Solving Trig Equations with Inverse Functions

Solving Trigonometric Equations with Inverse Functions

Recap: Previously, we solved trigonometric equations like \sin(x) = \frac{1}{2} using the unit circle to find special angles (e.g., \frac{\pi}{6}, \frac{5\pi}{6}).
Problem: What if we encounter equations like \sin(x) = \frac{1}{3}, where \frac{1}{3} isn't a standard value on the unit circle?
Solution: Introduce inverse trigonometric functions to solve for angles when the sine, cosine, or tangent is not a standard value.

General Concept: If f is a function, its inverse function, denoted as f^{-1}, "undoes" the effect of f.
Property: f^{-1}(f(a)) = a for any number a in the domain of f.
Application to Trigonometry -- if \sin(x) = \frac{1}{3}, then x = \sin^{-1}(\frac{1}{3}) (inverse sine of \frac{1}{3}).

Sine Function: The sine function, y = \sin(x), is a wave that oscillates between -1 and 1.
Inverse Function as Reflection: The inverse function is a reflection of the original function across the line y = x.
Issue: Reflecting the entire sine curve across y = x does not result in a function because it fails the vertical line test; a vertical line intersects the reflected curve multiple times.
Points on Sine Curve and Their Reflections: For example, the point \pi, 0 on the sine curve becomes 0, \pi on the reflected curve.

Solution: To obtain a valid inverse function, we restrict the domain of the original sine function.
Restricted Domain: The domain of the sine function is limited to [-\frac{\pi}{2}, \frac{\pi}{2}].
Sine Function in Restricted Domain: In this interval, the sine function ranges from -1 to 1 without repeating y-values.
Key Points: The sine function passes through the points (-\frac{\pi}{2}, -1), (0, 0), and (\frac{\pi}{2}, 1) in this restricted domain.

Reflection of Restricted Sine Function: Reflecting the restricted sine function across y = x yields the inverse sine function.
Domain and Range: The inverse sine function, denoted as \sin^{-1}(x) or arcsin(x), has a domain of [-1, 1] and a range of [-\frac{\pi}{2}, \frac{\pi}{2}] .
Graph of Inverse Sine: The graph starts at (-1, -\frac{\pi}{2}) and ends at (1, \frac{\pi}{2}).
Calculator Usage: Calculators only show the principal values of the inverse sine function due to the domain restriction.

Example: To solve \sin(x) = \frac{1}{3}, take the inverse sine of both sides: x = \sin^{-1}(\frac{1}{3}) .
Calculator Result: Using a calculator, \sin^{-1}(\frac{1}{3}) \approx 0.3398 radians.
Interpretation: This is the angle in radians whose sine is \frac{1}{3} .
Multiple Solutions: Infinitely many angles have a sine of \frac{1}{3}, but the inverse sine function only returns the angle within the range [-\frac{\pi}{2}, \frac{\pi}{2}] .

Inverse Cosine: The cosine function requires a different domain restriction to have an inverse.
- Restricted Domain: The domain of the cosine function it's limited to [0, \pi].
- Range of Inverse Cosine: The range of the inverse cosine function is [0, \pi].
Inverse Tangent: The tangent function has vertical asymptotes and a range of (-\infty, \infty).
- Appropriate Interval: [-\frac{\pi}{2}, \frac{\pi}{2}] is the interval to determine the inverse tangent function uniquely.
- Range of Inverse Tangent: The range of the inverse tangent function is (-\frac{\pi}{2}, \frac{\pi}{2}).

Equation Solutions: When solving trigonometric equations, remember that inverse trigonometric functions only provide one solution within a specific interval.
Multiple Solutions: Equations may have multiple solutions, but inverse functions only give the principal value within their defined range.