Inverse trigonometry Knowledge-Based Study

Study notes

1. Restricting the Domain

   Trigonometric functions are periodic, meaning they repeat their values over regular intervals. Because they are periodic, they are not one-to-one (they fail the horizontal line test) across their entire natural domains. Therefore, an inverse function cannot exist unless the domain of each trigonometric function is appropriately restricted.

   To define inverse functions, we choose standard intervals where each trigonometric function is strictly monotonic (either strictly increasing or strictly decreasing) and takes on every value in its range exactly once.

   • Sine: The domain is restricted to $[-\frac{\pi}{2}, \frac{\pi}{2}]$, where $\sin(x)$ attains all values in $[-1, 1]$.
   • Cosine: The domain is restricted to $[0, \pi]$, where $\cos(x)$ attains all values in $[-1, 1]$.
   • Tangent: The domain is restricted to $(-\frac{\pi}{2}, \frac{\pi}{2})$, where $\tan(x)$ attains all real values in $(-\infty, \infty)$.

   Principal Inverse Trigonometric Functions

   The inverse trigonometric functions yield the unique angle $\theta$ within the restricted range whose trigonometric ratio equals the input value.

   Arcsine Function

   $y = \arcsin(x)$ (or $\sin^{-1}(x)$) means $\sin(y) = x$, where:

   • Domain: $[-1, 1]$
   • Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$

   Arccosine Function

   $y = \arccos(x)$ (or $\cos^{-1}(x)$) means $\cos(y) = x$, where:

   • Domain: $[-1, 1]$
   • Range: $[0, \pi]$

   Arctangent Function

   $y = \arctan(x)$ (or $\tan^{-1}(x)$) means $\tan(y) = x$, where:

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

   Finding Exact Values

   To evaluate an inverse trigonometric expression such as $\theta = \arcsin(v)$, determine the angle $\theta$ lying strictly in the designated range of the function such that the original trigonometric function evaluated at $\theta$ produces $v$.

   Symmetry and Negative Arguments

   The ranges dictate how negative arguments behave:

   • Odd symmetry: $\arcsin(-x) = -\arcsin(x)$ and $\arctan(-x) = -\arctan(x)$
   • Complementary behavior: $\arccos(-x) = \pi - \arccos(x)$

   Example 1: Evaluate $\arcsin(-\frac{1}{2})$.
   Since $\sin(\frac{\pi}{6}) = \frac{1}{2}$ and the range of arcsine is $[-\frac{\pi}{2}, \frac{\pi}{2}]$, we have $\arcsin(-\frac{1}{2}) = -\frac{\pi}{6}$.

   Example 2: Evaluate $\arccos(-\frac{\sqrt{2}}{2})$.
   Because $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, and the range of arccosine is $[0, \pi]$, we obtain $\arccos(-\frac{\sqrt{2}}{2}) = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$.

   Composition Properties

   When composing trigonometric functions with their inverses, attention must be paid to domains and ranges.

   Trigonometric Function of Inverse Trigonometric Function

   When the inverse function is inside, the identity holds directly for all inputs in the domain:

   • $\sin(\arcsin(x)) = x$ for $x \in [-1, 1]$
   • $\cos(\arccos(x)) = x$ for $x \in [-1, 1]$
   • $\tan(\arctan(x)) = x$ for all real $x$

   Inverse Function of Trigonometric Function

   When the inverse function is outside, $\arcsin(\sin(\theta)) = \theta$ is true only when $\theta$ lies within the principal range.

   Example: Evaluate $\arcsin(\sin(\frac{5\pi}{6}))$.
   The angle $\frac{5\pi}{6}$ is not in $[-\frac{\pi}{2}, \frac{\pi}{2}]$. Since $\sin(\frac{5\pi}{6}) = \frac{1}{2}$, we find the angle in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ with a sine of $\frac{1}{2}$, which gives $\arcsin(\frac{1}{2}) = \frac{\pi}{6}$.

   Algebraic Expressions via Reference Triangles

   Expressions such as $\cos(\arcsin(x))$ can be evaluated algebraically by letting $\theta = \arcsin(x)$. Drawing a reference right triangle with opposite side $x$ and hypotenuse 1 gives an adjacent side of $\sqrt{1 - x^2}$. Thus, $\cos(\arcsin(x)) = \sqrt{1 - x^2}$ for $x \in [-1, 1]$.