Trigonometry is the branch of math that studies how the angles and the side lengths of triangles are related. Its three main ratios, sine, cosine and tangent, compare two sides of a right triangle. Knowing one side and one acute angle, or two sides, you can find every other side and angle without measuring them.
In Spanish
trigonometría
Also called
trig, right triangle trigonometry, triangle measurement
Every trigonometry problem starts the same way: stand at the angle, name the opposite, adjacent and hypotenuse, then pick the ratio that links the side you know to the side you want.
Formal definition and notation
For an acute angleθ in a right triangle, the trigonometric ratios are sin θ = opphyp, cos θ = adjhyp and tan θ = oppadj. All right triangles with the same acute angle are similar, so these ratios depend only on the angle, not on the size of the triangle. Using the unit circle, trigonometry extends sine and cosine to every angle, and to every real number when angles are measured in radians. That turns them into periodic functions that model waves, rotations and cycles.
Symbol
Read it as
Meaning
θ
theta
a Greek letter often used to name an angle
sin θ, cos θ, tan θ
sine theta, cosine theta, tangent theta
the three main trigonometric ratios of the angle θ
sin−1, cos−1, tan−1
inverse sine, inverse cosine, inverse tangent
work backward from a ratio to the angle; tan−1(1) = 45°. Also written arcsin, arccos, arctan
Key facts
There are six trigonometric functions: sine, cosine, tangent, and their reciprocals cosecant (1sin), secant(1cos) and cotangent (1tan).
Special angles are worth knowing: sin 30° = 12, cos 60° = 12, tan 45° = 1.
The Pythagorean identity, sin2θ + cos2θ = 1, is the Pythagorean theorem for a triangle with hypotenuse 1.
Angles can be measured in degrees or radians: 180° = π radians. Check your calculator’s mode before you press sin, cos or tan.
For triangles without a right angle, the law of sines and the law of cosines do the same job.
Trigonometry is used to find heights and distances you cannot measure directly, such as the height of a tree, the width of a river or the slope angle of a ramp.
The trigonometric ratios (SOH CAH TOA)
The trigonometric ratios (SOH CAH TOA)
sinθ = opphyp, cosθ = adjhyp, tanθ = oppadj
opp
the leg across from the angle θ
adj
the leg that touches θ (and is not the hypotenuse)
hyp
the hypotenuse, the longest side, across from the right angle
Opposite and adjacent depend on which acute angle you stand at. Switch to the other acute angle and the two legs swap names, but the hypotenuse never changes.
Try it: build a right triangle
Interactive
Drag the ends of legs a and b to change the triangle; the readout finds the hypotenuse c. For the angle where leg b meets c, leg a is opposite and leg b is adjacent, so its sine is a ÷ c and its tangent is a ÷ b. Double both legs and check that the ratios stay the same.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Trigonometry examples, step by step
Example 1: Write all three ratios
Warm-up
In a right triangle, the leg opposite angle θ is 8, the adjacent leg is 15 and the hypotenuse is 17. Find sin θ, cos θ and tan θ.
SOH: sin θ = opphyp = 817.
CAH: cos θ = adjhyp = 1517.
TOA: tan θ = oppadj = 815.
Answer:sin θ = 817, cos θ = 1517 and tan θ = 815.
Example 2: Find a missing side
Core
A 20-foot ladder leans against a wall and makes a 70° angle with the ground. How high up the wall does it reach, to the nearest tenth of a foot?
The ladder is the hypotenuse (20). The height is the side opposite the 70° angle.
Opposite and hypotenuse means sine: sin 70° = h20.
Multiply: h = 20 × sin 70° ≈ 20 × 0.9397.
h ≈ 18.8.
Answer: The ladder reaches about 18.8 feet up the wall.
Example 3: Find a missing angle
Test-style
A wheelchair ramp rises 2 feet over a horizontal distance of 24 feet. What angle does the ramp make with the ground, to the nearest tenth of a degree?
The rise is opposite the angle and the horizontal distance is adjacent, so use tangent: tan θ = 224.
Undo tangent with inverse tangent: θ = tan−1(224).
A calculator in degree mode gives θ ≈ 4.8°.
Answer: The ramp makes an angle of about 4.8 degrees with the ground.
Common mistakes
✗ MistakeFor angle A, calling the leg that touches A the opposite side.
✓ InsteadThe opposite side is the one across the triangle from A; the leg touching A is the adjacent side.
Opposite and adjacent are always named from the angle you are using, and they swap when you switch angles.
✗ MistakeGetting sin 30 ≈ −0.988 on a calculator.
✓ InsteadIn degree mode, sin 30° = 0.5.
The calculator was in radian mode and found the sine of 30 radians. Match the mode to the units of the angle.
✗ MistakeAssuming sin 60° = 2 × sin 30° = 1.
✓ Insteadsin 60° ≈ 0.866, not 1.
Sine is not proportional to the angle. Doubling the angle does not double the ratio.
✗ MistakeUsing SOH CAH TOA in a triangle with angles 50°, 60° and 70°.
✓ InsteadUse the law of sines or the law of cosines, or draw an altitude to make right triangles.
The ratios opp ÷ hyp and adj ÷ hyp are defined only in right triangles.
The Pythagorean theorem links only the three sides of a right triangle, a2 + b2 = c2. Trigonometry links sides and angles, so it can find an angle from two sides.
Sine is one tool of trigonometry, the ratio opposite ÷ hypotenuse. Trigonometry is the whole subject, which also uses cosine, tangent, identities and the laws of sines and cosines.
Trigonometryvs.Geometry
Geometry studies shapes, sizes and positions in general. Trigonometry is the part that measures triangles with ratios and then extends those ratios to circles and waves.
