write 1 as tan 45°: (tan 45° + tan x)/(1 − tan 45° tan x).
That is tan(45° + x).
$\frac{1 + t a n x}{1 - t a n x} = tan (\frac{π}{4} + x)$

Answer: tan(45° + x)

Prove

A and B are acute angles with tan A = ½ and tan B = ⅓. Prove that A + B = 45°.

Try it first, then show the working

tan(A + B) = (½ + ⅓)/(1 − ½ × ⅓) = (5/6)/(5/6) = 1.
A + B is between 0° and 180° and its tangent is 1, and both angles are acute with small tangents, so A + B = 45°.
$\frac{\frac{1}{2} + \frac{1}{3}}{1 - \frac{1}{6}} = 1$

Answer: tan(A + B) = 1, so A + B = 45°

Quick calculation

Find the exact value of tan 75°.

Try it first, then show the working

tan(45° + 30°) = (1 + 1/√3)/(1 − 1/√3)
= (√3 + 1)/(√3 − 1)
Multiply top and bottom by (√3 + 1): (4 + 2√3)/2 = 2 + √3.
$\frac{4 + 2 3}{2} = 2 + 3$

Answer: 2 + √3

Common mistake: Writing tan(A + B) = tan A + tan B, or putting a plus on the bottom. The bottom is 1 − tan A tan B.

Class 11

tan(A − B)

tan (A - B) = \frac{tan A - tan B}{1 + tan A tan B}

What each letter means
$A$	any angle (none of A, B or A − B an odd multiple of π/2)
$B$	any other angle

The tangent of a difference: the difference of the tangents divided by one plus their product.

Why it works

Put −B for B in the rule for tan(A + B). Since tan(−B) = −tan B, the top becomes tan A − tan B and the bottom 1 + tan A tan B.

When to use it

Exact values like tan 15°, the angle between two lines with slopes m₁ and m₂ (tan θ = (m₁ − m₂)/(1 + m₁m₂)), and identities.

How to use it

Find tan A and tan B.
Top: tan A minus tan B. Bottom: 1 plus their product.
Divide and simplify.

To remember: The signs on top and bottom are always opposite.

Other forms

Special case $tan (π - A) = - tan A$ When the first angle is π (a half turn).

Worked examples

Expand

Expand tan(x − 45°).

(tan x − tan 45°) ÷ (1 + tan x tan 45°)
= (tan x − 1)/(1 + tan x)
$tan (x - \frac{π}{4}) = \frac{t a n x - 1}{1 + t a n x}$

Answer: (tan x − 1)/(1 + tan x)

Factorise

Write (tan 70° − tan 25°)/(1 + tan 70° tan 25°) as one tangent, and find its value.

It has the pattern (tan A − tan B)/(1 + tan A tan B): tan(70° − 25°) = tan 45°.
tan 45° = 1.

Answer: tan 45° = 1

Simplify

Simplify (1 − tan x)/(1 + tan x) to a single tangent.

Write 1 as tan 45°: (tan 45° − tan x)/(1 + tan 45° tan x).
That is tan(45° − x).
$\frac{1 - t a n x}{1 + t a n x} = tan (\frac{π}{4} - x)$

Answer: tan(45° − x)

Prove

Two lines have slopes 3 and ½. Prove that the acute angle θ between them is 45°.

Try it first, then show the working

tan θ = (3 − ½)/(1 + 3 × ½) = (5/2)/(5/2) = 1.
The acute angle whose tangent is 1 is 45°.
$\frac{3 - \frac{1}{2}}{1 + \frac{3}{2}} = 1$

Answer: tan θ = 1, so θ = 45°

Quick calculation

Find the exact value of tan 15°.

Try it first, then show the working

tan(45° − 30°) = (1 − 1/√3)/(1 + 1/√3) = (√3 − 1)/(√3 + 1)
Multiply top and bottom by (√3 − 1): (4 − 2√3)/2 = 2 − √3.
$\frac{4 - 2 3}{2} = 2 - 3$

Answer: 2 − √3

Common mistake: Keeping the minus on the bottom. For a difference the bottom is 1 + tan A tan B.

Class 11

sin 2A

sin 2 A = 2 sin A cos A

What each letter means
$A$	any angle

The sine of a double angle is twice the sine times the cosine of the single angle.

Why it works

Put B = A in sin(A + B) = sin A cos B + cos A sin B: the two terms are the same, sin A cos A, so they add to 2 sin A cos A.

When to use it

Finding sin 2x from sin x and cos x, simplifying 2 sin x cos x, and solving equations with sin 2x.

How to use it

Find sin A and cos A (with sin²A + cos²A = 1 if only one is given).
Multiply them and double: sin 2A = 2 sin A cos A.
Read it backwards too: 2 sin x cos x is sin 2x.

To remember: Double the angle: twice sine times cosine.

Other forms

Same formula, another form $sin 2 A = \frac{2 t a n A}{1 + t a n ^{2} A}$ In terms of tan A alone (divide by sin²A + cos²A and then by cos²A).

