📐 Math Formulas
70 formulas in 20 sections. For each one you get the formula, what every symbol means, where it is used, an explanation in plain words and a worked example.
Sections: Ratios · Identities · Standard Values · Reciprocal Ratios · Heights & Distances · Compound Angles · Double Angle · Laws (Sine & Cosine) · Radians · Quadrants & Signs · Allied Angles · Periodicity · Triple Angle · Product to Sum · Sum to Product · Inverse Trigonometry · Maxima & Minima · Half Angle · Extra Standard Values · General Solutions
sin θ = Opposite ÷ Hypotenuse
| θ | the angle |
|---|---|
| Opposite | side facing the angle |
| Hypotenuse | longest side |
Where it is used: In a right triangle, to link an angle to two of its sides — heights, distances, physics, waves.
In simple words: For a given corner angle, sine is the side across from it divided by the longest slanting side.
Example: Opposite=3, Hyp=5 → sin θ = 3/5 = 0.6
cos θ = Adjacent ÷ Hypotenuse
| θ | the angle |
|---|---|
| Adjacent | side next to the angle |
| Hypotenuse | longest side |
Where it is used: To relate an angle to the side beside it and the hypotenuse — surveying, navigation, forces.
In simple words: Cosine is the side next to the angle divided by the longest slanting side.
Example: Adjacent=4, Hyp=5 → cos θ = 4/5 = 0.8
tan θ = Opposite ÷ Adjacent
| θ | the angle |
|---|---|
| Opposite | side facing the angle |
| Adjacent | side next to the angle |
Where it is used: To find an angle or height from the two shorter sides — ramps, slopes, heights of buildings.
In simple words: Tangent is the side across from the angle divided by the side next to it. Also sin ÷ cos.
Example: Opposite=3, Adjacent=4 → tan θ = 3/4 = 0.75
sin=O/H, cos=A/H, tan=O/A, cosec=H/O, sec=H/A, cot=A/O
| O | opposite side |
|---|---|
| A | adjacent side |
| H | hypotenuse |
Where it is used: The complete set of the six trig ratios in a right triangle, the base of all trigonometry.
In simple words: 'SOH-CAH-TOA' helps you remember the first three; the last three are just their flips.
Example: O=3, A=4, H=5 → sin=3/5, cos=4/5, tan=3/4
sec(90°−θ)=cosecθ, cosec(90°−θ)=secθ
| θ | the angle |
|---|
Where it is used: Cofunction identities for the reciprocal ratios.
In simple words: Secant of an angle equals cosecant of its complement, and vice versa.
Example: sec60° = cosec30° = 2
−1 ≤ sinθ ≤ 1, −1 ≤ cosθ ≤ 1
| θ | any angle |
|---|
Where it is used: To check that a sine or cosine answer is valid — they never leave [−1, 1].
In simple words: Sine and cosine always stay between −1 and 1, whatever the angle.
Example: sinθ = 2 has no solution
sin²θ + cos²θ = 1
| θ | any angle |
|---|
Where it is used: The most-used trig identity — to find one ratio from another and to simplify expressions.
In simple words: For any angle, the square of its sine plus the square of its cosine is always exactly 1.
Example: sin30°=0.5, cos30°≈0.866 → 0.25+0.75 = 1
tan θ = sin θ ÷ cos θ
| θ | the angle (cos θ ≠ 0) |
|---|
Where it is used: To rewrite tangent in terms of sine and cosine when simplifying or solving.
In simple words: Tangent is just sine divided by cosine for the same angle.
Example: tan45° = sin45°/cos45° = 1
sin(A + B) = sinA cosB + cosA sinB
| A, B | two angles |
|---|
Where it is used: To expand the sine of a combined angle — used in waves, oscillations and advanced trig.
In simple words: Breaks the sine of two added angles into a combination of their sines and cosines.
Example: sin(30°+60°) = sin90° = 1
cos(A + B) = cosA cosB − sinA sinB
| A, B | two angles |
|---|
Where it is used: To expand the cosine of a combined angle in trigonometry and physics.
