For an angle in a right triangle, the opposite side divided by the adjacent side.
Precisely: tan A = (side opposite A) ÷ (side adjacent to A) = sin A ÷ cos A, for an acute angle A of a right triangle.
How it works
Tan tells you how steep a slope is: rise over run. So from a known angle of elevation and a distance along the ground, tan gives the height.
Examples
tan 45° = 1: the opposite and adjacent sides are equal, so the tangent is 1.
Standing 30 m from a tower and looking up at 60°, its height is 30 × tan 60° = 30√3 ≈ 52 m, using the tangent.
A road that climbs 1 m for every 10 m across has tangent 1/10 for its angle of slope.
Do not confuse: The tangent ratio is not a tangent to a circle (a line touching it at one point): the two share a name and an old history, but are different ideas.
Used in
heights and distances, slopes of lines (the slope m = tan of the angle), engineering and ramps
A straight line that touches a circle at exactly one point without cutting into it.
Precisely: A line that meets a circle in exactly one point, the point of contact; it is perpendicular to the radius drawn to that point, and there is exactly one tangent at each point of the circle.
How it works
Think of a secant cutting the circle at two points and slide it outwards: the two points move together and finally meet; the secant has become a tangent. That is why the tangent is the special case of a secant whose chord has shrunk to a point.
Examples
A bicycle wheel standing on the road: the road is a tangent to the wheel, touching it at one point.
The radius to the point of contact is perpendicular to the tangent, so ∠OTP = 90°.
From a point outside a circle exactly two tangents can be drawn, and they have equal lengths.
Do not confuse: A tangent to a circle (a line) is different from the tangent ratio (tan A, a number). A line that crosses the circle twice is a secant.
Used in
circle theorems, belts and pulleys, gears, calculus (the tangent to a curve, Class 11 and 12)
One member of a sequence, or one part of an expression joined to the rest by + or −.
Precisely: In a sequence, each entry tₙ of the ordered list; in an algebraic expression, each summand, such as 5x², 20xy and 4y² in 5x² + 20xy + 4y².
How it works
In a sequence, a term has a position: first term, second term, nth term. In an expression, split at the + and − signs (the sign goes with the term that follows it) to find the terms.
Examples
In the square numbers 1, 4, 9, 16, 25, the fifth term is 25.
In 5x² + 20xy + 4y² there are three terms.
In x² − 4x − 96, the middle term is −4x.
In an AP with first term 11 and common difference −4, the next terms are 7, 3 and −1.
Do not confuse: A term is not a factor: in 3xy, the 3, x and y are factors of one term. In 3x + y there are two terms.
Used in
sequences and progressions, algebraic expressions, factorisation
OpenStax: Elementary Algebra 2e, 1.2 Use the Language of Algebra (terms)
terminating decimal
TER-mi-nay-ting DES-i-mulClass 9
A decimal that comes to an end, like 0.375 or 2.5.
Precisely: A decimal with finitely many digits after the point; a fraction in lowest terms gives one exactly when its denominator has no prime factors other than 2 and 5.
How it works
Write the fraction in lowest terms and factorise the bottom. Only 2s and 5s means it terminates (it can be scaled to a power of 10); any other prime means it repeats.
Examples
3/8 = 0.375 is a terminating decimal, since 8 = 2 × 2 × 2.
7/20 gives the terminating decimal 0.35, since 20 = 2 × 2 × 5.
1/3 is not a terminating decimal: 3 is neither 2 nor 5, so 0.333… goes on forever.
Do not confuse: A terminating decimal is still rational, and a repeating decimal is rational too; only non-repeating, non-ending decimals are irrational.
Used in
converting fractions and decimals, money and measurement, rational numbers
OpenStax: Prealgebra 2e, 5.3 Decimals and Fractions
theoretical probability
theer-uh-TET-i-kul prob-uh-BIL-i-teeClass 9
A probability worked out by counting, when every outcome is equally likely, with no experiment needed.
Precisely: P(E) = number of favourable outcomes ÷ number of possible outcomes, valid when all outcomes in the sample space are equally likely.
How it works
List the sample space, check that its outcomes are equally likely, count the outcomes in the event, and divide by the total. It is what happens in an ideal, perfectly fair situation.
Examples
The theoretical probability of an even number on a fair die is 3/6 = 1/2.
In PEACE, the theoretical probability of drawing a P, E or C is 4/5, and of not drawing an E is 3/5.
Three coins are tossed: the theoretical probability of exactly two heads is 3/8.
The theoretical probability that a pointer on a spinner numbered 1 to 8 stops at a number less than 9 is 8/8 = 1: certain.
Do not confuse: It needs equally likely outcomes. A bag of 3 red and 7 blue balls has 2 colours, but the probability of red is 3/10, not 1/2: count the balls, not the colours.
Used in
dice, coins and cards, games of chance, probability in Class 10 to 12
A branching picture that shows every possible result of an experiment done in steps.
Precisely: A diagram for a multi-step experiment in which each stage branches into its possible outcomes; each path from the start to an end is one outcome of the whole experiment, and probabilities are written on the branches.
How it works
From a starting point draw one branch for each result of step 1; from the end of each branch draw branches for step 2, and so on. Count the paths for the sample space, and multiply the probabilities along a path for that path's probability.
Examples
A tree diagram for tossing a coin twice has 4 paths: HH, HT, TH and TT, so P(HH) = 1/4.