Quick check: 3 questions
Question 1 of 3
In a right triangle, which ratio equals the adjacent side divided by the hypotenuse?
Show answer and explanation
Answer: B. CAH: cosine is adjacent over hypotenuse. Sine uses the opposite side, tangent is opposite over adjacent, and secant is the flip of cosine, hypotenuse over adjacent.
Question 2 of 3
In a right triangle, the side opposite angle A is 7, the other leg is 24 and the hypotenuse is 25. What is sinA?
Show answer and explanation
Answer: C. Sine is opposite over hypotenuse: 725. 2425 is the cosine, 724 is the tangent, and 257 is the ratio upside down.
Question 3 of 3
A kite string is 60 m long and makes a 40° angle with the flat ground. About how high is the kite?
Show answer and explanation
Answer: A. The string is the hypotenuse and the height is opposite the angle, so h = 60 × sin 40° ≈ 38.6 m. 46.0 uses cosine, 50.3 uses tangent, and 93.3 divides by sine.
Trigonometry: frequently asked questions
What is trigonometry used for?
Trigonometry finds lengths and angles that are hard to measure directly. Surveyors use it to find heights and distances, builders use it for roof pitches and ramps, and navigation uses it to find positions from angles. Because sine and cosine repeat, it also describes anything that cycles, such as sound waves, tides and a point on a turning wheel.
What are the six trigonometric functions?
They are sine, cosine and tangent, plus their reciprocals: cosecant = 1sin, secant = 1cos and cotangent = 1tan. In a right triangle, each one compares two sides as seen from one acute angle. Most problems need only sine, cosine and tangent; the other three are mainly a shorthand.
What should you know before learning trigonometry?
You need the parts of a right triangle, the Pythagorean theorem, similar triangles and solving simple equations such as x ÷ 12 = 0.6. Comfort with fractions, square roots and a scientific calculator also helps. Algebra skills matter most later, when trigonometry moves to the unit circle, graphs and identities.
Acute AngleAn acute angle is an angle that measures more than 0 degrees and less than 90 degrees. It is narrower than the square corner of a page. Angles of 15, 45 and 89 degrees are all acute; 90 degrees is a right angle, so it is not acute. Every triangle has at least two acute angles.Geometry · Grade 4, Grade 5, Grade 7Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8Similar FiguresSimilar figures have the same shape but not necessarily the same size. Corresponding angles match, and corresponding side lengths are proportional: each side of one figure is the same multiple, the scale factor, of the matching side of the other. A scale drawing is similar to the real object.Geometry · Grade 7, Grade 8, Algebra 1TriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5Unit CircleThe unit circle is the circle with radius 1 centered at the origin, x2 + y2 = 1. An angle θ measured from the positive x-axis meets the circle at the point (cos θ, sin θ). This lets sine and cosine work for every angle, including obtuse and negative angles, not just the acute angles of right triangles.Trigonometry · Algebra 2, PrecalculusSineSine is the trigonometric ratio that compares the side opposite an angle with the hypotenuse of a right triangle: sin θ = oppositehypotenuse. On the unit circle, sin θ is the y-coordinate of the point at angle θ, so sine values always stay between −1 and 1.Trigonometry · Geometry, Algebra 2, PrecalculusCosineCosine is the trigonometric ratio that compares the side adjacent to an angle with the hypotenuse of a right triangle: cos θ = adjacenthypotenuse. On the unit circle, cos θ is the x-coordinate of the point at angle θ. The cosine of an angle equals the sine of its complement.Trigonometry · Geometry, Algebra 2, PrecalculusReal NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1RadianA radian is a unit for measuring angles: one radian is the central angle that cuts off an arc exactly as long as the circle’s radius. A full turn is 2π radians, so π radians = 180° and 1 radian is about 57.3°. To change degrees to radians, multiply by π180.Trigonometry · Geometry, Algebra 2, PrecalculusRotationA rotation is a transformation that turns a figure around a fixed point, called the center, by a given angle and direction. Every point stays the same distance from the center, so size and shape do not change. About the origin, a 90° counterclockwise turn sends (x, y) to (−y, x).Geometry · Grade 8, GeometryReciprocalThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 34 is 43, and the reciprocal of 5 is 15. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Number System · Grade 5, Grade 6, Grade 7SecantSecant (sec) is the trigonometric function equal to 1 divided by cosine: sec θ = 1cos θ. In a right triangle it is the hypotenuse divided by the side adjacent to the angle. Secant is undefined wherever cosine is 0, such as at 90°, and its values are always at least 1 or at most −1.Trigonometry · Algebra 2, Precalculus, CalculusPythagorean TheoremThe Pythagorean theorem says that in a right triangle, a2 + b2 = c2. The legs are a and b, the two sides that form the right angle. The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Geometry · Grade 8, Algebra 1HypotenuseThe hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a2 + b2 = c2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.Geometry · Grade 8, Algebra 1AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6Special Right TrianglesSpecial right triangles are the two right triangles whose angles are 45°-45°-90° or 30°-60°-90°. Their sides always come in fixed ratios: x : x : x√2 for the 45-45-90 triangle and x : x√3 : 2x for the 30-60-90 triangle. From one side you can find the other two without the Pythagorean theorem.Geometry · Grade 8, Geometry, Algebra 2TangentTangent (tan) is the trigonometric ratio of the side opposite an acute angle to the side adjacent to it in a right triangle: tan θ = oppositeadjacent. For legs 3 and 4, tan θ = 34 = 0.75. It is the TOA in SOH CAH TOA, and it equals sin θcos θ.Trigonometry · Geometry, Algebra 2, Precalculus