Trigonometry Formulas
A Comprehensive Reference Guide
Basic Trigonometric Ratios (Right Triangle)
SOH-CAH-TOA Mnemonic:
\[
\begin{aligned}
\sin(\theta) &= \frac{\text{Opposite}}{\text{Hypotenuse}} \quad &
\cos(\theta) &= \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad &
\tan(\theta) &= \frac{\text{Opposite}}{\text{Adjacent}}
\end{aligned}
\]
Reciprocal Identities
\[
\begin{aligned}
\sin(\theta) &= \frac{1}{\csc(\theta)} \\[8pt]
\cos(\theta) &= \frac{1}{\sec(\theta)} \\[8pt]
\tan(\theta) &= \frac{1}{\cot(\theta)}
\end{aligned}
\]
\[
\begin{aligned}
\csc(\theta) &= \frac{1}{\sin(\theta)}\\[8pt]
\sec(\theta) &= \frac{1}{\cos(\theta)}\\[8pt]
\cot(\theta) &= \frac{1}{\tan(\theta)}
\end{aligned}
\]
Quotient Identities
\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \qquad
\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
\]
Pythagorean Identities
\[
\begin{aligned}
\sin^2(\theta) + \cos^2(\theta) &= 1 \\[10pt]
1 + \tan^2(\theta) &= \sec^2(\theta) \\[10pt]
1 + \cot^2(\theta) &= \csc^2(\theta)
\end{aligned}
\]
Trigonometric Table
| Angle |
\(0^\circ\) |
\(30^\circ\) |
\(45^\circ\) |
\(60^\circ\) |
\(90^\circ\) |
\(180^\circ\) |
\(270^\circ\) |
\(360^\circ\) |
| Radians |
\(0\) |
\(\frac{\pi}{6}\) |
\(\frac{\pi}{4}\) |
\(\frac{\pi}{3}\) |
\(\frac{\pi}{2}\) |
\(\pi\) |
\(\frac{3\pi}{2}\) |
\(2\pi\) |
| sin |
\(0\) |
\(\frac{1}{2}\) |
\(\frac{\sqrt{2}}{2}\) |
\(\frac{\sqrt{3}}{2}\) |
\(1\) |
\(0\) |
\(-1\) |
\(0\) |
| cos |
\(1\) |
\(\frac{\sqrt{3}}{2}\) |
\(\frac{\sqrt{2}}{2}\) |
\(\frac{1}{2}\) |
\(0\) |
\(-1\) |
\(0\) |
\(1\) |
| tan |
\(0\) |
\(\frac{1}{\sqrt{3}}\) |
\(1\) |
\(\sqrt{3}\) |
\(\infty\) |
\(0\) |
\(\infty\) |
\(0\) |
| cot |
\(\infty\) |
\(\sqrt{3}\) |
\(1\) |
\(\frac{1}{\sqrt{3}}\) |
\(0\) |
\(\infty\) |
\(0\) |
\(\infty\) |
| sec |
\(1\) |
\(\frac{2}{\sqrt{3}}\) |
\(\sqrt{2}\) |
\(2\) |
\(\infty\) |
\(-1\) |
\(\infty\) |
\(1\) |
| csc |
\(\infty\) |
\(2\) |
\(\sqrt{2}\) |
\(\frac{2}{\sqrt{3}}\) |
\(1\) |
\(\infty\) |
\(-1\) |
\(\infty\) |
Periodicity Identities
Note: Sine and cosine have period \(2\pi\), tangent and cotangent have period \(\pi\)
\[
\begin{aligned}
\sin\left(\frac{\pi}{2} - A\right) &= \cos(A) \qquad &
\cos\left(\frac{\pi}{2} - A\right) &= \sin(A)\\[10pt]
\sin\left(\frac{\pi}{2} + A\right) &= \cos(A) &
\cos\left(\frac{\pi}{2} + A\right) &= -\sin(A) \\[10pt]
\sin\left(\frac{3\pi}{2} - A\right) &= -\cos(A) &
\cos\left(\frac{3\pi}{2} - A\right) &= -\sin(A) \\[10pt]
\sin\left(\frac{3\pi}{2} + A\right) &= -\cos(A) &
\cos\left(\frac{3\pi}{2} + A\right) &= \sin(A) \\[10pt]
\sin(\pi - A) &= \sin(A) &
\cos(\pi - A) &= -\cos(A) \\[10pt]
