Answer Key: Trig Test: What Do You Know?

This answer key provides the correct answers and a brief explanation for each question on the Trig Test: What Do You Know? Test.

Question 1

Prompt: What is the value of $\sin(30^\circ)$?
Correct Answer: $\frac{1}{2}$
Thought Process: The sine of $30^\circ$ is a fundamental value derived from the unit circle or a 30-60-90 right triangle. In a 30-60-90 triangle, the side opposite the $30^\circ$ angle is half the hypotenuse.

Question 2

Prompt: Which trigonometric function is defined as the ratio of the adjacent side to the hypotenuse in a right triangle?
Correct Answer: Cosine
Thought Process: This is the definition of the cosine function (SOH CAH TOA).

Question 3

Prompt: If $\tan(\theta) = \frac{3}{4}$ and $\theta$ is in the first quadrant, what is $\sin(\theta)$?
Correct Answer: $\frac{3}{5}$
Thought Process:

1. Since $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}$, we can assume the opposite side is 3 and the adjacent side is 4.
2. Use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find the hypotenuse: $3^2 + 4^2 = c^2 \implies 9 + 16 = c^2 \implies 25 = c^2 \implies c = 5$.
3. Now, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}$.

Question 4

Prompt: What is the reference angle for $210^\circ$?
Correct Answer: $30^\circ$
Thought Process:

1. $210^\circ$ is in the third quadrant (between $180^\circ$ and $270^\circ$).
2. To find the reference angle in the third quadrant, subtract $180^\circ$ from the given angle: $210^\circ - 180^\circ = 30^\circ$.

Question 5

Prompt: A ladder leans against a wall, forming a $60^\circ$ angle with the ground. If the base of the ladder is 5 feet from the wall, how long is the ladder? (Round to two decimal places)
Correct Answer: 10 feet
Thought Process:

1. Draw a right triangle. The ladder is the hypotenuse, the distance from the wall is the adjacent side, and the angle with the ground is $60^\circ$.
2. We know the adjacent side (5 feet) and the angle ($60^\circ$), and we want to find the hypotenuse (ladder length).
3. The trigonometric function that relates adjacent and hypotenuse is cosine: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$.
4. So, $\cos(60^\circ) = \frac{5}{\text{hypotenuse}}$.
5. We know $\cos(60^\circ) = \frac{1}{2}$.
6. Therefore, $\frac{1}{2} = \frac{5}{\text{hypotenuse}} \implies \text{hypotenuse} = 5 \times 2 = 10$ feet.

Question 6

Prompt: Which of the following is equivalent to $\frac{\sin(x)}{\cos(x)}$?
Correct Answer: $\tan(x)$
Thought Process: This is the definition of the tangent identity: $\tan(x) = \frac{\sin(x)}{\cos(x)}$.

Question 7

Prompt: Find the exact value of $\cos(135^\circ)$.
Correct Answer: $-\frac{\sqrt{2}}{2}$
Thought Process:

1. $135^\circ$ is in the second quadrant.
2. The reference angle for $135^\circ$ is $180^\circ - 135^\circ = 45^\circ$.
3. The cosine of the reference angle, $\cos(45^\circ)$, is $\frac{\sqrt{2}}{2}$.
4. In the second quadrant, the cosine value is negative. Therefore, $\cos(135^\circ) = -\frac{\sqrt{2}}{2}$.