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0. Must Known Trigo Values

Key Values to Memorize:
Angle \(\sin\) \(\cos\) \(\tan\)
\(0\) \(0\) \(1\) \(0\)
\(\frac{\pi}{6}\) \(\frac{1}{2}\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{\sqrt{3}}\)
\(\frac{\pi}{4}\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{2}}{2}\) \(1\)
\(\frac{\pi}{3}\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{2}\) \(\sqrt{3}\)
\(\frac{\pi}{2}\) \(1\) \(0\) \(\infty\)

Unit Circle:

1. Basic Inverse Trigonometric Functions

Key Values to Memorize:

  • \(\sin^{-1}(1) = \frac{\pi}{2}\)
  • \(\sin^{-1}(\frac{1}{2}) = \frac{\pi}{6}\)
  • \(\sin^{-1}(\frac{\sqrt{2}}{2}) = \frac{\pi}{4}\)
  • \(\sin^{-1}(\frac{\sqrt{3}}{2}) = \frac{\pi}{3}\)
  • \(\tan^{-1}(1) = \frac{\pi}{4}\)
  • \(\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}\)

Remember: Negative inputs give negative outputs for these functions

Domains and Ranges:

  • \(\sin^{-1}x: [-1,1] \to [-\frac{\pi}{2}, \frac{\pi}{2}]\)
  • \(\cos^{-1}x: [-1,1] \to [0, \pi]\)
  • \(\tan^{-1}x: \mathbb{R} \to (-\frac{\pi}{2}, \frac{\pi}{2})\)
  • \(\cot^{-1}x: \mathbb{R} \to (0, \pi)\)

Essential Identities:

  • \(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\)
  • \(\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}\)
  • \(\sin^{-1}(-x) = -\sin^{-1}x\)
  • \(\tan^{-1}(-x) = -\tan^{-1}x\)

2. Hyperbolic Functions Definitions

Basic Definitions:

  • \(\sinh x = \frac{e^x - e^{-x}}{2}\)
  • \(\cosh x = \frac{e^x + e^{-x}}{2}\)
  • \(\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x - e^{-x}}{e^x + e^{-x}}\)
  • \(\coth x = \frac{\cosh x}{\sinh x} = \frac{e^x + e^{-x}}{e^x - e^{-x}}\)

Hyperbolic Functions Graph:

Hyperbolic Functions Domains and Ranges:

  • \(\sinh x: \mathbb{R} \to \mathbb{R}\)
  • \(\cosh x: \mathbb{R} \to [1, \infty)\)
  • \(\tanh x: \mathbb{R} \to (-1, 1)\)
  • \(\coth x: \mathbb{R} \to (-\infty, -1) \cup (1, \infty)\)

Inverse Hyperbolic Functions:

  • \(\sinh^{-1}x = \ln(x + \sqrt{x^2 + 1})\)
  • \(\cosh^{-1}x = \ln(x + \sqrt{x^2 - 1}), x \geq 1\)
  • \(\tanh^{-1}x = \frac{1}{2}\ln(\frac{1+x}{1-x}), |x| < 1\)

Graph of Inverse Hyperbolic Functions:

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Important Identities:

  • \(\cosh^2 x - \sinh^2 x = 1\)
  • \(1 - \tanh^2 x = \text{sech}^2 x\)
  • \(\coth^2 x - 1 = \text{csch}^2 x\)

3. Addition Formulas

Hyperbolic Addition Formulas:

  • \(\sinh(x \pm y) = \sinh x \cosh y \pm \cosh x \sinh y\)
  • \(\cosh(x \pm y) = \cosh x \cosh y \pm \sinh x \sinh y\)
  • \(\tanh(x \pm y) = \frac{\tanh x \pm \tanh y}{1 \pm \tanh x \tanh y}\)

4. Common Problem-Solving Strategies

When solving inverse function problems:

  1. Check domains first:
    • For \(\sin^{-1}\) and \(\cos^{-1}\): input must be in [-1,1]
    • For \(\tanh^{-1}\): input must be in (-1,1)
    • For \(\cosh^{-1}\): input must be ≥ 1
  2. For composite functions:
    • Work from inside out
    • Check each step's domain
    • Pay attention to range restrictions
  3. For equations:
    • Try to isolate the inverse function
    • Apply the corresponding direct function to both sides
    • Check for extraneous solutions

5. Common Mistakes to Avoid

  • Don't forget domain restrictions when solving equations
  • Check if negative inputs affect the result
  • Remember that \(\sin^{-1}(\sin x)\) ≠ x for all x
  • Pay attention to branch cuts in inverse hyperbolic functions