Worked examples

Expand

Write sin 4x using sin 2x and cos 2x.

sin 4x = sin(2 × 2x) = 2 sin 2x cos 2x.
$sin 4 x = 2 sin 2 x cos 2 x$

Answer: 2 sin 2x cos 2x

Factorise

Write 2 sin 15° cos 15° as one sine, and find its value.

2 sin A cos A = sin 2A with A = 15°: sin 30°.
sin 30° = ½.

Answer: sin 30° = ½

Simplify

Simplify sin 2A ÷ (1 + cos 2A).

sin 2A = 2 sin A cos A and 1 + cos 2A = 2 cos²A.
2 sin A cos A ÷ 2 cos²A = sin A ÷ cos A = tan A.
$\frac{s i n 2 A}{1 + c o s 2 A} = tan A$

Answer: tan A

Prove

Prove that (sin x + cos x)² = 1 + sin 2x.

Try it first, then show the working

(sin x + cos x)² = sin²x + 2 sin x cos x + cos²x
= (sin²x + cos²x) + 2 sin x cos x = 1 + sin 2x
$(sin x + cos x)^{2} = 1 + sin 2 x$

Answer: Both sides are 1 + sin 2x

Quick calculation

x is acute and sin x = 3/5. Find sin 2x.

Try it first, then show the working

cos x = √(1 − 9/25) = 4/5.
sin 2x = 2 × 3/5 × 4/5 = 24/25.
$2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25}$

Answer: 24/25

Common mistake: Writing sin 2A = 2 sin A. Test with A = 30°: sin 60° is about 0.87, but 2 sin 30° = 1.

Class 11

cos 2A

cos 2 A = cos^{2} A - sin^{2} A

What each letter means
$A$	any angle

The cosine of a double angle has three useful forms: cos²A − sin²A, 2cos²A − 1 and 1 − 2sin²A.

Why it works

Put B = A in cos(A + B) = cos A cos B − sin A sin B to get cos²A − sin²A. Then replace sin²A by 1 − cos²A, or cos²A by 1 − sin²A, to get the other two forms.

When to use it

Finding cos 2x, turning squares into double angles (useful for integrals later), and identities such as 1 − cos 2x = 2 sin²x.

How to use it

Pick the form that uses what you know: cos A only (2cos²A − 1), sin A only (1 − 2sin²A), or both.
Put in the value and simplify.
Read it backwards to remove squares: cos²A = (1 + cos 2A)/2 and sin²A = (1 − cos 2A)/2.

To remember: Cos double: cos² minus sin², and two more by swapping one square.

Other forms

Same formula, another form $cos 2 A = 2 cos^{2} A - 1$ Using sin²A = 1 − cos²A.
Same formula, another form $cos 2 A = 1 - 2 sin^{2} A$ Using cos²A = 1 − sin²A.
Same formula, another form $cos 2 A = \frac{1 - t a n ^{2} A}{1 + t a n ^{2} A}$ In terms of tan A alone.

Worked examples

Expand

Write cos 4x using cos 2x only.

cos 4x = cos(2 × 2x) = 2cos²(2x) − 1.

Answer: 2cos²2x − 1

Factorise

Write 1 − 2sin²22.5° as one cosine, and find its value.

1 − 2sin²A = cos 2A with A = 22.5°: cos 45°.
cos 45° = 1/√2.

Answer: cos 45° = 1/√2

Simplify

Simplify (1 − cos 2x) ÷ sin 2x.

1 − cos 2x = 2sin²x and sin 2x = 2 sin x cos x.
2sin²x ÷ 2 sin x cos x = sin x ÷ cos x = tan x.

Answer: tan x

Prove

Prove that cos⁴x − sin⁴x = cos 2x.

Try it first, then show the working

cos⁴x − sin⁴x = (cos²x + sin²x)(cos²x − sin²x)
= 1 × (cos²x − sin²x) = cos 2x
$cos^{4} x - sin^{4} x = cos 2 x$

Answer: Both sides are cos 2x

Quick calculation

x is acute and cos x = 3/5. Find cos 2x.

Try it first, then show the working

cos 2x = 2cos²x − 1
2 × 9/25 − 1 = 18/25 − 25/25 = −7/25
$2 \times \frac{9}{25} - 1 = - \frac{7}{25}$

Answer: −7/25

Common mistake: Writing cos 2A = 2 cos A, or mixing up the signs of the three forms. Check any form with A = 0: cos 0 = 1.

Class 11

tan 2A

tan 2 A = \frac{2 tan A}{1 - tan ^{2} A}

What each letter means
$A$	any angle (tan A ≠ ±1, and A not an odd multiple of π/2)

The tangent of a double angle is twice the tangent divided by one minus the tangent squared.

Why it works

Put B = A in tan(A + B) = (tan A + tan B)/(1 − tan A tan B): the top becomes 2 tan A and the bottom 1 − tan²A.

When to use it

Finding tan 2x from tan x, and identities with tangents.

How to use it

Find tan A.
Top: 2 tan A. Bottom: 1 − tan²A.
Divide and simplify.

