Trigonometric Formulas Cheat Sheet

Need the value of $\sin 75^\circ$ without a calculator, or want to collapse a messy expression with $\sin(2\theta)$ in it? That is what trig formulas are for. This sheet collects the identities students use most: the basic definitions, the reciprocal and Pythagorean identities, and the formulas for adding, subtracting, doubling, or halving angles.

The Formulas And What Each Symbol Means

For a right-triangle angle $\theta$, the three core ratios are:

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

Here "opposite" and "adjacent" are named relative to $\theta$, and "hypotenuse" is the side across from the right angle. If $\cos \theta \ne 0$, tangent is the quotient of sine and cosine:

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

The reciprocal functions invert each ratio:

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

The Pythagorean identities, which appear constantly when you simplify, are:

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

The angle sum and difference formulas let you rewrite one angle as two easier ones:

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

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

The double-angle formulas handle a repeated angle:

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

with equivalent cosine forms

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

and the 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 $\theta/2$.

Why These Formulas Hold

The formulas are connected, not random, and seeing where they come from makes them easier to trust and recall.

The right-triangle definitions explain the ratios directly, and the unit circle explains why the same relationships keep working for any angle: the point at angle $\theta$ is $(\cos\theta, \sin\theta)$, which sits on $x^2+y^2=1$, so $\sin^2\theta+\cos^2\theta=1$ immediately. Dividing that identity by $\cos^2\theta$ (when $\cos\theta\ne0$) gives $1+\tan^2\theta=\sec^2\theta$, and dividing by $\sin^2\theta$ gives $1+\cot^2\theta=\csc^2\theta$. The double-angle formulas are just the sum formulas with $\alpha=\beta=\theta$: setting both angles equal in $\sin(\alpha+\beta)$ turns it into $\sin(2\theta)=2\sin\theta\cos\theta$. So almost everything on this sheet traces back to one circle relationship and one substitution.

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:

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

The pattern behind exact-value problems is this: break a harder angle into familiar angles, then apply one formula carefully.

Now You Try

Find $\cos 15^\circ$ by writing $15^\circ$ as $45^\circ - 30^\circ$ and applying the cosine difference formula. Your answer should come out exact and positive; if it does, you read the sheet correctly. For a second round, redo $\sin 75^\circ$ as $30^\circ + 45^\circ$ in the opposite order and confirm you get the same value.

Calculation Traps To Watch

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 steady: identify the pattern, check the condition, choose the matching identity, then simplify slowly enough that the signs stay correct.