In simple words: Breaks the cosine of two added angles into their cosines and sines combined.
Example: cos(30°+60°) = cos90° = 0
1 + tan²θ = sec²θ
| θ | any angle |
|---|
Where it is used: A key identity used to switch between tangent and secant when simplifying or in calculus.
In simple words: One plus tangent-squared always equals secant-squared for the same angle.
Example: tan45°=1 → 1+1 = 2 = sec²45°
1 + cot²θ = cosec²θ
| θ | any angle |
|---|
Where it is used: Used to switch between cotangent and cosecant in proofs and integration.
In simple words: One plus cotangent-squared always equals cosecant-squared for the same angle.
Example: cot45°=1 → 1+1 = 2 = cosec²45°
cos2A = (1 − tan²A) ÷ (1 + tan²A)
| A | the angle |
|---|
Where it is used: A tangent form of the double-angle cosine — used in integration substitution.
In simple words: Gives cosine of double the angle using only the tangent of the single angle.
Example: A=45°, tanA=1 → cos90° = 0
sin2A = 2tanA ÷ (1 + tan²A)
| A | the angle |
|---|
Where it is used: A tangent form of the double-angle sine — used in integration substitution.
In simple words: Gives sine of double the angle using only the tangent of the single angle.
Example: A=45°, tanA=1 → sin90° = 1
tanθ × cotθ = 1, sinθ × cosecθ = 1, cosθ × secθ = 1
| θ | any angle |
|---|
Where it is used: The reciprocal pairs multiply to 1 — used to simplify expressions.
In simple words: Each ratio times its flip equals 1.
Example: tan30° × cot30° = 1
sin0°=0, sin30°=½, sin45°=1/√2, sin60°=√3/2, sin90°=1
| θ | common angles 0°–90° |
|---|
Where it is used: The key values memorised for quick trig calculations in exams and problems.
In simple words: These are the ready answers for sine of the common angles, no calculator needed.
Example: sin30° = ½ = 0.5
cos0°=1, cos30°=√3/2, cos45°=1/√2, cos60°=½, cos90°=0
| θ | common angles 0°–90° |
|---|
Where it is used: The memorised cosine values for quick trig calculations in exams.
In simple words: Ready answers for cosine of the common angles — notice they run opposite to the sine values.
Example: cos60° = ½ = 0.5
tan0°=0, tan30°=1/√3, tan45°=1, tan60°=√3, tan90°=∞
| θ | common angles 0°–90° |
|---|
Where it is used: The memorised tangent values, used in heights-and-distances and geometry problems.
In simple words: Ready answers for tangent of the common angles; tan45° = 1 is the easiest to remember.
Example: tan60° = √3 ≈ 1.732
cosec30°=2, sec60°=2, cot45°=1, cosec45°=√2, sec45°=√2
| θ | common angles |
|---|
Where it is used: The reciprocal-ratio values for the standard angles, for quick calculation.
In simple words: Just the flips of the sine, cosine and tangent standard values.
Example: cosec30° = 1/sin30° = 1/0.5 = 2
sin0°=0, sin90°=1, sin180°=0, sin270°=−1
| 0°,90°,180°,270° | the quadrant boundaries |
|---|
Where it is used: The sine values at the four main compass directions of the circle.
In simple words: Sine's values at the top, bottom and sides of the circle.
Example: sin270° = −1 (bottom of the circle)
cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
| θ | the angle |
|---|
Where it is used: The reciprocal trig ratios, used in identities, calculus and advanced problems.
In simple words: These are just 'one divided by' the three main ratios — the flipped versions.
Example: sin30°=0.5 → cosec30° = 1/0.5 = 2
cot θ = cos θ ÷ sin θ
| θ | the angle (sin θ ≠ 0) |
|---|
Where it is used: To rewrite cotangent using sine and cosine when simplifying trig expressions.
In simple words: Cotangent is cosine divided by sine — the upside-down of tangent.
Example: cot45° = cos45°/sin45° = 1
Height = Distance × tan θ
| θ | angle of elevation |
|---|---|
| Distance | horizontal distance to the object |
Where it is used: To find the height of a tower, tree or building from a distance and the angle looking up.
