Solved examples: Using inverse trigonometric functions for missing angle measures

📚 Quick Study Guide

• 📐 Inverse Trigonometric Functions: These functions ($\arcsin(x)$, $\arccos(x)$, $\arctan(x)$) help you find an angle when you know the ratio of the sides of a right triangle.
• 📝 Range of $\arcsin(x)$: The range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$ or $[-90^{\circ}, 90^{\circ}]$. It gives angles in the first or fourth quadrant.
• 🧭 Range of $\arccos(x)$: The range is $[0, \pi]$ or $[0^{\circ}, 180^{\circ}]$. It gives angles in the first or second quadrant.
• 🧭 Range of $\arctan(x)$: The range is $(-\frac{\pi}{2}, \frac{\pi}{2})$ or $(-90^{\circ}, 90^{\circ})$. It gives angles in the first or fourth quadrant.
• 💡 SOH CAH TOA: Remember this acronym: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. It's crucial for setting up your inverse trig problems.
• ➕ Using a Calculator: Make sure your calculator is in degree or radian mode, depending on the problem's requirements. A wrong mode will give you the wrong answer!

Practice Quiz

1. What is the value of $\arcsin(\frac{1}{2})$ in degrees?
   1. 30°
   2. 45°
   3. 60°
   4. 90°
2. What is the value of $\arccos(\frac{\sqrt{3}}{2})$ in radians?
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{4}$
   3. $\frac{\pi}{3}$
   4. $\frac{\pi}{2}$
3. What is the value of $\arctan(1)$ in degrees?
   1. 30°
   2. 45°
   3. 60°
   4. 90°
4. Find the value of $\arcsin(-\frac{\sqrt{2}}{2})$ in degrees.
   1. 45°
   2. -45°
   3. 135°
   4. -135°
5. What is the value of $\arccos(-\frac{1}{2})$ in radians?
   1. $\frac{\pi}{3}$
   2. $\frac{2\pi}{3}$
   3. $\frac{\pi}{6}$
   4. $\frac{5\pi}{6}$
6. Solve for $x$: $\sin(x) = 0.5$, where $x$ is in degrees and $0° ≤ x ≤ 90°$.
   1. 30°
   2. 45°
   3. 60°
   4. 90°
7. Solve for $x$: $\tan(x) = \sqrt{3}$, where $x$ is in radians and $0 ≤ x ≤ \frac{\pi}{2}$.
   1. $\frac{\pi}{6}$
   2. $\frac{\pi}{4}$
   3. $\frac{\pi}{3}$
   4. $\frac{\pi}{2}$