5.4 Inverse trigonometric functions

Sine, cosine, and tangent are restricted to intervals on which they are one-to-one before inverse functions are defined.

5.4.1 Principal values and solving angles

Inverse sine. The function \(\arcsin:[-1,1]\to[-\pi/2,\pi/2]\) satisfying \(\sin(\arcsin x)=x\).

Inverse cosine has range \([0,\pi]\), while inverse tangent has range \((-\pi/2,\pi/2)\). A calculator returns only a principal value. When solving over a larger interval, use reference angles and quadrant signs to find all solutions.

Worked Example: Evaluating an inverse function

\(\arcsin(1/2)=\pi/6\), while \(\arcsin(-1/2)=-\pi/6\), because both lie in the principal range.

Try It. Evaluate \(\arccos(-\sqrt2/2)\).
Check Your Work \(3\pi/4\).

Worked Example: Finding all angles in one revolution

If \(\cos\theta=-0.2250\) for \(0\le\theta<360^\circ\), the reference angle is approximately \(77.0^\circ\). Cosine is negative in quadrants II and III, so \(\theta\approx103.0^\circ\) or \(257.0^\circ\).

Try It. Solve \(\sin\theta=0.6\) for \(0\le\theta<360^\circ\).
Check Your Work \(\theta\approx36.87^\circ\) or \(143.13^\circ\).

Worked Example: Evaluating a composition exactly

If \(\alpha=\arcsin(9/11)\), then a right triangle gives adjacent side \(\sqrt{40}\) and hypotenuse 11. Therefore,

\[\cos(\arcsin(9/11))=\frac{2\sqrt{10}}{11}.\]

Try It. Evaluate \(\tan[\arccos(3/5)]\).
Check Your Work \(4/3\).

Worked Example: Finding a channel angle

A channel rises 1.2 m over 18 m horizontally. Its inclination is

\[\theta=\arctan(1.2/18)\approx3.81^\circ.\]

Try It. Find the inclination for a 0.9 m rise over 25 m.
Check Your Work \(\arctan(0.9/25)\approx2.06^\circ\).
Applied Problem: Locating two possible pipe directions. A pipe direction satisfies \(\sin\theta=-0.35\) for \(0\le\theta<360^\circ\). Find both directions to the nearest tenth of a degree.
Check Your Work The reference angle is \(\arcsin(0.35)\approx20.5^\circ\). Sine is negative in quadrants III and IV, giving \(200.5^\circ\) and \(339.5^\circ\).

5.4.2 Practice Problems

  1. Evaluate \(\arcsin(0)\).
    Check Your Work 0.
  2. Evaluate \(\arccos(1/2)\).
    Check Your Work \(\pi/3\).
  3. Evaluate \(\arctan(1)\).
    Check Your Work \(\pi/4\).
  4. Solve \(\cos\theta=0.4\) on \(0\le\theta<360^\circ\).
    Check Your Work \(\theta\approx66.4^\circ,293.6^\circ\).
  5. Evaluate \(\sin(\arccos(5/13))\).
    Check Your Work \(12/13\).
  6. A 2.5 m rise occurs over 40 m. Find the slope angle.
    Check Your Work \(\arctan(2.5/40)\approx3.58^\circ\).
  7. A tank roof rises 3 m over a half-span of 8 m. Find its angle.
    Check Your Work \(\arctan(3/8)\approx20.6^\circ\).
  8. A force has vertical component 45 N and magnitude 120 N. Find possible directions in one revolution.
    Check Your Work \(\sin\theta=0.375\), giving approximately \(22.0^\circ\) and \(158.0^\circ\).