In simple words: Stand back a known distance, measure the up-angle to the top, and this gives the height.
Example: Distance=30 m, θ=45° → Height = 30 × 1 = 30 m
Distance = Height ÷ tan θ
| θ | angle of elevation |
|---|---|
| Height | height of the object |
Where it is used: To find how far you are standing from a tower or tree from its height and the up-angle.
In simple words: Knowing the height and the angle you look up at, this gives how far away you are.
Example: Height=30 m, θ=45° → Distance = 30 ÷ 1 = 30 m
Angle of depression = Angle of elevation (alternate angles)
| — | measured downward from the horizontal |
|---|
Where it is used: To solve problems where you look DOWN at an object from a height.
In simple words: The down-angle from the top equals the up-angle from the bottom, so the same maths applies.
Example: From a cliff, the depression to a boat equals the boat's elevation to the cliff top.
sin(A − B) = sinA cosB − cosA sinB
| A, B | two angles |
|---|
Where it is used: To expand the sine of a difference of angles — trig proofs, waves, physics.
In simple words: Like the sum version, but with a minus in the middle.
Example: sin(60°−30°) = sin30° = 0.5
cos(A − B) = cosA cosB + sinA sinB
| A, B | two angles |
|---|
Where it is used: To expand the cosine of a difference of angles in trigonometry.
In simple words: Like the cosine sum version, but with a plus in the middle.
Example: cos(60°−30°) = cos30° ≈ 0.866
tan(A + B) = (tanA + tanB) ÷ (1 − tanA tanB)
| A, B | two angles |
|---|
Where it is used: To find the tangent of a combined angle — used in slopes and rotations.
In simple words: Combines the two tangents on top, with a correction underneath.
Example: tan(45°+45°) = tan90° = ∞ (denominator 0)
tan(A − B) = (tanA − tanB) ÷ (1 + tanA tanB)
| A, B | two angles |
|---|
Where it is used: To find the tangent of a difference of angles — slopes, angle between lines.
In simple words: Like the tangent sum, but signs swap: minus on top, plus underneath.
Example: tan(45°−45°) = tan0° = 0
sin 2A = 2 sinA cosA
| A | the angle |
|---|
Where it is used: To rewrite sin of twice an angle — used in waves, integration and simplification.
In simple words: Sine of double the angle is twice the sine times the cosine of the single angle.
Example: sin60° = 2·sin30°·cos30° = 2·½·(√3/2) = √3/2
cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1
| A | the angle |
|---|
Where it is used: Three useful forms for cos of a double angle — chosen to suit the problem.
In simple words: Different ways to write cosine of double an angle; pick whichever fits what you already know.
Example: cos60° = 1 − 2sin²30° = 1 − 2(¼) = ½
tan 2A = 2 tanA ÷ (1 − tan²A)
| A | the angle |
|---|
Where it is used: To find the tangent of twice an angle in trig and calculus.
In simple words: A ready formula for tangent of double an angle from the single-angle tangent.
Example: A=30°, tan30°=1/√3 → tan60° = (2/√3)/(1−1/3) = √3
sinθ = 2t/(1+t²), cosθ = (1−t²)/(1+t²), t = tan(θ/2)
| t | tan of the half angle |
|---|
Where it is used: To express sine and cosine through a single variable — used in integration substitutions.
In simple words: Everything can be written in terms of the tangent of half the angle.
Example: θ=90°, t=1 → sin = 2/2 = 1, cos = 0
a/sinA = b/sinB = c/sinC
| a, b, c | the three sides |
|---|---|
| A, B, C | the angles opposite them |
Where it is used: To find unknown sides or angles in ANY triangle (not just right-angled) — surveying, navigation.
In simple words: In any triangle, each side divided by the sine of its opposite angle gives the same value.
Example: If a=7, A=30°, B=45° → b = 7·sin45°/sin30°
c² = a² + b² − 2ab·cosC
| a, b, c | the three sides |
|---|---|
| C | angle opposite side c |
Where it is used: To find the third side of any triangle from two sides and the angle between them, or to find an angle from three sides.
