8.3 Inverse Trigonometric Functions

Inverse trigonometric functions flip the input and output of sine, cosine, and tangent. They're crucial for solving equations and finding angles when given trigonometric values. These functions have restricted domains and ranges to ensure unique solutions.

Exact values of inverse trig functions for common angles are essential to know. They help simplify expressions and solve problems without a calculator. Understanding these values and how to use technology for more complex calculations is key to mastering inverse trigonometry.

Inverse Trigonometric Functions

Inverse trigonometric functions

• Inverses of the trigonometric functions sine, cosine, and tangent
  • Denoted as $\arcsin$, $\arccos$, and $\arctan$ or $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$
  • Reverse the input and output values of the original functions ($\sin \theta = \frac{1}{2}$, then $\arcsin \frac{1}{2} = \theta$)
• Domain and range differ from the original trigonometric functions
  • $\arcsin x$: Domain $[-1, 1]$, Range $[-\frac{\pi}{2}, \frac{\pi}{2}]$
  • $\arccos x$: Domain $[-1, 1]$, Range $[0, \pi]$
  • $\arctan x$: Domain $(-\infty, \infty)$, Range $(-\frac{\pi}{2}, \frac{\pi}{2})$
• Solve equations by applying inverse trigonometric functions
  • Isolate the trigonometric function and apply its inverse to both sides ($\cos \theta = \frac{\sqrt{3}}{2}$, then $\theta = \arccos \frac{\sqrt{3}}{2}$)
• Principal values of inverse trigonometric functions are unique angles within their restricted ranges

Exact values of inverse expressions

• Common angles have exact values for their inverse trigonometric functions
  • $\arcsin \frac{1}{2} = \frac{\pi}{6}$, $\arccos \frac{1}{2} = \frac{\pi}{3}$, $\arctan 1 = \frac{\pi}{4}$
  • $\arcsin \frac{\sqrt{2}}{2} = \frac{\pi}{4}$, $\arccos \frac{\sqrt{2}}{2} = \frac{\pi}{4}$, $\arctan \sqrt{3} = \frac{\pi}{3}$
  • $\arcsin \frac{\sqrt{3}}{2} = \frac{\pi}{3}$, $\arccos \frac{\sqrt{3}}{2} = \frac{\pi}{6}$, $\arctan \frac{1}{\sqrt{3}} = \frac{\pi}{6}$
• Simplify expressions by recognizing these exact values
  • $\arcsin(\sin \frac{\pi}{3}) = \frac{\pi}{3}$ because $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$ and $\arcsin \frac{\sqrt{3}}{2} = \frac{\pi}{3}$
• These values can be visualized on the unit circle

Technology for inverse trig evaluation

• Calculators and software can evaluate inverse trigonometric functions for any input value
  • Find $\arccos(-0.8)$ using a calculator: $\arccos(-0.8) \approx 2.4980$ radians or $143.13^\circ$
• Graph inverse trigonometric functions using technology to visualize their behavior
  • Plot $y = \arctan x$ using a graphing calculator or software to see its domain, range, and asymptotes

Composite functions with inverse trig

• Trigonometric and inverse trigonometric functions cancel each other when composed
  • $\sin(\arcsin x) = x$ for $-1 \leq x \leq 1$
  • $\cos(\arccos x) = x$ for $-1 \leq x \leq 1$
  • $\tan(\arctan x) = x$ for all real numbers $x$
• Solve equations involving composite functions by working from the inside out
  1. $\tan(\arccos x) = 1$
  2. $\arccos x = \frac{\pi}{4}$ because $\tan \frac{\pi}{4} = 1$
  3. $x = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ because $\arccos \frac{\sqrt{2}}{2} = \frac{\pi}{4}$

Periodic functions and inverse relations

• Trigonometric functions are periodic, repeating their values at regular intervals
• Inverse trigonometric functions are not periodic, as they are derived from inverse relations of the original functions
• Radians are often used to express angles in inverse trigonometric functions, providing a more natural way to describe rotations