Inverse Trig Functions

Inverse Trig Functions

Inverse Functions: A Review

• Definition: Two functions $f(x)$ and $g(x)$ are inverses if $f(g(x)) = x$ and $g(f(x)) = x$.

• Example 1: If $f(x) = x^3$, then $f^{-1}(x) = \sqrt[3]{x}$.

  • $f(f^{-1}(x)) = f(\sqrt[3]{x}) = (\sqrt[3]{x})^3 = x$
  • $f^{-1}(f(x)) = f^{-1}(x^3) = \sqrt[3]{x^3} = x$

• Example 2: If $g(x) = \frac{1}{x - 3}$, then $g^{-1}(x) = \frac{1}{x} + 3$.

  • $g(g^{-1}(x)) = g(\frac{1}{x} + 3) = \frac{1}{(\frac{1}{x} + 3) - 3} = \frac{1}{\frac{1}{x}} = x$
  • $g^{-1}(g(x)) = g^{-1}(\frac{1}{x - 3}) = \frac{1}{\frac{1}{x - 3}} + 3 = (x - 3) + 3 = x$

Geometric Interpretation of Inverse Functions

• Horizontal Line Test: A function has an inverse if and only if every horizontal line intersects its graph at most once.

• Sketching the Inverse: To sketch the inverse of a function:

  • Draw the line $y = x$.
  • Reflect the graph of the original function across the line $y = x$.

Inverse Trigonometric Functions

Sine Function

• $sin(x)$ does not pass the horizontal line test over its entire domain.

• Restricting the Domain: To define an inverse, restrict the domain of $sin(x)$ to $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

• Inverse Sine Function: Denoted as $sin^{-1}(x)$ or $arcsin(x)$.

• Analogy: Similar to how $x^2$ has an inverse $\sqrt{x}$ only when the domain of $x^2$ is restricted to $[0, \infty)$.
  * The inverse is created by drawing the line $y=x$ and rotating the curve around the line. The solid blue curve is the $\sqrt{x}$.

Graph of Sine Inverse

• Passes through points $(0, 0)$, $(-1, -\frac{\pi}{2})$ and $(1, \frac{\pi}{2})$

• Obtained by restricting the original sine curve between $[-\frac{\pi}{2}, \frac{\pi}{2}]$

• The y values for the original sine function in that range went from $[-1, 1]$

• Domain and range get switched compared to the original function.

Properties of $sin^{-1}(x)$

• Domain: $[-1, 1]$

• Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$

• Symmetry: Odd function.

  • $sin^{-1}(-x) = -sin^{-1}(x)$

Cosine Function

• $cos(x)$ also fails the horizontal line test over its entire domain.

• Restricting the Domain: Restrict the domain of $cos(x)$ to $[0, \pi]$.

• On that region the function passes the horizontal line test, therefore an inverse function can be constructed by rotating it around the line $y=x$.

  • Remember that it's tough to visualize it, but if you tilt your head 45 degrees, you would see that the red curve and the black curve are symmetric around the blue dotted line.

Properties of $cos^{-1}(x)$

• Domain: $[-1, 1]$

• Range: $[0, \pi]$

• Symmetry: Neither even nor odd.

Tangent Function

• $tan(x)$ fails the horizontal line test over its entire domain.

• Restricting the Domain: Restrict the domain of $tan(x)$ to $(-\frac{\pi}{2}, \frac{\pi}{2})$.

• When you rotate the curve you have to consider that asymptotes are also properties of the function.

  • Every property of x gets switched with every property of y.
  • A point on the x axis would go to a point on the y axis. A vertical asymptote would go to a horizontal asymptote, etcetera.

Graph of Tan Inverse

• Curve goes through zero and tapers off a horizontal asymptotes at $\frac{\pi}{2}$ and $-\frac{\pi}{2}$

Properties of $tan^{-1}(x)$

• Domain: $(-\infty, \infty)$

• Range: $(-\frac{\pi}{2}, \frac{\pi}{2})$

• Symmetry: Odd function.

  • $tan^{-1}(-x) = -tan^{-1}(x)$

Other Inverse Trig Functions

• $sec^{-1}(x)$, $csc^{-1}(x)$, and $cot^{-1}(x)$ exist, but are less commonly used.