In simple words: A stronger Pythagoras that works for any triangle, adjusting for the angle between the sides.
Example: a=3, b=4, C=90° → c² = 9+16−0 = 25, c=5
Area = ½ × a × b × sinC
| a, b | two sides |
|---|---|
| C | the angle between them |
Where it is used: To find a triangle's area from two sides and the included angle — surveying, geometry.
In simple words: Half the product of two sides times the sine of the angle between them.
Example: a=6, b=8, C=30° → ½·6·8·0.5 = 12
a = b·cosC + c·cosB
| a, b, c | sides |
|---|---|
| B, C | angles opposite b and c |
Where it is used: Expresses one side of a triangle as the sum of projections of the other two — used in triangle proofs.
In simple words: One side equals the shadows the other two sides cast along it.
Example: Holds for any triangle's sides and angles.
cosC = (a² + b² − c²) ÷ 2ab
| a, b, c | the three sides |
|---|---|
| C | angle opposite side c |
Where it is used: To find an angle of a triangle when all three sides are known.
In simple words: Rearranges the cosine rule to give an angle from the three side lengths.
Example: a=b=c → cosC = 1/2, so C = 60° (equilateral)
Radians = Degrees × (π ÷ 180)
| π | pi ≈ 3.14159 |
|---|---|
| 180 | degrees in π radians |
Where it is used: To convert an angle from degrees to radians — needed in higher maths, calculus and physics.
In simple words: Multiply the degrees by π and divide by 180 to get the same angle in radians.
Example: 90° = 90 × π/180 = π/2 radians
s = r × θ
| s | arc length |
|---|---|
| r | radius |
| θ | angle in radians |
Where it is used: To find the length of a circular arc when the angle is in radians — physics, engineering.
In simple words: Multiply the radius by the angle (in radians) to get the arc length.
Example: r=5, θ=2 rad → s = 10 units
Degrees = Radians × (180 ÷ π)
| π | pi ≈ 3.14159 |
|---|
Where it is used: To convert an angle from radians back to degrees.
In simple words: Multiply radians by 180 and divide by π to get degrees.
Example: π/2 rad = 90°
Q1: All +, Q2: Sin +, Q3: Tan +, Q4: Cos +
| Q1–Q4 | the four quadrants (0–90, 90–180, 180–270, 270–360) |
|---|
Where it is used: To decide whether a trig ratio is positive or negative for angles beyond 90°.
In simple words: 'All-Silver-Tea-Cups': tells which ratios stay positive in each quarter turn.
Example: cos300° is positive (Q4), sin200° is negative (Q3)
sin(−θ)=−sinθ, cos(−θ)=cosθ, tan(−θ)=−tanθ
| θ | the angle |
|---|
Where it is used: To simplify trig of negative angles — cosine is even, sine and tangent are odd.
In simple words: A minus angle keeps cosine the same but flips the sign of sine and tangent.
Example: sin(−30°) = −sin30° = −0.5
sin(90°−θ)=cosθ, cos(90°−θ)=sinθ, tan(90°−θ)=cotθ
| θ | the angle |
|---|
Where it is used: To convert between sine/cosine and tangent/cotangent of complementary angles.
In simple words: The sine of an angle equals the cosine of its complement, and vice versa.
Example: sin60° = cos30°
sin(180°−θ)=sinθ, cos(180°−θ)=−cosθ, tan(180°−θ)=−tanθ
| θ | the angle |
|---|
Where it is used: To find trig ratios of second-quadrant angles from first-quadrant ones.
In simple words: For an angle and its supplement, sine stays the same but cosine and tangent flip sign.
Example: sin150° = sin30° = 0.5
sin(90°+θ)=cosθ, cos(90°+θ)=−sinθ, tan(90°+θ)=−cotθ
| θ | the angle |
|---|
Where it is used: To reduce second-quadrant angles to first-quadrant ratios.
In simple words: Adding 90° swaps sine and cosine and adjusts the sign by quadrant.