A tree diagram for one fruit from basket A (apple, orange, orange) and one from B (banana, mango) gives P(apple and banana) = 1/3 × 1/2 = 1/6.
For 4 red and 5 blue balls drawn one after the other without putting back, the tree diagram gives P(red then blue) = 4/9 × 5/8 = 5/18.
Picking pens (3 red, 4 black, 2 green) with replacement, the tree diagram gives P(same colour) = 9/81 + 16/81 + 4/81 = 29/81.
Do not confuse: Along one path you multiply the probabilities; for different paths that fit the same event you add them.
Used in
multi-step experiments, listing sample spaces, conditional probability (Class 12)
OpenStax: Introductory Statistics 2e, 3.5 Tree and Venn Diagrams
triangular number
try-ANG-gyoo-ler NUM-berFoundation
A number of dots that can be arranged in a triangle: 1, 3, 6, 10, 15, ...
Precisely: The nth triangular number is 1 + 2 + ... + n = n(n + 1)/2.
How it works
Each new row of the triangle has one more dot than the last, so the nth triangular number adds the first n natural numbers. Two copies of the triangle fit into an n by (n + 1) rectangle, which gives n(n + 1)/2.
Examples
The fifth triangular number is 1 + 2 + 3 + 4 + 5 = 15.
The 10th, 17th and 80th triangular numbers are 55, 153 and 3240.
25 rows of marbles with 1, 2, 3, ... in each use 325 marbles, the 25th triangular number.
The first triangular number greater than 1000 is the 45th, 1035.
Do not confuse: Triangular numbers (1, 3, 6, 10) are not an AP: their gaps (2, 3, 4) keep growing. The square numbers are sums of odd numbers instead.
Used in
patterns, the sum of the first n natural numbers, counting handshakes and pairs
An equation with trigonometric ratios that is true for every angle, like sin² A + cos² A = 1.
Precisely: An equation involving trigonometric ratios of an angle that holds for all values of the angle for which the ratios are defined; the three basic ones are sin² A + cos² A = 1, 1 + tan² A = sec² A and cot² A + 1 = cosec² A.
How it works
Start from Pythagoras in a right triangle (AB² + BC² = AC²) and divide by one side squared: dividing by AC² gives cos² A + sin² A = 1, by AB² gives 1 + tan² A = sec² A, by BC² gives cot² A + 1 = cosec² A.
Examples
sin² 30° + cos² 30° = 1/4 + 3/4 = 1, as the trigonometric identity says.
Using the trigonometric identity sin² A + cos² A = 1, if sin A = 3/5 then cos A = 4/5.
sec A (1 − sin A)(sec A + tan A) = 1 is proved from the trigonometric identity 1 + tan² A = sec² A.
Do not confuse: sin² A means (sin A)², not sin (A²). And an identity is true for every angle, not just for a particular one.
Used in
simplifying trigonometric expressions, proofs, calculus and physics (Class 11 and 12)
OpenStax: Algebra and Trigonometry 2e, 9.1 Verifying Trigonometric Identities
trigonometric ratio
trig-uh-nuh-MET-rik RAY-shee-ohClass 10
One of six fractions made from two sides of a right triangle, named after an angle: sine, cosine, tangent, cosecant, secant and cotangent.
Precisely: For an acute angle A of a right triangle, a ratio of two of its sides: sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent, and their reciprocals cosec A, sec A and cot A.
How it works
Label the sides from the angle's point of view: opposite, adjacent, hypotenuse. Similar right triangles have equal angles, so the ratios stay the same however big the triangle is. Knowing one ratio gives all the others through Pythagoras.
Examples
If tan A = 4/3, take sides 4 and 3; the hypotenuse is 5, so the other trigonometric ratios are sin A = 4/5, cos A = 3/5, cosec A = 5/4, sec A = 5/3, cot A = 3/4.
The trigonometric ratios of 30° are sin 30° = 1/2, cos 30° = √3/2 and tan 30° = 1/√3.
If sin (A − B) = 1/2 and cos (A + B) = 1/2, the trigonometric ratios give A − B = 30° and A + B = 60°, so A = 45° and B = 15°.
Do not confuse: sin A is one number, the sine of A; it is not sin × A. And the ratios depend on the angle, not the triangle's size.
Used in
heights and distances, trigonometric identities, physics and engineering
OpenStax: Algebra and Trigonometry 2e, 7.2 Right Triangle Trigonometry
trigonometry
trig-uh-NOM-uh-treeClass 10
The part of maths about the sides and angles of triangles, used to find heights and distances you cannot measure directly.
Precisely: The study of relationships between the angles and sides of triangles, through the trigonometric ratios; from the Greek tri (three), gon (side) and metron (measure).
How it works
In a right triangle the ratios of the sides depend only on the angle, not on the size of the triangle. So if you know an angle and one side, the ratios give the other sides.
Examples
Trigonometry finds the height of a tower from the angle you look up at it and your distance from its foot.
Astronomers used trigonometry to find distances to the Moon and nearby stars.
Aryabhata's Aryabhatiyam (about 500 CE) used the half-chord, the idea behind the sine in trigonometry.
Do not confuse: Trigonometry is not only about right triangles: in Class 11 its ratios become functions of any angle.
Used in
heights and distances (Class 10), navigation and surveying, physics: waves, forces and motion, engineering