To remember: Twice the tan over one minus its square.

Other forms

To remember itIt is tan(A + B) with B = A; no new rule to learn.

Worked examples

Expand

Write tan 4x using tan 2x.

tan 4x = tan(2 × 2x) = 2 tan 2x ÷ (1 − tan²2x).
$tan 4 x = \frac{2 t a n 2 x}{1 - ( t a n 2 x ) ^{2}}$

Answer: 2 tan 2x/(1 − tan²2x)

Factorise

Write 2 tan 22.5° ÷ (1 − tan²22.5°) as one tangent, and find its value.

It has the pattern 2 tan A/(1 − tan²A) with A = 22.5°: tan 45°.
tan 45° = 1.

Answer: tan 45° = 1

Simplify

Simplify 2 tan x ÷ (1 + tan²x).

1 + tan²x = sec²x, so 2 tan x ÷ sec²x = 2 tan x cos²x.
= 2 (sin x/cos x) cos²x = 2 sin x cos x = sin 2x.
$\frac{2 t a n x}{1 + t a n ^{2} x} = sin 2 x$

Answer: sin 2x

Prove

Prove that tan 2x = sin 2x ÷ cos 2x gives the same as 2 tan x ÷ (1 − tan²x).

Try it first, then show the working

sin 2x ÷ cos 2x = 2 sin x cos x ÷ (cos²x − sin²x).
Divide top and bottom by cos²x: 2 tan x ÷ (1 − tan²x).
$\frac{s i n 2 x}{c o s 2 x} = \frac{2 t a n x}{1 - t a n ^{2} x}$

Answer: Both are 2 tan x/(1 − tan²x)

Quick calculation

x is acute and tan x = ½. Find tan 2x.

Try it first, then show the working

tan 2x = 2 × ½ ÷ (1 − ¼)
= 1 ÷ ¾ = 4/3
$\frac{2 \times \frac{1}{2}}{1 - \frac{1}{4}} = \frac{4}{3}$

Answer: 4/3

Common mistake: Writing tan 2A = 2 tan A, or 1 + tan²A on the bottom (that is sec²A, from a different rule).

Class 11

sin 3A

sin 3 A = 3 sin A - 4 sin^{3} A

What each letter means
$A$	any angle

The sine of a triple angle can be written with sin A alone: 3 sin A − 4 sin³A.

Why it works

Write sin 3A = sin(2A + A) = sin 2A cos A + cos 2A sin A. Put sin 2A = 2 sin A cos A and cos 2A = 1 − 2sin²A, and replace cos²A by 1 − sin²A: everything collects into 3 sin A − 4 sin³A.

When to use it

Finding sin 3x from sin x, and identities or equations with sin 3x.

How to use it

Find sin A.
Work out 3 sin A − 4 sin³A.

To remember: Three sine, minus four sine cubed.

Other forms

To remember itThe cosine rule is the mirror image: cos 3A = 4cos³A − 3 cos A.

Worked examples

Expand

Write sin 3x using sin x only.

sin 3x = 3 sin x − 4 sin³x.
$sin 3 x = 3 sin x - 4 sin^{3} x$

Answer: 3 sin x − 4 sin³x

Factorise

Write 3 sin 10° − 4 sin³10° as one sine, and find its value.

It has the pattern 3 sin A − 4 sin³A with A = 10°: sin 30°.
sin 30° = ½.

Answer: sin 30° = ½

Simplify

Simplify (3 sin x − sin 3x) ÷ 4.

3 sin x − sin 3x = 3 sin x − (3 sin x − 4 sin³x) = 4 sin³x.
Divided by 4: sin³x.
$\frac{3 s i n x - s i n 3 x}{4} = sin^{3} x$

Answer: sin³x

Prove

Prove that sin 3x = sin(2x + x) gives 3 sin x − 4 sin³x.

Try it first, then show the working

sin(2x + x) = sin 2x cos x + cos 2x sin x = 2 sin x cos²x + (1 − 2sin²x) sin x.
= 2 sin x (1 − sin²x) + sin x − 2sin³x = 3 sin x − 4 sin³x.
$sin 2 x cos x + cos 2 x sin x = 3 sin x - 4 sin^{3} x$

Answer: Both are 3 sin x − 4 sin³x

Quick calculation

sin x = ½. Find sin 3x.

Try it first, then show the working

3 × ½ − 4 × ⅛
= 3/2 − ½ = 1
$3 \times \frac{1}{2} - 4 \times \frac{1}{8} = 1$

Answer: 1

Common mistake: Getting the signs or the numbers the wrong way round (4 sin A − 3 sin³A). Check with A = 90°: sin 270° = −1, and 3 − 4 = −1.

Class 11

cos 3A

cos 3 A = 4 cos^{3} A - 3 cos A

What each letter means
$A$	any angle

The cosine of a triple angle can be written with cos A alone: 4cos³A − 3 cos A.

Why it works

Write cos 3A = cos(2A + A) = cos 2A cos A − sin 2A sin A. Put cos 2A = 2cos²A − 1 and sin 2A = 2 sin A cos A, and replace sin²A by 1 − cos²A: everything collects into 4cos³A − 3 cos A.