Evaluating Inverse Trig Functions

• The trig functions in calc one, should be well known for calc two.
• The $tan^{-1}(x)$ function seems to pop up a little bit more than the other ones, a little more than $sin^{-1}(x)$ or $cos^{-1}(x)$.

General Approach

1. Check the Domain: Ensure the input value is within the domain of the inverse trig function.

2. Set Equal to $\theta$: Let the inverse trig function equal an angle $\theta$.

3. Apply Trig Function: Apply the corresponding trigonometric function to both sides.

4. Solve for $\theta$: Determine the angle $\theta$ that satisfies the equation, considering the range of the inverse trig function.

Example 1: Evaluate $sin^{-1}(\frac{1}{2})$

1. $\frac{1}{2}$ is in the domain of $sin^{-1}(x)$ which is $[-1, 1]$.

2. Let $sin^{-1}(\frac{1}{2}) = \theta$.

3. $sin(sin^{-1}(\frac{1}{2})) = sin(\theta)$

4. $\frac{1}{2} = sin(\theta)$. What angle, $\theta$, has a sine of $\frac{1}{2}$?

   • $\theta = \frac{\pi}{6}$

Example 2: Evaluate $cos^{-1}(\frac{\sqrt{2}}{2})$

1. $\frac{\sqrt{2}}{2}$ is in the domain of $cos^{-1}(x)$ which is $[-1, 1]$.

2. What angle, $\theta$, has a cosine of $\frac{\sqrt{2}}{2}$?

   • $\theta = \frac{\pi}{4}$

Example 3: Evaluate $tan^{-1}(\sqrt{3})$

1. $\sqrt{3}$ is in the domain of $tan^{-1}(x)$ which is $(-\infty, \infty)$.

2. What angle, $\theta$, has a tangent of $\sqrt{3}$?

   • $\theta = \frac{\pi}{3}$

Example 4: Evaluate $cos^{-1}(\sqrt{3})$

1. $\sqrt{3}$ is not in the domain of $cos^{-1}(x)$ which is $[-1, 1]$.

2. Therefore, $cos^{-1}(\sqrt{3})$ does not exist.

Example 5: Evaluate $cos^{-1}(\frac{\sqrt{3}}{2})$

1. $\frac{\sqrt{3}}{2}$ is in the domain of $cos^{-1}(x)$ which is $[-1, 1]$.

2. What angle, $\theta$, has a cosine of $\frac{\sqrt{3}}{2}$?

   • $\theta = \frac{\pi}{6}$

Example 6: Evaluate $sin^{-1}(-1)$

1. $-1$ is in the domain of $sin^{-1}(x)$ which is $[-1, 1]$.

2. What angle, $\theta$, has a sine of $-1$?

   • Range of $sin^{-1}(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
   • $\theta = -\frac{\pi}{2}$

Example 7: Evaluate $sin^{-1}(-\frac{\sqrt{3}}{2})$

1. $-\frac{\sqrt{3}}{2}$ is in the domain of $sin^{-1}(x)$ which is $[-1, 1]$.

2. What angle, $\theta$, has a sine of $-\frac{\sqrt{3}}{2}$?

   • Range of $sin^{-1}(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
   • $\theta = -\frac{\pi}{3}$

Example 8: Evaluate $cos^{-1}(-\frac{\sqrt{3}}{2})$

1. $-\frac{\sqrt{3}}{2}$ is in the domain of $cos^{-1}(x)$ which is $[-1, 1]$.

2. What angle, $\theta$, has a cosine of $-\frac{\sqrt{3}}{2}$?

   • Range of $cos^{-1}(x)$ is $[0, \pi]$.
   • $\theta = \frac{5\pi}{6}$

Example 9: Evaluate $tan^{-1}(-1)$

1. $-1$ is in the domain of $tan^{-1}(x)$ which is $(-\infty, \infty)$.

2. What angle, $\theta$, has a tangent of $-1$?

   • Range of $tan^{-1}(x)$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$.
   • $\theta = -\frac{\pi}{4}$