\sin(\pi + A) &= -\sin(A) &
\cos(\pi + A) &= -\cos(A) \\[10pt]
\sin(2\pi - A) &= -\sin(A) &
\cos(2\pi - A) &= \cos(A) \\[10pt]
\sin(2\pi + A) &= \sin(A) &
\cos(2\pi + A) &= \cos(A)
\end{aligned}
\]
Co-function Identities
\[
\begin{aligned}
\sin\left(\frac{\pi}{2} - \theta\right) &= \cos(\theta) \qquad &
\cos\left(\frac{\pi}{2} - \theta\right) &= \sin(\theta) \\[10pt]
\tan\left(\frac{\pi}{2} - \theta\right) &= \cot(\theta) &
\cot\left(\frac{\pi}{2} - \theta\right) &= \tan(\theta) \\[10pt]
\sec\left(\frac{\pi}{2} - \theta\right) &= \csc(\theta) &
\csc\left(\frac{\pi}{2} - \theta\right) &= \sec(\theta)
\end{aligned}
\]
Even-Odd Identities
\[
\begin{aligned}
\sin(-\theta) &= -\sin(\theta) \quad \text{(Odd)} \\[10pt]
\cos(-\theta) &= \cos(\theta) \quad \text{(Even)} \\[10pt]
\tan(-\theta) &= -\tan(\theta) \quad \text{(Odd)} \\[10pt]
\csc(-\theta) &= -\csc(\theta) \quad \text{(Odd)} \\[10pt]
\sec(-\theta) &= \sec(\theta) \quad \text{(Even)} \\[10pt]
\cot(-\theta) &= -\cot(\theta) \quad \text{(Odd)}
\end{aligned}
\]
Sum and Difference Identities
\[
\begin{aligned}
\sin(x \pm y) &= \sin(x)\cos(y) \pm \cos(x)\sin(y) \\[10pt]
\cos(x \pm y) &= \cos(x)\cos(y) \mp \sin(x)\sin(y) \\[10pt]
\tan(x \pm y) &= \frac{\tan(x) \pm \tan(y)}{1 \mp \tan(x)\tan(y)}
\end{aligned}
\]
Double Angle Identities
\[
\begin{aligned}
\sin(2x) &= 2\sin(x)\cos(x) = \frac{2\tan(x)}{1+\tan^2(x)} \\[10pt]
\cos(2x) &= \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x) = \frac{1-\tan^2(x)}{1+\tan^2(x)} \\[10pt]
\tan(2x) &= \frac{2\tan(x)}{1-\tan^2(x)} \\[10pt]
\sec(2x) &= \frac{\sec^2(x)}{2-\sec^2(x)} = \frac{1+\tan^2(x)}{1-\tan^2(x)} \\[10pt]
\csc(2x) &= \frac{\sec(x)\csc(x)}{2}
\end{aligned}
\]
Triple Angle Identities
\[
\begin{aligned}
\sin(3x) &= 3\sin(x) - 4\sin^3(x) \\[10pt]
\cos(3x) &= 4\cos^3(x) - 3\cos(x) \\[10pt]
\tan(3x) &= \frac{3\tan(x) - \tan^3(x)}{1 - 3\tan^2(x)}
\end{aligned}
\]
Half Angle Identities
\[
\begin{aligned}
\sin\left(\frac{x}{2}\right) &= \pm \sqrt{\frac{1 - \cos(x)}{2}} \\[10pt]
\cos\left(\frac{x}{2}\right) &= \pm \sqrt{\frac{1 + \cos(x)}{2}} \\[10pt]
\tan\left(\frac{x}{2}\right) &= \pm\sqrt{\frac{1 - \cos(x)}{1 + \cos(x)}} = \frac{\sin(x)}{1 + \cos(x)} = \frac{1 - \cos(x)}{\sin(x)}
\end{aligned}
\]
Sign Selection: Choose ± based on the quadrant of \(\frac{x}{2}\)
Power Reducing Identities
\[
\begin{aligned}
\sin^2(x) &= \frac{1 - \cos(2x)}{2} \\[10pt]
\cos^2(x) &= \frac{1 + \cos(2x)}{2} \\[10pt]
\tan^2(x) &= \frac{1 - \cos(2x)}{1 + \cos(2x)}
\end{aligned}
\]
Product Identities
\[
\begin{aligned}
\sin(x)\cos(y) &= \frac{1}{2}[\sin(x+y) + \sin(x-y)] \\[10pt]
\cos(x)\cos(y) &= \frac{1}{2}[\cos(x+y) + \cos(x-y)] \\[10pt]