When to use it

Finding cos 3x from cos x, and identities or equations with cos 3x.

How to use it

Find cos A.
Work out 4cos³A − 3 cos A.

To remember: Four cos cubed, minus three cos: the sine rule turned round.

Other forms

To remember itCompare sin 3A = 3 sin A − 4 sin³A: the same numbers, in the other order.

Worked examples

Expand

Write cos 3x using cos x only.

cos 3x = 4cos³x − 3 cos x.
$cos 3 x = 4 cos^{3} x - 3 cos x$

Answer: 4cos³x − 3 cos x

Factorise

Write 4cos³20° − 3 cos 20° as one cosine, and find its value.

It has the pattern 4cos³A − 3 cos A with A = 20°: cos 60°.
cos 60° = ½.

Answer: cos 60° = ½

Simplify

Simplify (cos 3x + 3 cos x) ÷ 4.

cos 3x + 3 cos x = 4cos³x − 3 cos x + 3 cos x = 4cos³x.
Divided by 4: cos³x.
$\frac{c o s 3 x + 3 c o s x}{4} = cos^{3} x$

Answer: cos³x

Prove

Prove that cos 3x = cos(2x + x) gives 4cos³x − 3 cos x.

Try it first, then show the working

cos(2x + x) = cos 2x cos x − sin 2x sin x = (2cos²x − 1) cos x − 2 sin²x cos x.
= 2cos³x − cos x − 2(1 − cos²x) cos x = 4cos³x − 3 cos x.
$cos 2 x cos x - sin 2 x sin x = 4 cos^{3} x - 3 cos x$

Answer: Both are 4cos³x − 3 cos x

Quick calculation

cos x = ½. Find cos 3x.

Try it first, then show the working

4 × ⅛ − 3 × ½
= ½ − 3/2 = −1
$4 \times \frac{1}{8} - 3 \times \frac{1}{2} = - 1$

Answer: −1

Common mistake: Copying the sine pattern (3 cos A − 4 cos³A). Check with A = 0: cos 0 = 1, and 4 − 3 = 1.

Class 11

cos x + cos y

cos x + cos y = 2 cos \frac{x + y}{2} cos \frac{x - y}{2}

What each letter means
$x$	any angle
$y$	any other angle

A sum of two cosines becomes a product: twice the cosine of the half-sum times the cosine of the half-difference.

Why it works

Add cos(A + B) = cos A cos B − sin A sin B and cos(A − B) = cos A cos B + sin A sin B: the sine parts cancel, leaving 2 cos A cos B. Now call A + B = x and A − B = y, so A = (x + y)/2 and B = (x − y)/2.

When to use it

Turning a sum of cosines into a product: to simplify, to factorise, or to solve equations like cos 3x + cos x = 0.

How to use it

Find the half-sum (x + y)/2 and the half-difference (x − y)/2.
Write 2 cos(half-sum) cos(half-difference).
Simplify the angles, then the values.

To remember: Cos plus cos: two cos cos, of the half-sum and the half-difference.

Other forms

Special case $1 + cos 2 A = 2 cos^{2} A$ When x is 2A and y is 0.

Worked examples

Expand

Write 2 cos 4x cos x as a sum.

2 cos A cos B = cos(A + B) + cos(A − B) with A = 4x and B = x.
$2 cos 4 x cos x = cos 5 x + cos 3 x$

Answer: cos 5x + cos 3x

Factorise

Factorise cos 5x + cos 3x.

Half-sum (5x + 3x)/2 = 4x; half-difference (5x − 3x)/2 = x.
$cos 5 x + cos 3 x = 2 cos 4 x cos x$

Answer: 2 cos 4x cos x

Simplify

Simplify (cos 7x + cos 5x) ÷ cos x.

cos 7x + cos 5x = 2 cos 6x cos x.
$\frac{2 c o s 6 x c o s x}{c o s x} = 2 cos 6 x$

Answer: 2 cos 6x

Prove

Prove that cos 75° + cos 15° = √6/2.

Try it first, then show the working

cos 75° + cos 15° = 2 cos 45° cos 30°
$2 \times \frac{1}{2} \times \frac{3}{2} = \frac{6}{2}$

Answer: √6/2

Quick calculation

Find cos 100° + cos 20° without tables.

Try it first, then show the working

2 cos 60° cos 40° = 2 × ½ × cos 40° = cos 40°.
$cos \frac{5 π}{9} + cos \frac{π}{9} = cos \frac{2 π}{9}$

Answer: cos 40°

Common mistake: Adding the angles without halving: cos 5x + cos 3x is 2 cos 4x cos x, not 2 cos 8x cos 2x.

Class 11

cos x − cos y

cos x - cos y = - 2 sin \frac{x + y}{2} sin \frac{x - y}{2}

What each letter means
$x$	any angle
$y$	any other angle

A difference of two cosines becomes minus twice the sine of the half-sum times the sine of the half-difference.

