Trigonometric Functions Formula Sheet
Quick reference for standard trigonometric functions and formula
Basic Trigonometric Functions
\( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
\( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
\( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)
\( \csc \theta = \frac{1}{\sin \theta} \)
\( \sec \theta = \frac{1}{\cos \theta} \)
\( \cot \theta = \frac{1}{\tan \theta} \)
Pythagorean Identities
\( \sin^2 \theta + \cos^2 \theta = 1 \)
\( 1 + \tan^2 \theta = \sec^2 \theta \)
\( 1 + \cot^2 \theta = \csc^2 \theta \)
Sum and Difference Formulas
\( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
\( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)
\( \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)
Double-Angle & Half-Angle Formulas
\( \sin(2\theta) = 2 \sin \theta \cos \theta \)
\( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta \)
\( \tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta} \)
\( \sin^2(\frac{\theta}{2}) = \frac{1 - \cos \theta}{2} \)
\( \cos^2(\frac{\theta}{2}) = \frac{1 + \cos \theta}{2} \)
\( \tan^2(\frac{\theta}{2}) = \frac{1 - \cos \theta}{1 + \cos \theta} \)
Product-to-Sum & Sum-to-Product Formulas
\( \sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)] \)
\( \cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)] \)
\( \sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)] \)
\( \cos A \sin B = \frac{1}{2}[\sin(A + B) - \sin(A - B)] \)
Common Sine, Cosine, and Tangent Values
| Degree (°) | Radian (rad) | \( \sin \theta \) | \( \cos \theta \) | \( \tan \theta \) |
|---|---|---|---|---|
| 0° | \( 0 \) | \( 0 \) | \( 1 \) | \( 0 \) |
| 30° | \( \frac{\pi}{6} \) | \( \frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{\sqrt{3}} \) |
| 45° | \( \frac{\pi}{4} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( 1 \) |
| 60° | \( \frac{\pi}{3} \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) | \( \sqrt{3} \) |
| 90° | \( \frac{\pi}{2} \) | \( 1 \) | \( 0 \) | — |
| 120° | \( \frac{2\pi}{3} \) | \( \frac{\sqrt{3}}{2} \) | \( -\frac{1}{2} \) | \( -\sqrt{3} \) |
| 135° | \( \frac{3\pi}{4} \) | \( \frac{\sqrt{2}}{2} \) | \( -\frac{\sqrt{2}}{2} \) | \( -1 \) |
| 150° | \( \frac{5\pi}{6} \) | \( \frac{1}{2} \) | \( -\frac{\sqrt{3}}{2} \) | \( -\frac{1}{\sqrt{3}} \) |
| 180° | \( \pi \) | \( 0 \) | \( -1 \) | \( 0 \) |
About the Trigonometric Functions Formula Sheet
This page provides a comprehensive reference for trigonometric functions and identities, including sine, cosine, tangent, cosecant, secant, and cotangent. It covers essential formulas such as Pythagorean identities, sum and difference formulas, double-angle and half-angle formulas, and product-to-sum conversions.
The formula sheet is an essential resource for students, educators, engineers, and scientists working with mathematics, physics, or engineering problems. It provides a quick reference to compute trigonometric values, solve equations, analyze waveforms, and perform geometric calculations efficiently.
Key Benefits and Applications
- Quick access to accurate trigonometric formulas and identities for problem-solving in mathematics and engineering.
- Supports algebraic manipulation, geometry, and physics calculations involving angles and rotational motion.
- Includes common values in degrees and radians for standard angles, making reference and computation faster.
This Trigonometric Functions Formula Sheet is designed to save time and reduce errors for anyone who needs reliable, high-quality trigonometric references for mathematics, engineering, physics, and technical problem-solving.