Practice Problems

1. Bob said that

   $$\sin(30^\circ) = 0.5^\circ$$

   His teacher gave him a weird look. What did Bob say incorrectly? Rewrite the statement so it is correct.

2. True or False: All of the angles in a triangle add up to $180^\circ$ only for right triangles.

3. Write a general expression for all coterminal angles of the angle $\theta$:
   a) in degrees
   b) in radians

4. True or False: The reference angle of $-30^\circ$ is $-30^\circ$.

5. True or False: The reference angle of $89^\circ$ is $89^\circ$.

Solutions

1.
Bob’s mistake: He wrote $0.5^\circ$ instead of $0.5$.
Trig functions output numbers without units, not angles.
Correct statement:

$$\sin(30^\circ) = 0.5$$

Inverse trig functions, like $\arcsin$, do the opposite: they take a number (no units) and return an angle.

2.
False. The sum of the angles in ANY triangle is $180^\circ$, not just right triangles.

3.
Degrees:

$$\theta + 360^\circ n, \quad n \in \mathbb{Z}$$

Radians:

$$\theta + 2\pi n, \quad n \in \mathbb{Z}$$

4.
False. Reference angles are always positive acute angles between the terminal side and the x-axis.
The reference angle of $-30^\circ$ is $30^\circ$.

5.
True. $89^\circ$ is already acute, so its reference angle is itself.

$$\frac{\pi}{3} + 2\pi n, \quad n \in \mathbb{Z}$$