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⁡(arccos⁡x)=1−x2,x∈[−1,1].\sin\left(\arccos x\right) = \sqrt{1 - x^2}, \quad x \in [-1,1].
    • cos⁡(arcsin⁡x)=1−x2,x∈[−1,1].\cos\left(\arcsin x\right) = \sqrt{1 - x^2}, \quad x \in [-1,1].
    • tan⁡(arccos⁡x)=1−x2x,x∈[−1,1],x≠0.\tan\left(\arccos x\right) = \dfrac{\sqrt{1 - x^2}}{x}, \quad x \in [-1,1], x \neq 0.
    • sin⁡(arctan⁡x)=x1+x2,x∈R.\sin\left(\arctan x\right) = \dfrac{x}{\sqrt{1 + x^2}}, \quad x \in \mathbb{R}.
    • cos⁡(arctan⁡x)=11+x2,x∈R.\cos\left(\arctan x\right) = \dfrac{1}{\sqrt{1 + x^2}}, \quad x \in \mathbb{R}.
    • tan⁡(arctan⁡x)=x,x∈R.\tan\left(\arctan x\right) = x, \quad x \in \mathbb{R}.
  • Example: tan⁡(arcsin⁡(12))=1/21−(1/2)2=13.\tan\left(\arcsin\left(\tfrac{1}{2}\right)\right) = \dfrac{1/2}{\sqrt{1 - (1/2)^2}} = \dfrac{1}{\sqrt{3}}.
  • Caution: arccos⁡(−3)\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.