Why it works

Subtract cos(A − B) = cos A cos B + sin A sin B from cos(A + B) = cos A cos B − sin A sin B: the cosine parts cancel, leaving −2 sin A sin B. With A + B = x and A − B = y, A = (x + y)/2 and B = (x − y)/2.

When to use it

Factorising a difference of cosines, and equations like cos 3x − cos x = 0.

How to use it

Find the half-sum and the half-difference.
Write −2 sin(half-sum) sin(half-difference); keep the minus sign in front.
Simplify.

To remember: Cos minus cos: minus two sin sin.

Other forms

Same formula, another form $cos x - cos y = 2 sin \frac{x + y}{2} sin \frac{y - x}{2}$ The same, with the half-difference turned round instead of a minus in front.

Worked examples

Expand

Write −2 sin 3x sin x as a difference of cosines.

−2 sin A sin B = cos(A + B) − cos(A − B) with A = 3x and B = x.
$- 2 sin 3 x sin x = cos 4 x - cos 2 x$

Answer: cos 4x − cos 2x

Factorise

Factorise cos 4x − cos 2x.

Half-sum 3x, half-difference x.
$cos 4 x - cos 2 x = - 2 sin 3 x sin x$

Answer: −2 sin 3x sin x

Simplify

Simplify (cos x − cos 3x) ÷ sin x.

cos x − cos 3x = −2 sin 2x sin(−x) = 2 sin 2x sin x.
$\frac{2 s i n 2 x s i n x}{s i n x} = 2 sin 2 x$

Answer: 2 sin 2x

Prove

Prove that cos 20° − cos 100° = sin 40° × √3.

Try it first, then show the working

cos 20° − cos 100° = −2 sin 60° sin(−40°) = 2 sin 60° sin 40°.
2 sin 60° = √3, so it is √3 sin 40°.
$cos \frac{π}{9} - cos \frac{5 π}{9} = 3 sin \frac{2 π}{9}$

Answer: √3 sin 40°

Quick calculation

Find cos 105° − cos 15° exactly.

Try it first, then show the working

−2 sin 60° sin 45°
$- 2 \times \frac{3}{2} \times \frac{1}{2} = - \frac{3}{2}$

Answer: −√3/√2

Common mistake: Dropping the minus sign in front, or writing cosines instead of sines.

Class 11

sin x + sin y

sin x + sin y = 2 sin \frac{x + y}{2} cos \frac{x - y}{2}

What each letter means
$x$	any angle
$y$	any other angle

A sum of two sines becomes twice the sine of the half-sum times the cosine of the half-difference.

Why it works

Add sin(A + B) = sin A cos B + cos A sin B and sin(A − B) = sin A cos B − cos A sin B: the cos A sin B parts cancel, leaving 2 sin A cos B. With A + B = x and A − B = y, A = (x + y)/2 and B = (x − y)/2.

When to use it

Factorising a sum of sines, simplifying ratios like (sin 3x + sin x)/(cos 3x + cos x), and solving equations.

How to use it

Find the half-sum and the half-difference.
Write 2 sin(half-sum) cos(half-difference).
Simplify.

To remember: Sin plus sin: two sin cos.

Other forms

Special case $sin 2 A = 2 sin A cos A$ When x and y are both A.

Worked examples

Expand

Write 2 sin 2x cos x as a sum.

2 sin A cos B = sin(A + B) + sin(A − B).
$2 sin 2 x cos x = sin 3 x + sin x$

Answer: sin 3x + sin x

Factorise

Factorise sin 3x + sin x.

Half-sum 2x, half-difference x.
$sin 3 x + sin x = 2 sin 2 x cos x$

Answer: 2 sin 2x cos x

Simplify

Simplify (sin 3x + sin x) ÷ (cos 3x + cos x).

Top: 2 sin 2x cos x. Bottom: 2 cos 2x cos x.
$\frac{2 s i n 2 x c o s x}{2 c o s 2 x c o s x} = tan 2 x$

Answer: tan 2x

Prove

Prove that sin 75° + sin 15° = √6/2.

Try it first, then show the working

sin 75° + sin 15° = 2 sin 45° cos 30°
$2 \times \frac{1}{2} \times \frac{3}{2} = \frac{6}{2}$

Answer: √6/2

Quick calculation

Find sin 50° + sin 10° using one cosine.

Try it first, then show the working

2 sin 30° cos 20° = 2 × ½ × cos 20° = cos 20°.
$sin \frac{5 π}{18} + sin \frac{π}{18} = cos \frac{π}{9}$

Answer: cos 20°

Common mistake: Putting sin on both: sin plus sin gives sin cos, not sin sin.

Class 11

sin x − sin y

sin x - sin y = 2 cos \frac{x + y}{2} sin \frac{x - y}{2}

What each letter means
$x$	any angle
$y$	any other angle

A difference of two sines becomes twice the cosine of the half-sum times the sine of the half-difference.

Why it works

Subtract sin(A − B) = sin A cos B − cos A sin B from sin(A + B) = sin A cos B + cos A sin B: the sin A cos B parts cancel, leaving 2 cos A sin B. With A + B = x and A − B = y, A = (x + y)/2 and B = (x − y)/2.

