Complete Trigonometric Formulas Cheat Sheet: Identities, Double Angle, Half Angle & More

Trigonometric formulas form the backbone of mathematics, physics, and engineering. Whether you’re preparing for exams, tackling complex calculus problems, or writing code for signal processing, having a solid grasp of these formulas is essential. This article provides a systematic and complete collection of all key trigonometric formulas, along with memory tips and practical applications.

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1. Basic Definitions and Reciprocal Relations

1.1 Right Triangle Definitions

In a right triangle, let $\theta$ be one of the acute angles, with opposite side $a$, adjacent side $b$, and hypotenuse $c$:

$\sin\theta = \frac{a}{c}, \quad \cos\theta = \frac{b}{c}, \quad \tan\theta = \frac{a}{b}$

$\cot\theta = \frac{b}{a}, \quad \sec\theta = \frac{c}{b}, \quad \csc\theta = \frac{c}{a}$

1.2 Reciprocal Identities

The six trigonometric functions come in three reciprocal pairs:

• $\sin\theta \cdot \csc\theta = 1$
• $\cos\theta \cdot \sec\theta = 1$
• $\tan\theta \cdot \cot\theta = 1$

1.3 Quotient Identities

• $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
• $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$

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2. Pythagorean Identities

These are the most fundamental identities in trigonometry, derived directly from the Pythagorean theorem:

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

From this, two more identities are obtained:

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

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

Memory Tip: The first identity is the base. Dividing both sides by $\cos^2\theta$ gives the second; dividing by $\sin^2\theta$ gives the third.

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3. Co-function and Symmetry Identities

These identities allow us to convert trig functions of related angles:

Common Angle Transformations

Original Angle	$\sin$	$\cos$	$\tan$
$-\theta$	$-\sin\theta$	$\cos\theta$	$-\tan\theta$
$\pi - \theta$	$\sin\theta$	$-\cos\theta$	$-\tan\theta$
$\pi + \theta$	$-\sin\theta$	$-\cos\theta$	$\tan\theta$
$\frac{\pi}{2} - \theta$	$\cos\theta$	$\sin\theta$	$\cot\theta$
$\frac{\pi}{2} + \theta$	$\cos\theta$	$-\sin\theta$	$-\cot\theta$
$2\pi - \theta$	$-\sin\theta$	$\cos\theta$	$-\tan\theta$

Key Insight: For angles involving $\frac{\pi}{2}$ (90°), the function name changes (sin ↔ cos); for angles involving $\pi$ (180°) or $2\pi$ (360°), it stays the same. The sign is determined by the quadrant.

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4. Angle Sum and Difference Formulas

These formulas are the foundation from which double angle, half angle, and many other formulas are derived:

4.1 Sine Sum/Difference

$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$

4.2 Cosine Sum/Difference

$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$

Note: The sign on the right side of the cosine formula is opposite to the sign of the angle operation.

4.3 Tangent Sum/Difference

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

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5. Double Angle Formulas

The double angle formulas are a special case of the sum formulas, setting $\alpha = \beta$:

5.1 Double Angle

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

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

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

Memory Tip: The cosine double angle has three equivalent forms, each obtained by substituting $\sin^2\theta + \cos^2\theta = 1$.

5.2 Triple Angle Formulas

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

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

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6. Half Angle Formulas

Half angle formulas are derived from the double angle formulas by replacing $2\theta$ with $\theta$ and $\theta$ with $\frac{\theta}{2}$:

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

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

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

Note: The ± sign in the radical forms depends on the quadrant in which $\frac{\theta}{2}$ lies.

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7. Product-to-Sum and Sum-to-Product Formulas

7.1 Product-to-Sum

These convert a product of trig functions into a sum or difference:

$\sin\alpha\cos\beta = \frac{1}{2}[\sin(\alpha+\beta) + \sin(\alpha-\beta)]$

$\cos\alpha\sin\beta = \frac{1}{2}[\sin(\alpha+\beta) - \sin(\alpha-\beta)]$

$\cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha-\beta) + \cos(\alpha+\beta)]$

$\sin\alpha\sin\beta = -\frac{1}{2}[\cos(\alpha+\beta) - \cos(\alpha-\beta)]$

7.2 Sum-to-Product

These convert a sum or difference into a product:

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

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

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

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

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8. Weierstrass Substitution (Universal Formula)

The Weierstrass substitution (also called the tangent half-angle substitution) expresses all trig functions as rational functions of $t = \tan\frac{\theta}{2}$, which is extremely useful in integration:

$\sin\theta = \frac{2t}{1 + t^2}$

$\cos\theta = \frac{1 - t^2}{1 + t^2}$

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

where $t = \tan\dfrac{\theta}{2}$.

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9. Auxiliary Angle Formula (Harmonic Addition)

The auxiliary angle formula combines expressions of the form $a\sin\theta + b\cos\theta$ into a single sinusoidal function, which is very helpful for finding extreme values and solving equations:

$a\sin\theta + b\cos\theta = \sqrt{a^2 + b^2}\sin(\theta + \varphi)$

where $\tan\varphi = \dfrac{b}{a}$.

Equivalent form:

$a\sin\theta + b\cos\theta = \sqrt{a^2 + b^2}\cos(\theta - \psi)$

where $\tan\psi = \dfrac{a}{b}$.

Example: What is the maximum value of $y = 3\sin x + 4\cos x$? Using the auxiliary angle formula, $\sqrt{3^2+4^2} = 5$, so the maximum value is $5$.

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10. Power Reduction Formulas

Power reduction formulas lower the degree of trigonometric expressions, which is particularly useful for integration:

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

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

$\sin^2\theta \cdot \cos^2\theta = \frac{1 - \cos 4\theta}{8}$

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11. Common Special Angle Values

Angle	$0°$	$30°$	$45°$	$60°$	$90°$	$120°$	$180°$
Radians	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\frac{2\pi}{3}$	$\pi$
$\sin$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$	$\frac{\sqrt{3}}{2}$	$0$
$\cos$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$	$-\frac{1}{2}$	$-1$
$\tan$	$0$	$\frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$	Undefined	$-\sqrt{3}$	$0$

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12. Law of Sines and Law of Cosines

These two laws apply to any triangle (not just right triangles) and relate sides to angles.

12.1 Law of Sines

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$

where $R$ is the circumradius (radius of the circumscribed circle).

12.2 Law of Cosines

$c^2 = a^2 + b^2 - 2ab\cos C$

Equivalent form:

$\cos C = \frac{a^2 + b^2 - c^2}{2ab}$

Application: Given three sides, find the angles; or given two sides and the included angle, find the third side.

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Whether you’re studying for exams, developing software, or tackling engineering problems, this comprehensive trigonometric formula reference has you covered. Bookmark this page and never let trig formulas slow you down again!