Trigonometry Formulas - Full list

What Are the Basic Trigonometry Formulas?

Primary Ratios (SOHCAHTOA)

For a right triangle with angle $\theta$:

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

Reciprocal Identities

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

Quotient Identities

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

What Are the Pythagorean Identities?

Three identities derived directly from the Pythagorean theorem applied to the unit circle:

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

$$1 + \tan^2\theta = \sec^2\theta$$

$$1 + \cot^2\theta = \csc^2\theta$$

The first is the most-used identity in all of trigonometry. The other two come from dividing the first by $\cos^2\theta$ and $\sin^2\theta$ respectively.

Worked example. If $\sin\theta = 3/5$ and $\theta$ is in Q1, find $\cos\theta$.

$\cos^2\theta = 1 - \sin^2\theta = 1 - 9/25 = 16/25$, so $\cos\theta = 4/5$ (positive in Q1).

What Are the Sum and Difference Formulas?

Sine

$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$ $$\sin(A - B) = \sin A \cos B - \cos A \sin B$$

Cosine

$$\cos(A + B) = \cos A \cos B - \sin A \sin B$$ $$\cos(A - B) = \cos A \cos B + \sin A \sin B$$

Tangent

$$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$

$$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$

Worked example. Compute $\sin(75°) = \sin(45° + 30°)$.

$$\sin 75° = \sin 45° \cos 30° + \cos 45° \sin 30° = \tfrac{1}{\sqrt{2}} \cdot \tfrac{\sqrt{3}}{2} + \tfrac{1}{\sqrt{2}} \cdot \tfrac{1}{2} = \tfrac{\sqrt{3} + 1}{2\sqrt{2}}$$

What Are the Double-Angle Formulas?

$$\sin 2\theta = 2 \sin\theta \cos\theta$$

$$\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$$

$$\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$$

The cosine double-angle has three equivalent forms — each useful in different contexts. The $2\cos^2\theta - 1$ form is the key to deriving half-angle formulas.

What Are the Half-Angle Formulas?

$$\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}$$

$$\cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}$$

$$\tan\frac{\theta}{2} = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}$$

The $\pm$ sign depends on which quadrant $\theta/2$ is in.

What Are the Triple-Angle Formulas?

The triple-angle formulas express trig of $3\theta$ in terms of trig of $\theta$. Derive them by writing $3\theta = 2\theta + \theta$ and applying the sum + double-angle formulas.

$$\sin 3\theta = 3\sin\theta - 4\sin^3\theta$$

$$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$$

$$\tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$$

Worked example. Compute $\sin 90°$ via the triple-angle formula with $\theta = 30°$.

$\sin 90° = 3 \sin 30° - 4 \sin^3 30° = 3 \cdot \tfrac{1}{2} - 4 \cdot \tfrac{1}{8} = \tfrac{3}{2} - \tfrac{1}{2} = 1$ ✓.

What Are the Even-Odd (Negative Angle) Identities?

Sine, tangent, cosecant, and cotangent are odd functions (negating the input negates the output). Cosine and secant are even functions (negating the input leaves the output unchanged).

$$\sin(-\theta) = -\sin\theta \quad \tan(-\theta) = -\tan\theta \quad \csc(-\theta) = -\csc\theta \quad \cot(-\theta) = -\cot\theta$$

$$\cos(-\theta) = \cos\theta \quad \sec(-\theta) = \sec\theta$$

Geometric reason: reflecting a point on the unit circle across the $x$-axis (which is what $\theta \to -\theta$ does) flips the $y$-coordinate (so $\sin$ flips) but leaves the $x$-coordinate alone (so $\cos$ stays).

Use. Even/odd identities simplify integrals and Fourier-series derivations — odd-function integrals over symmetric intervals vanish.

What Are the Cofunction Identities?

The cofunction identities express trig of complementary angles in terms of the co-function. Two angles are complementary when they sum to $90°$ (or $\pi/2$ radians).

$$\sin(90° - \theta) = \cos\theta \quad \cos(90° - \theta) = \sin\theta$$

$$\tan(90° - \theta) = \cot\theta \quad \cot(90° - \theta) = \tan\theta$$

$$\sec(90° - \theta) = \csc\theta \quad \csc(90° - \theta) = \sec\theta$$

In radians:

$$\sin\left(\tfrac{\pi}{2} - \theta\right) = \cos\theta, \quad \cos\left(\tfrac{\pi}{2} - \theta\right) = \sin\theta, \quad \ldots$$

Geometric reason. In a right triangle, the two non-right angles are complementary — and the opposite-vs-adjacent pair swaps roles between them. Hence "co-" in cosine, cotangent, cosecant — they're the trig ratios of the complementary angle.

What Are the Sum-to-Product and Product-to-Sum Formulas?

Sum-to-Product

$$\sin A + \sin B = 2 \sin\frac{A+B}{2} \cos\frac{A-B}{2}$$

$$\sin A - \sin B = 2 \cos\frac{A+B}{2} \sin\frac{A-B}{2}$$

$$\cos A + \cos B = 2 \cos\frac{A+B}{2} \cos\frac{A-B}{2}$$

$$\cos A - \cos B = -2 \sin\frac{A+B}{2} \sin\frac{A-B}{2}$$

Product-to-Sum

$$2 \sin A \cos B = \sin(A + B) + \sin(A - B)$$

$$2 \cos A \cos B = \cos(A + B) + \cos(A - B)$$

$$2 \sin A \sin B = \cos(A - B) - \cos(A + B)$$

What Are the Inverse Trigonometric Formulas?

