Trigonometry Formulas
Reciprocal Identities
\[
\begin{aligned}
\sin(\theta) &= \frac{1}{\csc(\theta)} \\
\cos(\theta) &= \frac{1}{\sec(\theta)} \\
\tan(\theta) &= \frac{1}{\cot(\theta)}
\end{aligned}
\]
\[
\begin{aligned}
\csc(\theta) &= \frac{1}{\sin(\theta)}\\
\sec(\theta) &= \frac{1}{\cos(\theta)}\\
\cot(\theta) &= \frac{1}{\tan(\theta)}
\end{aligned}
\]
Trigonometric Table
| Angle |
\(0^\circ\) |
\(30^\circ\) |
\(45^\circ\) |
\(60^\circ\) |
\(90^\circ\) |
\(180^\circ\) |
\(270^\circ\) |
\(360^\circ\) |
| Angle (in 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 |
\(0\) |
\(\frac{\sqrt{3}}{3}\) |
\(1\) |
\(\sqrt{3}\) |
\(\infty\) |
\(0\) |
\(\infty\) |
\(0\) |
| cot |
\(\infty\) |
\(\sqrt{3}\) |
\(1\) |
\(\frac{\sqrt{3}}{3}\) |
\(0\) |
\(\infty\) |
\(0\) |
\(\infty\) |
| sec |
\(1\) |
\(2\) |
\(\sqrt{2}\) |
\(\frac{2}{\sqrt{3}}\) |
\(\infty\) |
\(-1\) |
\(\infty\) |
\(1\) |
| cosec |
\(\infty\) |
\(\frac{2}{\sqrt{3}}\) |
\(\sqrt{2}\) |
\(2\) |
\(1\) |
\(\infty\) |
\(-1\) |
\(\infty\) |
Periodicity Identities (in Radians)
\[\begin{aligned}
{2}
\sin\left(\frac{\pi}{2} - A\right) &= \cos(A) \qquad &
\cos\left(\frac{\pi}{2} - A\right) &= \sin(A)\\[10pt]
\sin\left(\frac{\pi}{2} + A\right) &= \cos(A) &
\cos\left(\frac{\pi}{2} + A\right) &= -\sin(A) \\[10pt]
\sin\left(\frac{3\pi}{2} - A\right) &= -\cos(A) &
\cos\left(\frac{3\pi}{2} - A\right) &= -\sin(A) \\[10pt]
\sin\left(\frac{3\pi}{2} + A\right) &= -\cos(A) &
\cos\left(\frac{3\pi}{2} + A\right) &= \sin(A) \\[10pt]
\sin(\pi - A) &= \sin(A) & \cos(\pi - A) &= -\cos(A)
\\[10pt]
\sin(\pi + A) &= -\sin(A) & \cos(\pi + A) &= -\cos(A)
\\[10pt]
\sin(2\pi - A) &= -\sin(A) & \cos(2\pi - A) &= \cos(A)
\\[10pt]
\sin(2\pi + A) &= \sin(A) & \cos(2\pi + A) &= \cos(A)
\end{aligned}\]
Sum and Difference Identities (in Radians)
\[\sin(x+y) =
\sin(x)\cos(y)+\cos(x)\sin(y)\]
\[\cos(x+y) =
\cos(x)\cos(y)-\sin(x)\sin(y)\]
\[\tan(x + y) = \frac{\tan(x) + \tan(y)}{1
- \tan(x)\tan(y)}\]
\[\sin(x-y) =
\sin(x)\cos(y)-\cos(x)\sin(y)\]
\[\cos(x-y) = \cos(x)\cos(y) +
\sin(x)\sin(y)\]
\[\tan(x - y) = \frac{\tan(x) - \tan(y)}{1
+ \tan(x)\tan(y)}\]
Double Angle Identities (in Radians)
\[
\begin{aligned}
\sin(2x) &= 2\sin(x)\cos(x) =
\frac{2\tan(x)}{1+\tan^2(x)}\\
\cos(2x)& = \cos^2(x) - \sin^2(x) =
\frac{1-\tan^2(x)}{1+\tan^2(x)}\\
\tan(2x) &= \frac{2\tan(x)}{1-\tan^2(x)}\\
\sec(2x) &= \frac{1}{\cos(2x)} =
\frac{1+\tan^2(x)}{1-\tan^2(x)}\\
\csc(2x) &= \frac{\sec(x)\csc(x)}{2}
\end{aligned}
\]
Triple Angle Identities
\[\sin(3x) = 3\sin(x) -
4\sin^3(x)\]
\[\cos(3x) = 4\cos^3(x) -
3\cos(x)\]
\[\tan(3x) = \frac{3\tan(x) - \tan^3(x)}{1
- 3\tan^2(x)}\]
Half Angle Identities
\[
\begin{aligned}
\sin\left(\frac{x}{2}\right) &= \pm \sqrt{\frac{1 - \cos(x)}{2}}\\
\cos\left(\frac{x}{2}\right) & = \pm \sqrt{\frac{1 + \cos(x)}{2}}\\
\tan\left(\frac{x}{2}\right) &=\sqrt{\frac{1 - \cos(x)}{1 + \cos(x)}} = \frac{1 -
\cos(x)}{\sin(x)}
\end{aligned}
\]
Product Identities
\[\sin(x)\cos(y) = \frac{\sin(x+y) +
\sin(x-y)}{2}\]
Sum to Product Identities
\[\sin(x) + \sin(y) =
2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)\]
Inverse Trigonometry Formulas
\[\sin^{-1}(-x) =
-\sin^{-1}(x)\]
\[\cos^{-1}(-x) = \pi -
\cos^{-1}(x)\]
\[\tan^{-1}(-x) =
-\tan^{-1}(x)\]
\[\csc^{-1}(-x) =
-\csc^{-1}(x)\]
\[\sec^{-1}(-x) = \pi -
\sec^{-1}(x)\]
\[\cot^{-1}(-x) = \pi -
\cot^{-1}(x)\]