Inverse Trigonometric Functions - Study Notes

Inverse Trigonometric Functions

• Remember that only one-to-one functions have inverses. To obtain inverse functions for sine, cosine, and tangent, we restrict their domains to intervals where they are one-to-one.
• Inverse sine: restrict the domain of sine to the interval $[-\pi/2,\pi/2]$.
• Inverse cosine: restrict the domain of cosine to the interval $[0,\pi]$.
• Inverse tangent: restrict the domain of tangent to the interval $(-\pi/2,\pi/2)$.

Inverse Sine (arcsin)

• Definition: arcsin x is the ANGLE in the interval $[-\pi/2, \pi/2]$ whose sine is x.
• Domain: $x \in [-1, 1]$.
• Notation: $\arcsin x$ or $\sin^{-1} x$.
• Examples:
  • $\arcsin\left(\dfrac{\sqrt{3}}{2}\right) = \dfrac{\pi}{3}$.
  • $\arcsin\left(-\dfrac{1}{2}\right) = -\dfrac{\pi}{6}$.
• Quick check: For x outside [-1,1], arcsin x is undefined.

Inverse Cosine (arccos)

• Definition: arccos x is the ANGLE in the interval $[0, \pi]$ whose cosine is x.
• Domain: $x \in [-1, 1]$.
• Notation: $\arccos x$ or $\cos^{-1} x$.
• Examples:
  • $\arccos(0) = \dfrac{\pi}{2}$.
  • $\arccos\left(-\dfrac{\sqrt{2}}{2}\right) = \dfrac{3\pi}{4}$.
• Important: $\arccos x$ is undefined for $x \notin [-1, 1]$.

Inverse Tangent (arctan)

• Definition: arctan x is the ANGLE in the interval $(-\pi/2, \pi/2)$ whose tangent is x.
• Domain: $x \in \mathbb{R}$.
• Notation: $\arctan x$ or $\tan^{-1} x$.
• Examples:
  • $\arctan(1) = \dfrac{\pi}{4}$.
  • $\arctan(-\sqrt{3}) = -\dfrac{\pi}{3}$.
• Note: the range is open at the endpoints, so ±π/2 are not included.

Combining Trig and Inverse Trig

• Inverse trig expressions return ANGLES. When you compose trig with inverse trig, you are often evaluating a trig function at a special angle.
• Examples of common exact values:
  • $\tan\left(\arcsin x\right) = \dfrac{x}{\sqrt{1 - x^2}}, \quad |x|<1.$
  • $\sin\left(\arccos x\right) = \sqrt{1 - x^2}, \quad x \in [-1,1].$
  • $\cos\left(\arcsin x\right) = \sqrt{1 - x^2}, \quad x \in [-1,1].$
  • $\tan\left(\arccos x\right) = \dfrac{\sqrt{1 - x^2}}{x}, \quad x \in [-1,1], x \neq 0.$
  • $\sin\left(\arctan x\right) = \dfrac{x}{\sqrt{1 + x^2}}, \quad x \in \mathbb{R}.$
  • $\cos\left(\arctan x\right) = \dfrac{1}{\sqrt{1 + x^2}}, \quad x \in \mathbb{R}.$
  • $\tan\left(\arctan x\right) = x, \quad x \in \mathbb{R}.$
• Example: $\tan\left(\arcsin\left(\tfrac{1}{2}\right)\right) = \dfrac{1/2}{\sqrt{1 - (1/2)^2}} = \dfrac{1}{\sqrt{3}}.$
• Caution: $\arccos(-3)$ is undefined, since $-3 \notin [-1,1]$.
• Naming reminder: inverse trig functions produce ANGLES; when you see expressions like (\tan(\sin^{-1} x)) or (\sin(\cos^{-1} y)), evaluate the inner inverse-trig first to get an angle, then apply the outer trig function.