Example: sin120° = sin(90°+30°) = cos30° ≈ 0.866
sin(360°−θ)=−sinθ, cos(360°−θ)=cosθ, tan(360°−θ)=−tanθ
| θ | the angle |
|---|
Where it is used: To find ratios of fourth-quadrant angles from first-quadrant ones.
In simple words: An angle just short of a full turn keeps cosine but flips sine and tangent.
Example: cos330° = cos30° ≈ 0.866
sin(θ+360°)=sinθ, cos(θ+360°)=cosθ, tan(θ+180°)=tanθ
| θ | the angle |
|---|
Where it is used: To reduce very large angles by removing whole turns before finding the ratio.
In simple words: Trig ratios repeat every full turn (360°); tangent repeats every half turn.
Example: cos400° = cos40°
sin3A = 3sinA − 4sin³A, cos3A = 4cos³A − 3cosA
| A | the angle |
|---|
Where it is used: To expand sine and cosine of three times an angle — used in equations and proofs.
In simple words: Ready formulas for the sine and cosine of triple an angle.
Example: sin90° = 3sin30° − 4sin³30° = 1.5 − 0.5 = 1
tan3A = (3tanA − tan³A) ÷ (1 − 3tan²A)
| A | the angle |
|---|
Where it is used: To expand the tangent of three times an angle.
In simple words: A ready formula for tangent of triple an angle from the single-angle tangent.
Example: Used to solve tan-based triple-angle equations.
2sinA cosB = sin(A+B) + sin(A−B)
| A, B | two angles |
|---|
Where it is used: To turn a product of ratios into a sum — used in integration and wave problems.
In simple words: Rewrites a multiplication of sine and cosine as an addition, which is easier to handle.
Example: 2sin45°cos15° = sin60° + sin30°
2cosA cosB = cos(A−B)+cos(A+B), 2sinA sinB = cos(A−B)−cos(A+B)
| A, B | two angles |
|---|
Where it is used: To turn products of two cosines or two sines into sums — used in integration.
In simple words: Rewrites a product of two cosines (or two sines) as a sum of cosines.
Example: 2cos45°cos15° = cos30° + cos60°
2cosA sinB = sin(A+B) − sin(A−B)
| A, B | two angles |
|---|
Where it is used: The fourth product-to-sum identity, completing the set.
In simple words: Turns cosine times sine into a difference of sines.
Example: 2cos60°sin30° = sin90° − sin30°
sinC + sinD = 2 sin((C+D)/2) cos((C−D)/2)
| C, D | two angles |
|---|
Where it is used: To turn a sum of sines into a product — used in solving trig equations.
In simple words: Rewrites the sum of two sines as a neat product of a sine and a cosine.
Example: sin50° + sin10° = 2 sin30° cos20°
sinC − sinD = 2 cos((C+D)/2) sin((C−D)/2)
| C, D | two angles |
|---|
Where it is used: To turn a difference of sines into a product — solving trig equations.
In simple words: Rewrites one sine minus another as a neat product.
Example: sin50° − sin10° = 2 cos30° sin20°
cosC+cosD = 2cos((C+D)/2)cos((C−D)/2); cosC−cosD = −2sin(..)sin(..)
| C, D | two angles |
|---|
Where it is used: To turn a sum or difference of cosines into a product.
In simple words: Rewrites two cosines added or subtracted as a product form.
Example: cos40° + cos20° = 2 cos30° cos10°
sin⁻¹x + cos⁻¹x = 90°, tan⁻¹x + cot⁻¹x = 90°
| x | the value (−1 ≤ x ≤ 1 for sin⁻¹, cos⁻¹) |
|---|
Where it is used: Standard inverse-trig identities used in Class 12 calculus and equations.
In simple words: The inverse-sine and inverse-cosine of the same number always add up to 90°.
Example: sin⁻¹(0.5)=30°, cos⁻¹(0.5)=60° → 90°
tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)), xy < 1
| x, y | the two values |
|---|
Where it is used: To combine two inverse tangents into one — used in Class 12 problems.
In simple words: Adds two inverse-tangent angles into a single inverse tangent.
