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 [−π/2,π/2][-\pi/2,\pi/2].
  • Inverse cosine: restrict the domain of cosine to the interval [0,π][0,\pi].
  • Inverse tangent: restrict the domain of tangent to the interval (−π/2,π/2)(-\pi/2,\pi/2).

Inverse Sine (arcsin)

  • Definition: arcsin x is the ANGLE in the interval [−π/2,π/2][-\pi/2, \pi/2] whose sine is x.
  • Domain: x∈[−1,1]x \in [-1, 1].
  • Notation: arcsin⁡x\arcsin x or sin⁡−1x\sin^{-1} x.
  • Examples:
    • arcsin⁡(32)=π3\arcsin\left(\dfrac{\sqrt{3}}{2}\right) = \dfrac{\pi}{3}.
    • arcsin⁡(−12)=−π6\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,π][0, \pi] whose cosine is x.
  • Domain: x∈[−1,1]x \in [-1, 1].
  • Notation: arccos⁡x\arccos x or cos⁡−1x\cos^{-1} x.
  • Examples:
    • arccos⁡(0)=π2\arccos(0) = \dfrac{\pi}{2}.
    • arccos⁡(−22)=3π4\arccos\left(-\dfrac{\sqrt{2}}{2}\right) = \dfrac{3\pi}{4}.
  • Important: arccos⁡x\arccos x is undefined for x∉[−1,1]x \notin [-1, 1].

Inverse Tangent (arctan)

  • Definition: arctan x is the ANGLE in the interval (−π/2,π/2)(-\pi/2, \pi/2) whose tangent is x.
  • Domain: x∈Rx \in \mathbb{R}.
  • Notation: arctan⁡x\arctan x or tan⁡−1x\tan^{-1} x.
  • Examples:
    • arctan⁡(1)=π4\arctan(1) = \dfrac{\pi}{4}.
    • arctan⁡(−3)=−π3\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∉[−1,1]-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.