Free Trig Identities Calculator

Trigonometric Identities Verification Tool

This free online Trig Identities Calculator (Trigonometric Identities Calculator) enables users to verify a broad range of trig identities quickly. By entering any angle (in degrees or radians), you can instantly see the relevant identities—including Pythagorean identities, double‑angle identities, half‑angle identities, sum‑and‑difference identities, and rotation/reflection formulas—applied to that specific value. Designed as both a verification tool and a learning aid, it helps students, educators, and professionals check and understand trig identity relationships.

What Are Trigonometric Identities?

Trigonometric identities are mathematical equalities that remain true for all values within the domain of the variable. Unlike equations that require solving for an unknown, identities state an inherent relationship that is always valid. Common categories include Pythagorean identities, reciprocal identities, and angle‑sum formulas. They are fundamental for simplifying trigonometric expressions, solving equations, and transforming functions.

The Pythagorean Identity: The Cornerstone

The most fundamental trig identity comes from the unit circle. For any angle $\theta$, the point on the unit circle has coordinates $(\cos\theta, \sin\theta)$, leading to:

$$\sin^{2}\theta + \cos^{2}\theta = 1$$

From this, you can express sine in terms of cosine or vice versa:

$$\sin\theta = \pm\sqrt{1 - \cos^{2}\theta}, \qquad \cos\theta = \pm\sqrt{1 - \sin^{2}\theta}$$

The sign depends on the quadrant in which $\theta$ lies. This Pythagorean identity forms the basis of many other trigonometric identities.

Rotation and Reflection Identities

Because sine, cosine, and tangent are periodic, shifting the angle by certain amounts yields predictable results. The period of sine and cosine is $2\pi$, while tangent has period $\pi$.

Shift (Rotation) Identities

The following table summarizes the effect of shifting an angle by a quarter‑period, half‑period, or full periods.

Shift	Sine	Cosine	Tangent
$\theta \pm \frac{\pi}{2}$	$\sin(\theta \pm \frac{\pi}{2}) = \pm\cos\theta$	$\cos(\theta \pm \frac{\pi}{2}) = \mp\sin\theta$	$\tan(\theta \pm \frac{\pi}{2}) = -\cot\theta$
$\theta \pm \pi$	$\sin(\theta \pm \pi) = -\sin\theta$	$\cos(\theta \pm \pi) = -\cos\theta$	$\tan(\theta \pm \pi) = \tan\theta$
$\theta \pm k\cdot2\pi$	$\sin(\theta \pm k\cdot2\pi) = \sin\theta$	$\cos(\theta \pm k\cdot2\pi) = \cos\theta$	$\tan(\theta \pm k\pi) = \tan\theta$

Note that tangent repeats every $\pi$ rather than $2\pi$.

Reflection Identities

Reflecting an angle about a line transforms the trigonometric functions in specific ways. The four common reflection axes are the horizontal axis ($\theta \rightarrow -\theta$), the first‑quadrant bisector ($\theta \rightarrow \frac{\pi}{2} - \theta$), the vertical axis ($\theta \rightarrow \pi - \theta$), and the second‑quadrant bisector ($\theta \rightarrow \frac{3\pi}{2} - \theta$).

Reflection	Sine	Cosine	Tangent
$\theta \rightarrow -\theta$	$\sin(-\theta) = -\sin\theta$	$\cos(-\theta) = \cos\theta$	$\tan(-\theta) = -\tan\theta$
$\theta \rightarrow \frac{\pi}{2} - \theta$	$\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta$	$\cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta$	$\tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta$
$\theta \rightarrow \pi - \theta$	$\sin(\pi - \theta) = \sin\theta$	$\cos(\pi - \theta) = -\cos\theta$	$\tan(\pi - \theta) = -\tan\theta$
$\theta \rightarrow \frac{3\pi}{2} - \theta$	$\sin\left(\frac{3\pi}{2} - \theta\right) = -\cos\theta$	$\cos\left(\frac{3\pi}{2} - \theta\right) = -\sin\theta$	$\tan\left(\frac{3\pi}{2} - \theta\right) = \cot\theta$

These identities, together with the shift identities, allow you to derive many other useful relationships.

Composite and Multi‑Angle Identities

Sum and Difference Identities

For two angles $\alpha$ and $\beta$, the trig functions of their sum or difference can be expressed as:

$$\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta$$ $$\cos(\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha \sin\beta$$ $$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta}$$

These formulas can be derived geometrically by placing two right triangles in a rectangle and comparing side lengths.

Double‑Angle Identities

Setting $\beta = \alpha$ turns the sum identities into double‑angle formulas:

$$\sin(2\alpha) = 2\sin\alpha \cos\alpha$$ $$\cos(2\alpha) = \cos^{2}\alpha - \sin^{2}\alpha = 2\cos^{2}\alpha - 1 = 1 - 2\sin^{2}\alpha$$ $$\tan(2\alpha) = \frac{2\tan\alpha}{1 - \tan^{2}\alpha}$$

Double‑angle identities are widely used in integration, simplifying expressions, and solving trigonometric equations.

Triple‑Angle Identities

Further extension yields triple‑angle formulas:

$$\sin(3\alpha) = 3\sin\alpha - 4\sin^{3}\alpha$$ $$\cos(3\alpha) = 4\cos^{3}\alpha - 3\cos\alpha$$ $$\tan(3\alpha) = \frac{3\tan\alpha - \tan^{3}\alpha}{1 - 3\tan^{2}\alpha}$$

Half‑Angle Identities

Half‑angle formulas involve square roots, so the sign must be chosen according to the quadrant of $\alpha/2$:

$$\sin\frac{\alpha}{2} = \pm\sqrt{\frac{1 - \cos\alpha}{2}}$$ $$\cos\frac{\alpha}{2} = \pm\sqrt{\frac{1 + \cos\alpha}{2}}$$ $$\tan\frac{\alpha}{2} = \pm\sqrt{\frac{1 - \cos\alpha}{1 + \cos\alpha}} = \frac{\sin\alpha}{1 + \cos\alpha} = \frac{1 - \cos\alpha}{\sin\alpha}$$

The forms using $\sin\alpha$ and $\cos\alpha$ avoid the sign ambiguity altogether.

How to Use This Trig Identities Verification Tool

Using the calculator is straightforward:

1. Choose the identity category (e.g., Pythagorean, double‑angle, half‑angle, rotation, reflection).
2. Enter the angle value in degrees or radians.
3. The tool displays the identity formula and its numeric evaluation for the entered angle.

This makes it easy to confirm that the identity holds for your specific input and to see the exact values of the trig functions involved. The calculator covers all major identity families, serving as a reliable reference for anyone working with trigonometric identities.