Trigonometry is the branch of mathematics that studies the relationships between angles and the sides of triangles. The trigonometric functions — sine, cosine, tangent and cotangent — are essential in geometry, physics, engineering and computer science. In this guide you'll learn their definitions, formulas and practical applications.
Definitions of the trigonometric functions
In a right triangle with acute angle α, opposite side a, adjacent side b and hypotenuse c:
- sin α = a / c — the ratio of the opposite side to the hypotenuse
- cos α = b / c — the ratio of the adjacent side to the hypotenuse
- tan α = a / b = sin α / cos α — the ratio of the opposite side to the adjacent side
- cot α = b / a = cos α / sin α — the ratio of the adjacent side to the opposite side
Trigonometric values for common angles
It's worth knowing the function values for angles of 0°, 30°, 45°, 60° and 90°:
- 0°: sin=0, cos=1, tan=0
- 30°: sin=1/2, cos=√3/2, tan=1/√3 ≈ 0.577
- 45°: sin=√2/2, cos=√2/2, tan=1
- 60°: sin=√3/2, cos=1/2, tan=√3 ≈ 1.732
- 90°: sin=1, cos=0, tan=undefined
Converting degrees to radians and back
Scientific calculators often work in radians. To convert:
- Degrees to radians:
rad = degrees × π / 180 - Radians to degrees:
degrees = rad × 180 / π
Example: 90° = π/2 ≈ 1.571 rad. Always check your calculator's mode (DEG or RAD) before calculating.
Basic trigonometric formulas
The Pythagorean identity
sin²α + cos²α = 1 — this is the most important trigonometric formula, following directly from the Pythagorean theorem.
Sum and difference formulas
sin(α + β) = sin α · cos β + cos α · sin βcos(α + β) = cos α · cos β − sin α · sin βtan(α + β) = (tan α + tan β) / (1 − tan α · tan β)
Double-angle formulas
sin(2α) = 2 · sin α · cos αcos(2α) = cos²α − sin²α = 1 − 2sin²α = 2cos²α − 1
Trigonometric functions on the unit circle
On a circle of radius 1, for an angle α measured from the x-axis:
- The point's x-coordinate = cos α
- The point's y-coordinate = sin α
This approach extends the definition to obtuse angles, reflex angles, and angles greater than 360°, which is essential in the physics of waves and signals.
The law of sines and the law of cosines
For any triangle with sides a, b, c and opposite angles A, B, C:
- Law of sines:
a/sin A = b/sin B = c/sin C - Law of cosines:
c² = a² + b² − 2ab · cos C
The law of cosines is a generalization of the Pythagorean theorem for non-right triangles.
Practical applications of trigonometry
- Architecture and construction — calculating roof pitch, rafter length, and structural strength.
- Navigation — a ship's or aircraft's heading is determined trigonometrically.
- Physics — force, velocity and acceleration components in a coordinate system.
- Electronics — sinusoidal signals, filtering, Fourier analysis.
- Computer graphics — rotating 2D and 3D objects uses sin and cos.
Use our trigonometric calculator to quickly compute sin, cos and tan for any angle.
FAQ — frequently asked questions about trigonometry
How do you calculate sin 45°?
sin 45° = √2/2 ≈ 0.7071. This comes from an isosceles right triangle, where both acute angles are 45° and both legs are equal.
Why is tan 90° undefined?
tan α = sin α / cos α. For α = 90°: cos 90° = 0, and dividing by zero isn't allowed. The tangent approaches infinity as the angle approaches 90°.
What are the inverse functions — arcsin, arccos, arctan?
Inverse functions let you find the angle when you know the value of the function. E.g. if sin α = 0.5, then α = arcsin(0.5) = 30°. On a calculator these are the sin⁻¹, cos⁻¹, tan⁻¹ buttons.
How do you calculate a triangle's side length using trigonometry?
Knowing one acute angle α and the hypotenuse c: opposite side = c · sin α, adjacent side = c · cos α. You can also use the law of sines or the law of cosines for triangles that aren't necessarily right triangles.
How can you memorize trigonometric formulas?
A popular mnemonic is SOH-CAH-TOA (Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent). Another method is memorizing the unit circle.
What's the difference between radians and degrees?
Degrees divide a full turn into 360 parts. Radians are an arc-length measure — a full turn is 2π rad ≈ 6.283 rad. Scientific calculators can work in either mode, so always check which one is active.
What is the Pythagorean trigonometric identity?
The Pythagorean identity is sin²α + cos²α = 1. It follows directly from the Pythagorean theorem applied to a triangle on the unit circle.
Is trigonometry needed in everyday life?
Yes. It's used by architects, engineers, surveyors, navigators, musicians (sound analysis) and graphics programmers. Even calculating a ramp's slope or a ladder's leaning angle is trigonometry.
How do you calculate a triangle's area using sin?
Triangle area = (1/2) · a · b · sin C, where a and b are two sides and C is the angle between them. This formula works for any triangle, not just right triangles.
What are hyperbolic trigonometric functions?
Sinh, cosh, and tanh are functions similar to the trigonometric ones, but derived from the exponential function e rather than the circle. They're used in relativistic physics, engineering and machine learning.