Chapter 7 Lesson 2: Inverse Trigonometric Functions and Equations

Welcome back to Professor Baker's Math Class! In this lesson, we'll be exploring the fascinating world of inverse trigonometric functions and how to solve various trigonometric equations. Get ready to expand your mathematical toolkit!

Topics Covered:

  • Values of Inverse Trigonometric Functions: Understanding how to find the principal values of $\arcsin(x)$, $\arccos(x)$, and $\arctan(x)$. Remember that the range of $\arcsin(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$, the range of $\arccos(x)$ is $[0, \pi]$, and the range of $\arctan(x)$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$.
  • Composition of Trigonometric and Inverse Trigonometric Functions:
    • Problem Type 1: Simplifying expressions like $\sin(\arcsin(x))$ or $\cos(\arccos(x))$. When the function and its inverse are composed, they often cancel each other out, but be mindful of the domain restrictions! For example, $\sin(\arcsin(x)) = x$ for $-1 \le x \le 1$.
    • Composition with Different Functions (Type 1 & 2): Evaluating expressions such as $\cos(\arcsin(x))$. This often requires using a right triangle to visualize the relationship and then applying trigonometric identities.
  • Composition with Variable Expressions: Working with expressions like $\sin(\arccos(x))$, $\tan(\arcsin(x))$, etc., where you need to use trigonometric identities and right triangle geometry to simplify. For example, to find $\cos(\arcsin(x))$, let $\theta = \arcsin(x)$, then $\sin(\theta) = x$. Draw a right triangle, and use the Pythagorean theorem to find the adjacent side.
  • Calculator Approximations: Using your calculator to find approximate values of inverse trigonometric functions in both degrees and radians. Make sure your calculator is in the correct mode!

Solving Trigonometric Equations:

  • Basic Sine and Cosine Equations: Finding solutions to equations like $\sin(x) = a$ or $\cos(x) = b$ within a specified interval. Remember to consider all possible solutions within the given range by using the unit circle and properties of sine and cosine.
  • Basic Tangent, Cotangent, Secant, and Cosecant Equations: Solving equations involving the other trigonometric functions. You may need to rewrite these equations in terms of sine and cosine first.
  • Equations with Squared Functions: Solving equations like $\sin^2(x) = c$ or $\cos^2(x) = d$. Remember to take both the positive and negative square roots when solving.
  • Equations with Angle Multiples: Solving equations like $\sin(2x) = e$ or $\cos(3x) = f$. First, solve for the multiple of the angle (e.g., $2x$), then divide to find $x$. Be sure to consider all solutions within the interval, as multiplying the angle increases the number of solutions. For example, when solving $\sin(2x) = 1/2$ on the interval $[0, 2\pi)$, we first solve for $2x$, which gives us $2x = \pi/6$ and $2x=5\pi/6$ on the interval $[0, 2\pi)$, and $2x= 13\pi/6$ and $2x = 17\pi/6$ on the interval $[2\pi, 4\pi)$. After dividing by 2, this yields $x = \pi/12, 5\pi/12, 13\pi/12, 17\pi/12$.

Key Concepts:

  • Inverse Trigonometric Functions: The inverse trigonometric functions, denoted as $\arcsin(x)$, $\arccos(x)$, and $\arctan(x)$, are used to find the angle whose sine, cosine, or tangent is a given value.
  • Domain and Range: Understanding the domain and range of each inverse trigonometric function is crucial for finding correct solutions.
  • Unit Circle: The unit circle is an invaluable tool for visualizing trigonometric functions and their inverses.
  • Trigonometric Identities: Mastering trigonometric identities is essential for simplifying expressions and solving equations.

Remember to practice regularly and don't hesitate to ask questions. Good luck, and happy solving!