When to use it

Factorising a difference of sines, and simplifying ratios of sums and differences.

How to use it

Find the half-sum and the half-difference, in that order (x first).
Write 2 cos(half-sum) sin(half-difference).
Simplify.

To remember: Sin minus sin: two cos sin.

Other forms

Special case $sin x - sin (- x) = 2 sin x$ When y is −x.

Worked examples

Expand

Write 2 cos 3x sin x as a difference of sines.

2 cos A sin B = sin(A + B) − sin(A − B).
$2 cos 3 x sin x = sin 4 x - sin 2 x$

Answer: sin 4x − sin 2x

Factorise

Factorise sin 5x − sin 3x.

Half-sum 4x, half-difference x.
$sin 5 x - sin 3 x = 2 cos 4 x sin x$

Answer: 2 cos 4x sin x

Simplify

Simplify (sin 5x − sin 3x) ÷ (cos 5x + cos 3x).

Top: 2 cos 4x sin x. Bottom: 2 cos 4x cos x.
$\frac{2 c o s 4 x s i n x}{2 c o s 4 x c o s x} = tan x$

Answer: tan x

Prove

Prove that sin 75° − sin 15° = 1/√2.

Try it first, then show the working

sin 75° − sin 15° = 2 cos 45° sin 30°
$2 \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{2}$

Answer: 1/√2

Quick calculation

Find sin 70° − sin 10° using one cosine.

Try it first, then show the working

2 cos 40° sin 30° = 2 × ½ × cos 40° = cos 40°.
$sin \frac{7 π}{18} - sin \frac{π}{18} = cos \frac{2 π}{9}$

Answer: cos 40°

Common mistake: Swapping the functions: sin minus sin is cos of the half-sum times sin of the half-difference, not the other way round.

Class 11

2 sin x cos y

2 sin x cos y = sin (x + y) + sin (x - y)

What each letter means
$x$	any angle
$y$	any other angle

Twice a sine times a cosine is a sum of two sines: of the sum of the angles and of their difference.

Why it works

Add sin(x + y) = sin x cos y + cos x sin y and sin(x − y) = sin x cos y − cos x sin y: the second parts cancel and 2 sin x cos y is left.

When to use it

Turning a product into a sum, to simplify it or (later, in calculus) to integrate it.

How to use it

Find x + y and x − y.
Write sin(x + y) + sin(x − y).
If the sine's angle is the smaller one, x − y is negative: use sin(−t) = −sin t.

To remember: Two sin cos gives sin plus sin.

Other forms

Rearranged $sin x cos y = \frac{s i n ( x + y ) + s i n ( x - y )}{2}$ Without the 2, halve the right side.

Worked examples

Expand

Write 2 sin 5x cos 2x as a sum.

x + y = 7x and x − y = 3x.
$2 sin 5 x cos 2 x = sin 7 x + sin 3 x$

Answer: sin 7x + sin 3x

Factorise

Write sin 7x + sin 3x as a product.

Read the formula backwards: it is 2 sin 5x cos 2x.
$sin 7 x + sin 3 x = 2 sin 5 x cos 2 x$

Answer: 2 sin 5x cos 2x

Simplify

Simplify 2 sin x cos 3x.

sin(x + 3x) + sin(x − 3x) = sin 4x + sin(−2x).
$2 sin x cos 3 x = sin 4 x - sin 2 x$

Answer: sin 4x − sin 2x

Prove

Prove that 2 sin 75° cos 15° = 1 + √3/2.

Try it first, then show the working

sin 90° + sin 60° = 1 + √3/2.
$2 sin \frac{5 π}{12} cos \frac{π}{12} = 1 + \frac{3}{2}$

Answer: 1 + √3/2

Quick calculation

Find 2 sin 45° cos 15° exactly.

Try it first, then show the working

sin 60° + sin 30°
$\frac{3}{2} + \frac{1}{2} = \frac{3 + 1}{2}$

Answer: (√3 + 1)/2

Common mistake: Forgetting the 2: sin x cos y alone is half of sin(x + y) + sin(x − y).

Class 11

2 cos x cos y

2 cos x cos y = cos (x + y) + cos (x - y)

What each letter means
$x$	any angle
$y$	any other angle

Twice a cosine times a cosine is the cosine of the sum plus the cosine of the difference.

Why it works

Add cos(x + y) = cos x cos y − sin x sin y and cos(x − y) = cos x cos y + sin x sin y: the sine parts cancel and 2 cos x cos y is left.

When to use it

Turning a product of cosines into a sum, to simplify or later to integrate.

How to use it

Find x + y and x − y.
Write cos(x + y) + cos(x − y). The order of x and y does not matter, since cos(−t) = cos t.
Simplify.

To remember: Two cos cos gives cos plus cos.

Other forms

Special case $2 cos^{2} x = 1 + cos 2 x$ When y is the same angle as x.