\sin(x)\sin(y) &= \frac{1}{2}[\cos(x-y) - \cos(x+y)]
\end{aligned}
\]
Sum to Product Identities
\[
\begin{aligned}
\sin(x) + \sin(y) &= 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) \\[10pt]
\sin(x) - \sin(y) &= 2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right) \\[10pt]
\cos(x) + \cos(y) &= 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) \\[10pt]
\cos(x) - \cos(y) &= -2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)
\end{aligned}
\]
Inverse Trigonometry Formulas
Domain and Range:
- \(\sin^{-1}(x)\): Domain \([-1, 1]\), Range \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
- \(\cos^{-1}(x)\): Domain \([-1, 1]\), Range \([0, \pi]\)
- \(\tan^{-1}(x)\): Domain \((-\infty, \infty)\), Range \((-\frac{\pi}{2}, \frac{\pi}{2})\)
\[
\begin{aligned}
\sin^{-1}(-x) &= -\sin^{-1}(x) \\[10pt]
\cos^{-1}(-x) &= \pi - \cos^{-1}(x) \\[10pt]
\tan^{-1}(-x) &= -\tan^{-1}(x) \\[10pt]
\csc^{-1}(-x) &= -\csc^{-1}(x) \\[10pt]
\sec^{-1}(-x) &= \pi - \sec^{-1}(x) \\[10pt]
\cot^{-1}(-x) &= \pi - \cot^{-1}(x)
\end{aligned}
\]
Inverse Trigonometry Complementary Relations
\[
\begin{aligned}
\sin^{-1}(x) + \cos^{-1}(x) &= \frac{\pi}{2} \\[10pt]
\tan^{-1}(x) + \cot^{-1}(x) &= \frac{\pi}{2} \\[10pt]
\sec^{-1}(x) + \csc^{-1}(x) &= \frac{\pi}{2}
\end{aligned}
\]
Inverse Trigonometry Sum Formulas
\[
\begin{aligned}
\sin^{-1}(x) + \sin^{-1}(y) &= \sin^{-1}\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right) \\[10pt]
\tan^{-1}(x) + \tan^{-1}(y) &= \tan^{-1}\left(\frac{x+y}{1-xy}\right) \quad (xy < 1) \\[10pt]
2\tan^{-1}(x) &= \tan^{-1}\left(\frac{2x}{1-x^2}\right) \quad (|x| < 1)
\end{aligned}
\]
Law of Sines and Cosines
Law of Sines:
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = 2R
\]
where \(R\) is the circumradius of the triangle
Law of Cosines:
\[
\begin{aligned}
c^2 &= a^2 + b^2 - 2ab\cos(C) \\[10pt]
\cos(C) &= \frac{a^2 + b^2 - c^2}{2ab}
\end{aligned}
\]
Area of Triangle
\[
\begin{aligned}
\text{Area} &= \frac{1}{2}ab\sin(C) \\[10pt]
\text{Area} &= \sqrt{s(s-a)(s-b)(s-c)} \quad \text{(Heron's Formula)}
\end{aligned}
\]
where \(s = \frac{a+b+c}{2}\) is the semi-perimeter
Important Notes
- Angle Conversion: \(1 \text{ radian} = \frac{180°}{\pi} \approx 57.2958°\) and \(1° = \frac{\pi}{180} \text{ radians}\)
- Quadrant Rules: All trig functions positive in Q1; Sin positive in Q2; Tan positive in Q3; Cos positive in Q4 (ASTC rule)
- Reference Angles: Always measured from the x-axis to the terminal side
- Unit Circle: \(x = \cos(\theta)\), \(y = \sin(\theta)\) for any point on the unit circle
Master these formulas and conquer trigonometry!