Trigonometric Formulas Cheat Sheet

This trigonometric formulas cheat sheet gives the identities students use most: the basic definitions, reciprocal and Pythagorean identities, and the formulas for adding, subtracting, doubling, or halving angles.

The useful way to think about trig formulas is that they are connected, not random. The right-triangle definitions explain ratios, and the unit circle explains why the same relationships keep showing up for any angle.

Core Trig Formulas You Actually Use

For a right-triangle angle $\theta$:

$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$

If $\cos \theta \ne 0$, tangent is the quotient of sine and cosine:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

The reciprocal functions are:

$$\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$$

These basic definitions work directly in right triangles. For wider trig problems, the same relationships are usually interpreted on the unit circle instead.

Pythagorean Identities

These are the identities that show up constantly when you simplify expressions:

$$\sin^2 \theta + \cos^2 \theta = 1$$ $$1 + \tan^2 \theta = \sec^2 \theta \quad \text{when } \cos \theta \ne 0$$ $$1 + \cot^2 \theta = \csc^2 \theta \quad \text{when } \sin \theta \ne 0$$

The first identity is the main one. The other two come from dividing by $\cos^2 \theta$ or $\sin^2 \theta$, so the denominator condition matters.

Angle Sum And Difference Formulas

Use these when you rewrite an angle as two easier angles added or subtracted:

$$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$ $$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$ $$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$$ $$\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$$ $$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$$ $$\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$$

For tangent, the denominator must not be $0$, and each tangent value used must be defined.

Double-Angle And Half-Angle Formulas

Double-angle formulas are useful when the same angle appears twice:

$$\sin(2\theta) = 2 \sin \theta \cos \theta$$ $$\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$$

Equivalent cosine forms are:

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

Half-angle formulas are:

$$\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$$ $$\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$$ $$\tan\left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta}$$

For the square-root half-angle formulas, the sign depends on the quadrant of $\theta/2$.

Worked Example: Find $\sin 75^\circ$

Write $75^\circ$ as $45^\circ + 30^\circ$ and use the angle-sum formula:

$$\sin(75^\circ) = \sin(45^\circ + 30^\circ)$$ $$= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$$

Now substitute the known special-angle values:

$$= \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)$$ $$= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}$$

This is the main pattern behind exact-value trig problems: break a harder angle into familiar angles, then apply one formula carefully.

Common Mistakes With Trig Formulas

1. Mixing up identities and equations. An identity is true for every angle where both sides are defined. An equation like $\sin \theta = \frac{1}{2}$ is true only for specific angles.
2. Forgetting domain conditions. $\tan \theta$, $\sec \theta$, and formulas built from them are not defined when $\cos \theta = 0$.
3. Dropping the sign in half-angle formulas. The $\pm$ is resolved by the quadrant of $\theta/2$, not by the sign of $\theta$ alone.
4. Copying the wrong sign in angle formulas. The cosine formulas are especially easy to swap by mistake.
5. Using right-triangle definitions outside their setting without switching to the unit-circle view. For angles beyond acute triangle angles, the unit circle is the safer interpretation.

When Trigonometric Formulas Are Used

These formulas are used to simplify trig expressions, solve trig equations, find exact values, and support calculus work such as derivatives, integrals, and substitutions. They also appear in physics and engineering whenever a problem involves rotation, waves, oscillation, or periodic motion.

In practice, the workflow is usually: identify the pattern, check the condition, choose the matching identity, and then simplify slowly enough that the signs stay correct.

Try A Similar Problem

Try finding $\cos 15^\circ$ by writing $15^\circ$ as $45^\circ - 30^\circ$. If your result is exact and positive, you used the sheet correctly.