About the Trigonometric Identities Calculator
Trigonometric identities are equations that hold for every angle, such as sin²θ + cos²θ = 1. They are the tools used to simplify expressions, solve equations, integrate in calculus and derive formulas in physics and engineering. Learning them means memorising a handful of core formulas and knowing how to rearrange them, and it helps a great deal to see them working with real numbers.
This trigonometric identities calculator evaluates the main identities at an angle θ, and at a second angle φ for the sum and difference formulas. It shows the value of each side so you can confirm the identity holds, gives exact values such as √3/2 for multiples of 15°, and checks the sum formula against a direct calculation.
How to Use the Trigonometric Identities Calculator
Enter the angle θ.
Enter a second angle φ for the sum and difference formulas.
Choose degrees or radians.
The Formulas
Pythagorean: sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = csc²θ
quotient: tan θ = sin θ / cos θ
double angle: sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
half angle: sin(θ/2) = ±√((1 − cos θ)/2) cos(θ/2) = ±√((1 + cos θ)/2)
sum and difference:
sin(θ ± φ) = sin θ cos φ ± cos θ sin φ
cos(θ ± φ) = cos θ cos φ ∓ sin θ sin φStep-by-Step Example
θ = 30° and φ = 45°.
sin 30° = 1/2, cos 30° = √3/2
Pythagorean: (1/2)² + (√3/2)² = 1/4 + 3/4 = 1
double angle: sin 60° = 2 × 1/2 × √3/2 = √3/2 ≈ 0.866025
cos 60° = 3/4 − 1/4 = 1/2
half angle: sin 15° = √((1 − √3/2)/2) ≈ 0.258819
sum: sin 75° = (1/2)(√2/2) + (√3/2)(√2/2) = (√2 + √6)/4 ≈ 0.965926
difference: sin(−15°) ≈ −0.258819, cos(−15°) ≈ 0.965926The Pythagorean Identities
The basic identity sin²θ + cos²θ = 1 comes straight from the unit circle: a point at angle θ has coordinates (cos θ, sin θ), and its distance from the origin is 1, so by Pythagoras the squares add to 1. Dividing every term by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = csc²θ. These are the most used identities for simplifying expressions and for substitutions in integrals.
Double and Half Angle Formulas
The double-angle formulas follow from the sum formulas with φ = θ. The three forms of cos 2θ are interchangeable thanks to the Pythagorean identity, and choosing the right one is often the key step: 1 − 2sin²θ when an expression contains only sines, 2cos²θ − 1 when it contains only cosines.
Rearranging the cos 2θ forms gives the power-reducing formulas sin²θ = (1 − cos 2θ)/2 and cos²θ = (1 + cos 2θ)/2, which are used to integrate sin²x and cos²x. Replacing θ with θ/2 gives the half-angle formulas. The ± sign is chosen by the quadrant in which θ/2 lies; the calculator computes sin(θ/2) and cos(θ/2) directly, so the sign shown is the correct one.
Sum and Difference Formulas
The sum formulas let you find exact values for angles that are sums or differences of special angles, such as 75° = 30° + 45° or 15° = 45° − 30°. They also explain phase shifts in waves: a sin x + b cos x can be written as a single sine wave R sin(x + α), where R = √(a² + b²). A useful check is that the sum formula should match the direct calculation, and the calculator shows both.
Exact Values
For angles that are multiples of 15°, sine and cosine have exact forms built from square roots. The table below covers the first quadrant; other quadrants follow from symmetry with signs set by the quadrant.
angle 0° 15° 30° 45° 60° 75° 90°
sin 0 (√6 − √2)/4 1/2 √2/2 √3/2 (√6 + √2)/4 1
cos 1 (√6 + √2)/4 √3/2 √2/2 1/2 (√6 − √2)/4 0Using Identities to Prove Equations
To prove an identity, start from one side and transform it step by step into the other using known identities; do not treat the equation as true and work on both sides at once. A numerical check at one or two angles, which this calculator provides, is a quick way to catch a sign error or a misremembered formula before you write a proof, but it is not a proof itself.
Where the Identities Are Used
In calculus, identities turn hard integrals into easy ones. The integral of sin²x is found by rewriting it as (1 − cos 2x)/2, and substitutions such as x = sin θ rely on 1 − sin²θ = cos²θ to remove square roots. In physics, the sum formulas combine waves and describe interference and beats, while the double-angle formulas appear in projectile range, R = v² sin 2θ / g, which is greatest at 45°. In engineering, AC power calculations use the product of sines and cosines, simplified with these same identities.
Solving trigonometric equations also leans on identities. An equation such as cos 2x = sin x becomes 1 − 2sin²x = sin x, a quadratic in sin x, once the right form of cos 2x is chosen. That choice is often the whole trick: picking the form that leaves only one trigonometric function.
Reciprocal and Co-function Identities
The reciprocal identities define the other three functions: csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ. The co-function identities link complementary angles: sin(90° − θ) = cos θ and cos(90° − θ) = sin θ, which is why the sine and cosine rows in the exact-value table mirror each other.
Understanding Your Result
The headline gives sin θ and cos θ, exact when θ is a multiple of 15°.
The Pythagorean line shows sin²θ + cos²θ = 1 and the value of sec²θ.
The tangent, double angle and half angle lines give each identity's value.
The sum and difference lines apply the formulas with φ.
When Should You Use This Calculator?
Use it to check identity work and exact values in trigonometry homework.
Use it to find exact values for angles such as 15° or 75°.
Use it to confirm which form of cos 2θ or which half-angle sign applies.
Common Mistakes
Splitting functions like brackets. sin(θ + φ) is not sin θ + sin φ.
Writing sin 2θ as 2 sin θ. The double angle needs 2 sin θ cos θ.
Choosing the half-angle sign from θ. It depends on the quadrant of θ/2.
Mixing degrees and radians. Check the unit before entering angles.