Trigonometric Identity Calculator
Evaluate common trigonometric identities — Pythagorean, double-angle, and half-angle — at any angle. Ideal for verifying textbook results or checking homework answers instantly.
About this calculator
Trigonometric identities are equations that hold true for all valid angle values. The Pythagorean identity states sin²(θ) + cos²(θ) = 1. The double-angle identities give sin(2θ) = 2·sin(θ)·cos(θ) and cos(2θ) = cos²(θ) − sin²(θ). Half-angle identities express sin(θ/2) = √((1 − cos θ)/2) and cos(θ/2) = √((1 + cos θ)/2). The secant identity 1 + tan²(θ) = sec²(θ) is another Pythagorean form. This calculator evaluates each of these identities numerically for a given angle, allowing you to confirm that both sides of an identity are equal or to find the resulting value when one expression form is preferred over another.
How to use
Let's evaluate the double-angle sine identity at θ = 30°. The formula is sin(2θ) = 2 × sin(θ) × cos(θ). Enter angle = 30, select degrees, choose 'Double Angle – Sine'. The calculator computes: 2 × sin(30°) × cos(30°) = 2 × 0.5 × 0.8660 = 0.8660. This matches sin(60°) = 0.8660, confirming the identity. Set decimal places to 4 for a precise result. Try the Pythagorean identity at any angle to always get 1.
Frequently asked questions
Why does the Pythagorean identity always equal 1 regardless of the angle?
The Pythagorean identity sin²(θ) + cos²(θ) = 1 is derived directly from the unit circle, where any point (cos θ, sin θ) lies at a distance of exactly 1 from the origin. Because the radius is always 1, the sum of squares of the coordinates is always 1, no matter what angle θ is. This identity is fundamental to almost every branch of trigonometry and is frequently used to simplify expressions in calculus and physics.
How are double-angle formulas used in real-world applications?
Double-angle formulas appear in signal processing, optics, and mechanical engineering. For example, in AC circuit analysis, the identity sin(2θ) = 2·sin(θ)·cos(θ) helps simplify power calculations. In optics, double-angle relationships describe polarization and interference. In robotics and computer graphics, they help compose rotation matrices efficiently. Understanding these identities allows engineers to reduce computational complexity in simulations and hardware implementations.
What is the difference between a half-angle identity and a double-angle identity in trigonometry?
Double-angle identities express a trig function of 2θ in terms of sin(θ) and cos(θ), effectively doubling the angle. Half-angle identities do the reverse: they express sin(θ/2) or cos(θ/2) using cos(θ), effectively halving the angle. Half-angle formulas are especially useful when integrating trigonometric functions or finding exact values for angles like 15° or 22.5° that do not appear directly in standard tables. Both sets of identities are derived from the cosine addition formula.