Worked examples

Expand

Write 2 cos 3x cos x as a sum.

x + y = 4x, x − y = 2x.
$2 cos 3 x cos x = cos 4 x + cos 2 x$

Answer: cos 4x + cos 2x

Factorise

Write cos 4x + cos 2x as a product.

Backwards: 2 cos 3x cos x.
$cos 4 x + cos 2 x = 2 cos 3 x cos x$

Answer: 2 cos 3x cos x

Simplify

Simplify 2 cos 2x cos x − cos 3x.

2 cos 2x cos x = cos 3x + cos x.
$2 cos 2 x cos x - cos 3 x = cos x$

Answer: cos x

Prove

Prove that 2 cos 45° cos 15° = (√3 + 1)/2.

Try it first, then show the working

cos 60° + cos 30° = ½ + √3/2.
$2 cos \frac{π}{4} cos \frac{π}{12} = \frac{3 + 1}{2}$

Answer: (√3 + 1)/2

Quick calculation

Find 2 cos 75° cos 15° exactly.

Try it first, then show the working

cos 90° + cos 60° = 0 + ½
$2 cos \frac{5 π}{12} cos \frac{π}{12} = \frac{1}{2}$

Answer: ½

Common mistake: Writing a minus between the two cosines: that is the rule for two sines.

Class 11

2 sin x sin y

2 sin x sin y = cos (x - y) - cos (x + y)

What each letter means
$x$	any angle
$y$	any other angle

Twice a sine times a sine is the cosine of the difference minus the cosine of the sum.

Why it works

Subtract cos(x + y) = cos x cos y − sin x sin y from cos(x − y) = cos x cos y + sin x sin y: the cosine parts cancel and 2 sin x sin y is left.

When to use it

Turning a product of sines into a difference of cosines.

How to use it

Find x − y and x + y.
Write cos(x − y) − cos(x + y): the difference angle comes first.
Simplify.

To remember: Two sin sin: cos of the difference, minus cos of the sum.

Other forms

Special case $2 sin^{2} x = 1 - cos 2 x$ When y is the same angle as x.

Worked examples

Expand

Write 2 sin 3x sin x as a difference.

x − y = 2x, x + y = 4x.
$2 sin 3 x sin x = cos 2 x - cos 4 x$

Answer: cos 2x − cos 4x

Factorise

Write cos 2x − cos 4x as a product.

Backwards: 2 sin 3x sin x.
$cos 2 x - cos 4 x = 2 sin 3 x sin x$

Answer: 2 sin 3x sin x

Simplify

Simplify cos 2x − 2 sin 3x sin x.

2 sin 3x sin x = cos 2x − cos 4x.
$cos 2 x - 2 sin 3 x sin x = cos 4 x$

Answer: cos 4x

Prove

Prove that 2 sin 75° sin 15° = ½.

Try it first, then show the working

cos 60° − cos 90° = ½ − 0.
$2 sin \frac{5 π}{12} sin \frac{π}{12} = \frac{1}{2}$

Answer: ½

Quick calculation

Find 2 sin 45° sin 15° exactly.

Try it first, then show the working

cos 30° − cos 60°
$\frac{3}{2} - \frac{1}{2} = \frac{3 - 1}{2}$

Answer: (√3 − 1)/2

Common mistake: Putting the sum first: cos(x + y) − cos(x − y) is −2 sin x sin y, the negative.

Class 11

cos²A from cos 2A

cos^{2} A = \frac{1 + cos 2 A}{2}

What each letter means
$A$	any angle

A cosine squared can be written without a square, using the double angle: cos²A = (1 + cos 2A)/2.

Why it works

From cos 2A = 2cos²A − 1, add 1 to both sides and divide by 2.

When to use it

Finding cos of half an angle (cos 15° from cos 30°), and removing squares before integrating in Class 12.

How to use it

Double the angle and find its cosine.
Add 1, divide by 2, and take the square root (with the right sign) if cos A itself is wanted.

To remember: Cos squared: one plus, over two.

Other forms

Same formula, another form $cos^{2} \frac{A}{2} = \frac{1 + c o s A}{2}$ The half-angle form: the same rule with A/2 in place of A.

Worked examples

Expand

Write cos²3x without a square.

Double the angle: 6x.
$(cos 3 x)^{2} = \frac{1 + c o s 6 x}{2}$

Answer: (1 + cos 6x)/2

Factorise

Write (1 + cos 4x)/2 as a square.

It has the pattern (1 + cos 2A)/2 with A = 2x.
$\frac{1 + c o s 4 x}{2} = (cos 2 x)^{2}$

Answer: cos²2x

Simplify

Simplify 2cos²x − cos 2x.

2cos²x = 1 + cos 2x.
$2 cos^{2} x - cos 2 x = 1$

Answer: 1

Prove

Prove that cos 15° = √(2 + √3)/2.

Try it first, then show the working

cos²15° = (1 + cos 30°)/2 = (1 + √3/2)/2 = (2 + √3)/4.
cos 15° is positive, so it is √(2 + √3)/2.
$\frac{1 + \frac{3}{2}}{2} = \frac{2 + 3}{4}$

Answer: √(2 + √3)/2

Quick calculation

cos 2A = 7/25. Find cos²A.