The inverse functions answer the question "what angle has this trig ratio?"

$$\arcsin(\sin\theta) = \theta \quad (\theta \in [-\pi/2, \pi/2])$$

$$\arccos(\cos\theta) = \theta \quad (\theta \in [0, \pi])$$

$$\arctan(\tan\theta) = \theta \quad (\theta \in (-\pi/2, \pi/2))$$

Common derivative formulas:

$$\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arctan x = \frac{1}{1+x^2}$$

What Is Euler's Identity? (The Most Beautiful Equation)

$$e^{i\pi} + 1 = 0$$

This single equation connects five of the most important constants in mathematics: $e$, $i$, $\pi$, $1$, and $0$. It comes from the more general Euler's formula:

$$e^{i\theta} = \cos\theta + i \sin\theta$$

When $\theta = \pi$, this gives $e^{i\pi} = -1$, or rearranged: $e^{i\pi} + 1 = 0$.

The physicist Richard Feynman called it "the most remarkable formula in mathematics."

Why Are Trigonometry Formulas Important? (The Real-World GROUND)

"Trigonometry is the gateway to mathematical physics." — adapted from numerous textbooks.

Trig formulas don't just live in textbook problems — they run measurable swaths of modern technology:

• GPS satellites. Position triangulation uses sum/difference formulas continuously.

• Audio compression (MP3, AAC). Built on the discrete cosine transform — applied to every 20-millisecond chunk of audio.

• Image compression (JPEG, MPEG). The 2D discrete cosine transform partitions image energy into low and high frequency components.

• Signal processing. The fast Fourier transform (FFT) — used in everything from cellular networks to MRI scans to radio astronomy — is built on sum-to-product identities.

• Computer graphics. 3D rotations chain together using sum/difference formulas. Each chained rotation uses these identities to avoid recomputing trig from scratch.

• Quantum mechanics. Wave functions are expressed as $e^{i\theta}$ — Euler's formula in physics.

The modern formula list comes mostly from Leonhard Euler's 1748 Introductio in Analysin Infinitorum — which unified centuries of trigonometric identities into the symbolic system we still use.

A Worked Example

Prove that $\sin(2\theta) = 2 \sin\theta \cos\theta$.

The intuitive (wrong) approach. A student in a hurry assumes $\sin(2\theta) = 2 \sin\theta$.

$$\sin(2\theta) \stackrel{?}{=} 2\sin\theta$$

Why it fails. Sine is not a linear function — multiplying the input by 2 doesn't multiply the output by 2. Check with $\theta = 30°$: $\sin 60° = \tfrac{\sqrt{3}}{2}$ but $2 \sin 30° = 2 \cdot \tfrac{1}{2} = 1$. Different values.

The correct method. Start from the sum formula $\sin(A + B) = \sin A \cos B + \cos A \sin B$ with $A = B = \theta$:

$$\sin(\theta + \theta) = \sin\theta \cos\theta + \cos\theta \sin\theta = 2 \sin\theta \cos\theta$$

Check. With $\theta = 30°$: $2 \sin 30° \cos 30° = 2 \cdot \tfrac{1}{2} \cdot \tfrac{\sqrt{3}}{2} = \tfrac{\sqrt{3}}{2} = \sin 60°$ ✓.

At Bhanzu, our trainers teach trig identities by deriving them from the sum/difference formulas — once the student sees how $\sin(2\theta) = \sin(\theta + \theta)$, the formula stops being a memorisation burden and becomes a one-line derivation.

What Are the Most Common Mistakes With Trigonometry Formulas?

Mistake 1: Treating $\sin(A + B)$ as $\sin A + \sin B$

Where it slips in: Confusing trig with linear functions.

Don't do this: $\sin(A + B) = \sin A + \sin B$.

The correct way: $\sin(A + B) = \sin A \cos B + \cos A \sin B$. The sum doesn't distribute. Same warning for $\cos(A + B)$.

Mistake 2: Forgetting the $\pm$ in half-angle formulas

Where it slips in: Computing $\sin(\theta/2)$ as a positive square root automatically.

Don't do this: Always taking the positive root.

The correct way: Determine which quadrant $\theta/2$ is in, then choose the sign accordingly. The half-angle formula has $\pm$ built in.

Mistake 3: Using the wrong form of the cosine double angle

Where it slips in: Three forms of $\cos 2\theta$ — picking the wrong one for the context.

Don't do this: Forcing $\cos 2\theta = \cos^2\theta - \sin^2\theta$ when only $\sin\theta$ is known.

The correct way: If only $\sin\theta$ is known, use $\cos 2\theta = 1 - 2\sin^2\theta$. If only $\cos\theta$ is known, use $\cos 2\theta = 2\cos^2\theta - 1$. Pick the form whose right-hand side uses what you have.

The Mathematicians Who Shaped Trigonometry Formulas

Hipparchus of Nicaea (c. 190–c. 120 BCE, Greece) — Built the first systematic trig table around 130 BCE.

Leonhard Euler (1707–1783, Switzerland) — Standardised modern notation and unified trigonometry with complex numbers via Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$, leading to Euler's identity.

Jean-Baptiste Joseph Fourier (1768–1830, France) — Showed that any periodic function can be written as a sum of sines and cosines (Fourier series), which became the foundation of signal processing.

A Practical Next Step

Try these three before moving on to trigonometric equations.

1. Use the double-angle formula to compute $\cos 60°$ from $\cos 30° = \tfrac{\sqrt{3}}{2}$.

2. Use $\sin(A+B)$ to compute $\sin 105°$.

3. If $\cos\theta = \tfrac{4}{5}$ and $\theta$ is in Q1, find $\sin 2\theta$.