6.1 The Inverse of Sin, Cos, and Tan

Overview of Class Structure

• Last week's completion: Chapter 5 covering trigonometric functions.

• This week's focus: Chapter 6 - Analytic Trigonometry.

• Upcoming Test:

  • To be held at the test center during the designated week.

  • Flexible timing for students; can choose any day in the week for the test.

  • Confirmation needed if testing center option was not submitted.

  • Communication via email for any inquiries about the testing process.

Chapter 6: Analytic Trigonometry

Introduction to Analytic Trigonometry

• Focus shift: Instead of providing an angle to find sine, cosine, and tangent, the goal is to find the angle from given trigonometric values.

Section 6.1: Inverse of Sine, Cosine, and Tangent Functions

Concept of One-to-One Functions

• Definition: A one-to-one function has an inverse if each $x$ corresponds to one $y$ and each $y$ corresponds to one $x$.

• To verify a function's one-to-one nature, use the horizontal line test: A horizontal line should intersect the graph at most once.

• If a function is one-to-one:

  • For every $x$ in the domain of $f$, $f^{-1}(f(x)) = x$.

  • For every $x$ in the domain of $f^{-1}$, $f(f^{-1}(x)) = x$.

  • The domain of $f$ is the range of $f^{-1}$, and vice versa.

  • Graphs of $f$ and $f^{-1}$ are reflections over the line $y = x$.

  • To find the inverse, rearranging and solving: if $y = f(x)$, then rearranging to $x = f(y)$ can yield the inverse function.

Inverse Sine Function

• To define the inverse sine function: Limit the sine function to ensure it’s one-to-one by restricting the domain: - \frac{\pi}{2} < x < \frac{\pi}{2}.

• Representation: $y = \sin^{-1}(x)$ means $x = \sin(y)$.

• Read as: "$y$ is the angle whose sine equals $x$."

• The restricted range:

  • Domain for $y = \sin^{-1}(x)$: $-1 \leq x \leq 1$.

  • Range: $- \frac{\pi}{2} \leq y \leq \frac{\pi}{2}$.

• Graphing establishes symmetry about $y = x$.

Working With Inverse Sine Examples

1. To find $\sin^{-1}(1)$:

   • What angle makes sine equal to $1$ within the range limit?

   • Answer: $\frac{\pi}{2}$. Hence, $\sin^{-1}(1) = \frac{\pi}{2}$.

2. To solve $\sin^{-1}(- \frac{1}{2})$:

   • Angle: between $- \frac{\pi}{2}$ and $0$?

   • Answer: $- \frac{\pi}{6}$. Hence, $\sin^{-1}(- \frac{1}{2}) = - \frac{\pi}{6}$.

3. To find $\sin^{-1}(\frac{\sqrt{2}}{2})$:

   • Answer: $\frac{\pi}{4}$.

4. For the function $\sin(\sin^{-1}(0.8))$:

   • Answer: $0.8$ exists within the range, so it equals $0.8$.

5. For $\sin(\sin^{-1}(1.3))$:

   • Not within the domain; answer does not exist.

6. To evaluate $\sin^{-1}(\sin(\frac{-\pi}{4}))$:

   • Confirmed back within domain limits resulting in $\frac{-\pi}{4}$.

Inverse Cosine Function

• Restriction for cosine to ensure one-to-one is from $0 \leq x \leq \pi$.

• Definition: $y = \cos^{-1}(x)$ means $x = \cos(y)$.

• Represented as $y$ is the angle whose cosine equals $x$, with:

  • Domain: $-1 \leq x \leq 1$.

  • Range: $0 \leq y \leq \pi$.

Working with Inverse Cosine Examples

1. For $\cos^{-1}(0)$:

   • Angle is $\frac{\pi}{2}$.

2. Find $\cos^{-1}(- \frac{\sqrt{2}}{2})$:

   • Result: $\frac{3\pi}{4}$.

3. Find $\cos^{-1}(\frac{\sqrt{3}}{2})$:

   • Result: $\frac{\pi}{6}$.

4. For $\cos(\cos^{-1}(-0.4))$:

   • Result: $-0.4$.

5. For $\cos^{-1}(\cos(- \frac{2\pi}{3}))$:

   • Result is $\frac{2\pi}{3}$ after confirming the even nature across the interval.

6. For $\cos^{-1}(\pi)$:

   • Not valid; does not exist in the domain.

Inverse Tangent Function

• For $y$ equal to tangent of $x$, note the asymptotes:

  • Asymptotes at odd multiples of $\frac{\pi}{2}$.

  • Define domain between $- \frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ with range: all real numbers ($-\infty$ to $\infty$).

• Working with inverse tangent examples:

  1. For $\tan^{-1}(1)$:

  • Value is $\frac{\pi}{4}$.

  1. For $\tan^{-1}(-\sqrt{3})$:

  • Would yield angle $- \frac{\pi}{3}$.

  1. Find $\tan^{-1}(- \frac{\sqrt{3}}{3})$:

  • Result is $- \frac{\pi}{6}$.

  1. Evaluate $\tan(\tan^{-1}(0))$:

  • Determines angle yielding $0$.

  1. For $\tan(\tan^{-1}(-5))$:

  • Result is $-5$.

Solving for Inverse Functions and Equations

1. Finding the inverse of a function (Example: $f(x) = 2 \sin(x) -1$):

   • Find inverse by swapping $x$ and $y$: $x = 2 \sin(y) -1$.

   • Solve for $y$ to yield: $f^{-1}(x) = \sin^{-1}(\frac{x+1}{2})$.

2. Domain and Range:

   • Domain follows the assigned limit, range calculated from the sine function properties.

3. Equation setup example: $3 \sin^{-1}(x) = \pi$

   • Divide both sides yielding $\sin^{-1}(x) = \frac{\pi}{3}$.

   • Solve resulting in $x = \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$.

Conclusion

• Chapter 6 concludes next class with practical applications of inverse functions and calculations for future assessments.

• Focus on review of previous material and understanding angle values, domains, and ranges for preparation of the upcoming test.