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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:
- 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
- For composite functions:
- Work from inside out
- Check each step's domain
- Pay attention to range restrictions
- 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