Inverse Trigonometric Functions Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Simplify sin⁡(arccos⁡(x))\sin(\arccos(x)) for x∈[−1,1]x\in[-1,1] without trigonometric functions in the final answer.

Solution

  1. 1
    Let θ=arccos⁡(x)\theta=\arccos(x), so cos⁡θ=x\cos\theta=x and θ∈[0,π]\theta\in[0,\pi].
  2. 2
    Use the Pythagorean identity: sin⁡2θ+cos⁡2θ=1⇒sin⁡2θ=1−x2⇒sin⁡θ=±1−x2\sin^2\theta+\cos^2\theta=1 \Rightarrow \sin^2\theta=1-x^2 \Rightarrow \sin\theta=\pm\sqrt{1-x^2}.
  3. 3
    Since θ∈[0,π]\theta\in[0,\pi], sin⁡θ≥0\sin\theta\geq0. Therefore sin⁡(arccos⁡(x))=1−x2\sin(\arccos(x))=\sqrt{1-x^2}.

Answer

sin⁡(arccos⁡(x))=1−x2\sin(\arccos(x))=\sqrt{1-x^2}
To simplify trig composed with inverse trig, introduce a variable for the angle, use the defining equation, then apply a Pythagorean identity. The range of the inverse function is crucial for choosing the correct sign of the square root.

About Inverse Trigonometric Functions

Functions that reverse the trigonometric functions: given a ratio, they return the corresponding angle. arcsin⁡\arcsin, arccos⁡\arccos, and arctan⁡\arctan are the inverses of sin⁡\sin, cos⁡\cos, and tan⁡\tan on restricted domains.

Learn more about Inverse Trigonometric Functions →

More Inverse Trigonometric Functions Examples

Example 1 easy

Evaluate [formula], [formula], and [formula]. State the range of each inverse trig function.

Example 3 easy

A right triangle has opposite side [formula] and hypotenuse [formula]. Find the angle [formula] oppo

Example 4 medium

Solve [formula] on [formula] using [formula], and explain why there are two solutions.

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