Try it first, then show the working

$(1 + 7/25) /2$
$\frac{1 + \frac{7}{25}}{2} = \frac{16}{25}$

Answer: 16/25

Common mistake: Forgetting to halve, or using a minus: the minus belongs to sin²A = (1 − cos 2A)/2.

Class 11

sin²A from cos 2A

sin^{2} A = \frac{1 - cos 2 A}{2}

What each letter means
$A$	any angle

A sine squared can be written without a square, using the double angle: sin²A = (1 − cos 2A)/2.

Why it works

From cos 2A = 1 − 2sin²A, move terms: 2sin²A = 1 − cos 2A, then divide by 2.

When to use it

Finding sin of half an angle (sin 15° from cos 30°), and removing squares before integrating.

How to use it

Double the angle and find its cosine.
Take it from 1, divide by 2, and take the square root (with the right sign) if sin A itself is wanted.

To remember: Sin squared: one minus, over two.

Other forms

Same formula, another form $sin^{2} \frac{A}{2} = \frac{1 - c o s A}{2}$ The half-angle form.

Worked examples

Expand

Write sin²2x without a square.

Double the angle: 4x.
$(sin 2 x)^{2} = \frac{1 - c o s 4 x}{2}$

Answer: (1 − cos 4x)/2

Factorise

Write (1 − cos 6x)/2 as a square.

It has the pattern (1 − cos 2A)/2 with A = 3x.
$\frac{1 - c o s 6 x}{2} = (sin 3 x)^{2}$

Answer: sin²3x

Simplify

Simplify cos²x − sin²x using these forms.

(1 + cos 2x)/2 − (1 − cos 2x)/2
$\frac{1 + c o s 2 x}{2} - \frac{1 - c o s 2 x}{2} = cos 2 x$

Answer: cos 2x

Prove

Prove that sin 15° = √(2 − √3)/2.

Try it first, then show the working

sin²15° = (1 − cos 30°)/2 = (2 − √3)/4.
sin 15° is positive, so it is √(2 − √3)/2.
$\frac{1 - \frac{3}{2}}{2} = \frac{2 - 3}{4}$

Answer: √(2 − √3)/2

Quick calculation

cos 2A = 7/25. Find sin²A.

Try it first, then show the working

(1 − 7/25)/2
$\frac{1 - \frac{7}{25}}{2} = \frac{9}{25}$

Answer: 9/25

Common mistake: Using a plus: 1 + cos 2A belongs to cos²A.

Class 11

tan 3A

tan 3 A = \frac{3 tan A - tan ^{3} A}{1 - 3 tan ^{2} A}

What each letter means
$A$	any angle (with 1 − 3tan²A not zero)

The tangent of a triple angle, written with tan A alone.

Why it works

Write tan 3A = tan(2A + A) and use the sum rule with tan 2A = 2 tan A/(1 − tan²A); clearing the fractions gives (3 tan A − tan³A)/(1 − 3tan²A).

When to use it

Finding tan 3x from tan x, and identities with tan 3x.

How to use it

Find tan A.
Top: 3 tan A − tan³A. Bottom: 1 − 3tan²A.
Divide.

To remember: Like sin 3A on top (3 minus cubed), over one minus three squares.

Other forms

To remember itIt is tan(2A + A): the sum rule used twice.

Worked examples

Expand

Write tan 3x using tan x only.

$tan 3 x = \frac{3 t a n x - t a n ^{3} x}{1 - 3 t a n ^{2} x}$

Answer: (3 tan x − tan³x)/(1 − 3tan²x)

Factorise

Write (3 tan 15° − tan³15°)/(1 − 3tan²15°) as one tangent, and find its value.

It has the pattern with A = 15°: tan 45°.
tan 45° = 1.
$tan \frac{3 π}{12} = 1$

Answer: tan 45° = 1

Simplify

Show that the rule gives tan 3x = sin 3x/cos 3x.

$\frac{3 s i n x - 4 s i n ^{3} x}{4 c o s ^{3} x - 3 c o s x} = \frac{3 t a n x - t a n ^{3} x}{1 - 3 t a n ^{2} x}$

Answer: Both are tan 3x

Prove

Prove that tan 3x = tan(2x + x) gives the rule.

Try it first, then show the working

$\frac{t a n 2 x + t a n x}{1 - t a n 2 x t a n x} = \frac{3 t a n x - t a n ^{3} x}{1 - 3 t a n ^{2} x}$

Answer: Both are (3 tan x − tan³x)/(1 − 3tan²x)

Quick calculation

tan x = ½. Find tan 3x.

Try it first, then show the working

Top: 3/2 − 1/8 = 11/8. Bottom: 1 − 3/4 = 1/4.
$\frac{\frac{11}{8}}{\frac{1}{4}} = \frac{11}{2}$

Answer: 11/2

Common mistake: Mixing the 3s: the top has 3 tan A and the bottom 3 tan²A.