Example: tan⁻¹1 + tan⁻¹(1/2) combine via this rule
sin⁻¹: [−90°,90°], cos⁻¹: [0°,180°], tan⁻¹: (−90°,90°)
| — | the allowed output angles |
|---|
Where it is used: The standard output ranges that make inverse trig single-valued (Class 12).
In simple words: Each inverse trig gives an angle only within a fixed range, so the answer is unique.
Example: sin⁻¹(0.5) = 30°, not 150°
2 tan⁻¹x = tan⁻¹(2x/(1−x²)), |x| < 1
| x | the value |
|---|
Where it is used: To simplify twice an inverse tangent into a single inverse tangent.
In simple words: Combines two equal inverse-tangent angles into one.
Example: 2tan⁻¹(1/3) = tan⁻¹(3/4)
max = √(a²+b²), min = −√(a²+b²)
| a, b | the coefficients |
|---|
Where it is used: To find the largest and smallest values of a sine-plus-cosine expression — used in optimisation and waves.
In simple words: A sine and cosine added together swings between plus and minus the root of a²+b².
Example: 3sinθ+4cosθ ranges from −5 to 5
sinθ cosθ = ½ sin2θ, so max = ½, min = −½
| θ | the angle |
|---|
Where it is used: To find the largest value of a sine-times-cosine product — optimisation.
In simple words: Sine times cosine is just half of sin(2θ), so it never exceeds one-half.
Example: Greatest value of sinθcosθ is ½ (at 45°)
sin²(A/2) = (1−cosA)/2, cos²(A/2) = (1+cosA)/2
| A | the angle |
|---|
Where it is used: To find sine or cosine of half an angle from the cosine of the full angle.
In simple words: Gives the half-angle's sine and cosine using the full angle's cosine.
Example: cos²(30°) = (1+cos60°)/2 = 3/4
tan(A/2) = (1 − cosA) ÷ sinA = sinA ÷ (1 + cosA)
| A | the angle |
|---|
Where it is used: To find the tangent of half an angle — used in integration substitutions.
In simple words: Two equivalent ways to get the tangent of half the angle.
Example: tan45° = (1−cos90°)/sin90° = 1
sin15° = (√3−1)/(2√2), sin75° = (√3+1)/(2√2)
| 15°, 75° | obtained from 45°−30° and 45°+30° |
|---|
Where it is used: Exact values for non-standard angles, found using compound-angle formulas.
In simple words: The exact values for 15° and 75°, worked out from the 45° and 30° values.
Example: sin15° ≈ 0.259
sin18° = (√5 − 1) ÷ 4
| 18° | a standard exact value |
|---|
Where it is used: An exact special value used in pentagon geometry and some proofs.
In simple words: The precise value of sine 18°, linked to the golden ratio.
Example: sin18° ≈ 0.309
θ = nπ + (−1)ⁿ α, n = 0, ±1, ±2, …
| α | a known solution |
|---|---|
| n | any integer |
Where it is used: To write all angles that share the same sine — solving trig equations (Class 11).
In simple words: Gives every angle, not just one, that has the same sine value.
Example: sinθ = ½ → θ = nπ + (−1)ⁿ·30°
θ = 2nπ ± α, n = 0, ±1, ±2, …
| α | a known solution |
|---|---|
| n | any integer |
Where it is used: To list all angles with the same cosine.
In simple words: Every angle with the same cosine as α, going round and round.
Example: cosθ = ½ → θ = 2nπ ± 60°
θ = nπ + α, n = 0, ±1, ±2, …
| α | a known solution |
|---|---|
| n | any integer |
Where it is used: To list all angles with the same tangent.
In simple words: Tangent repeats every half turn, so add any multiple of 180° to a solution.
Example: tanθ = 1 → θ = nπ + 45°
sinθ=0 → θ=nπ; cosθ=0 → θ=(2n+1)π/2
| n | any integer |
|---|
Where it is used: The base cases for solving trig equations equal to zero.
In simple words: Sine is zero at whole multiples of 180°; cosine is zero at odd multiples of 90°.
Example: sinθ=0 → θ = 0°, 180°, 360°, …