Trigonometric and Inverse Trigonometric Formulas

Trigonometric and Inverse Trigonometric Formulas

Trigonometric and Inverse Trigonometric Formulas

1. Relations Between Trigonometric Ratios

\[ \tan\theta = \frac{\sin\theta}{\cos\theta} \quad \tan\theta = \frac{1}{\cot\theta} \quad \tan\theta \cdot \cot\theta = 1 \]

\[ \cot\theta = \frac{\cos\theta}{\sin\theta} \quad \csc\theta = \frac{1}{\sin\theta} \quad \sec\theta = \frac{1}{\cos\theta} \]

2. Fundamental Identities

\[ \sin^2\theta + \cos^2\theta = 1 \quad 1 + \tan^2\theta = \sec^2\theta \quad 1 + \cot^2\theta = \csc^2\theta \]

3. Addition and Subtraction Formulae

\[ \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \]

\[ \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \]

\[ \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \]

4. Transformation Formulae

\[ \sin C + \sin D = 2 \sin\left(\frac{C+D}{2}\right) \cos\left(\frac{C-D}{2}\right) \]

\[ \cos C + \cos D = 2 \cos\left(\frac{C+D}{2}\right) \cos\left(\frac{C-D}{2}\right) \]

\[ \cos C - \cos D = -2 \sin\left(\frac{C+D}{2}\right) \sin\left(\frac{C-D}{2}\right) \]

5. Multiple Angle Formulae

\[ \sin 2A = 2 \sin A \cos A \quad \cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A \]

\[ \tan 2A = \frac{2 \tan A}{1 - \tan^2 A} \]

\[ \sin 3A = 3 \sin A - 4 \sin^3 A \quad \cos 3A = 4 \cos^3 A - 3 \cos A \]

6. Angle Measure Conversions

\[ \text{Radians} = \frac{\pi}{180} \times \text{Degrees} \quad \text{Degrees} = \frac{180}{\pi} \times \text{Radians} \]

\[ \theta (\text{radians}) = \frac{l}{r} \]

7. Standard Angle Values

Anglesincostancosecseccot
0°010∞1∞
30°1/2√3/21/√322/√3√3
45°1/√21/√21√2√21
60°√3/21/2√32/√321/√3
90°10∞1∞0

8. Inverse Trigonometric Functions

FunctionDomainRange
\( \sin^{-1}x \)[−1, 1][−π/2, π/2]
\( \cos^{-1}x \)[−1, 1][0, π]
\( \tan^{-1}x \)R(−π/2, π/2)
\( \cot^{-1}x \)R(0, π)
\( \sec^{-1}x \)R − (−1,1)[0, π] \ {π/2}
\( \csc^{-1}x \)R − (−1,1)[−π/2, π/2] \ {0}

9. Substitutions to Simplify Inverse Trigonometric Expressions

  • \( x = a \tan\theta \) or \( x = a \cot\theta \) for \( \sqrt{a^2 + x^2} \)
  • \( x = a \sin\theta \) or \( x = a \cos\theta \) for \( \sqrt{a^2 - x^2} \)
  • \( x = a \sec\theta \) or \( x = a \csc\theta \) for \( \sqrt{x^2 - a^2} \)
  • \( x = a \cos 2\theta \) for rational expressions like \( \frac{a + x}{a - x} \)

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