Trigonometric Functions in terms of each other
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Theorem
Sine in terms of Cosine
| \(\ds \sin x\) | \(=\) | \(\ds +\sqrt {1 - \cos ^2 x}\) | if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$ | |||||||||||
| \(\ds \sin x\) | \(=\) | \(\ds -\sqrt {1 - \cos ^2 x}\) | if there exists an integer $n$ such that $\paren {2 n + 1} \pi < x < \paren {2 n + 2} \pi$ |
Sine in terms of Tangent
| \(\ds \sin x\) | \(=\) | \(\ds +\frac {\tan x} {\sqrt {1 + \tan^2 x} }\) | if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \sin x\) | \(=\) | \(\ds -\frac {\tan x} {\sqrt {1 + \tan^2 x} }\) | if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ |
Sine in terms of Cotangent
Sine in terms of Secant
| \(\ds \sin x\) | \(=\) | \(\ds + \frac {\sqrt{\sec ^2 x - 1} } {\sec x}\) | if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \sin x\) | \(=\) | \(\ds - \frac {\sqrt{\sec ^2 x - 1} } {\sec x}\) | if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ |
Sine is Reciprocal of Cosecant
- $\sin \theta = \dfrac 1 {\csc \theta}$
Cosine in terms of Sine
| \(\ds \cos x\) | \(=\) | \(\ds +\sqrt {1 - \sin^2 x}\) | if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \cos x\) | \(=\) | \(\ds -\sqrt {1 - \sin^2 x}\) | if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ |
Cosine in terms of Tangent
| \(\ds \cos x\) | \(=\) | \(\ds +\frac 1 {\sqrt {1 + \tan^2 x} }\) | if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \cos x\) | \(=\) | \(\ds -\frac 1 {\sqrt {1 + \tan^2 x} }\) | if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ |
Cosine in terms of Cotangent
| \(\ds \cos x\) | \(=\) | \(\ds +\frac {\cot x} {\sqrt {1 + \cot^2 x} }\) | if there exists an integer $n$ such that $2 n \pi < x < \paren {2 n + 1} \pi$ | |||||||||||
| \(\ds \cos x\) | \(=\) | \(\ds -\frac {\cot x} {\sqrt {1 + \cot^2 x} }\) | if there exists an integer $n$ such that $\paren {2 n - 1} \pi < x < 2 n \pi$ |
Cosine is Reciprocal of Secant
- $\cos \theta = \dfrac 1 {\sec \theta}$
Cosine in terms of Cosecant
Tangent in terms of Sine
Tangent in terms of Cosine
Tangent is Reciprocal of Cotangent
- $\tan \theta = \dfrac 1 {\cot \theta}$
Tangent in terms of Secant
| \(\ds \tan x\) | \(=\) | \(\ds +\sqrt {\sec^2 x - 1}\) | if there exists an integer $n$ such that $n \pi < x < \paren {n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \tan x\) | \(=\) | \(\ds -\sqrt {\sec^2 x - 1}\) | if there exists an integer $n$ such that $\paren {n + \dfrac 1 2} \pi < x < \paren {n + 1} \pi$ |
Tangent in terms of Cosecant
Cotangent in terms of Sine
Cotangent in terms of Cosine
Cotangent is Reciprocal of Tangent
- $\cot \theta = \dfrac 1 {\tan \theta}$
Cotangent in terms of Secant
Cotangent in terms of Cosecant
Cotangent in terms of Cosecant
Secant in terms of Sine
Secant is Reciprocal of Cosine
- $\sec \theta = \dfrac 1 {\cos \theta}$
Secant in terms of Tangent
| \(\ds \sec x\) | \(=\) | \(\ds +\sqrt {\tan ^2 x + 1}\) | if there exists an integer $n$ such that $\paren {2 n - \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 1 2} \pi$ | |||||||||||
| \(\ds \sec x\) | \(=\) | \(\ds -\sqrt {\tan ^2 x + 1}\) | if there exists an integer $n$ such that $\paren {2 n + \dfrac 1 2} \pi < x < \paren {2 n + \dfrac 3 2} \pi$ |
Secant in terms of Cotangent
Secant in terms of Cosecant
Cosecant is Reciprocal of Sine
- $\csc \theta = \dfrac 1 {\sin \theta}$
Cosecant in terms of Cosine
Cosecant in terms of Tangent
Cosecant in terms of Cotangent
Cosecant in terms of Cotangent