Question

question 10 of 10
according to this diagram, what is tan 67°?
13
67°
5
23°
90°
12
a. 13/12
b. 13/5
c. 5/12
d. 12/5
e. 12/13
f. 5/13

Explanation:

Step1: Recall tangent definition

In a right triangle, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$ to $\theta$.

Step2: Identify sides for $67^\circ$

For $67^\circ$ angle, opposite side is $12$, adjacent side is $5$? Wait, no—wait, the triangle: right angle, $23^\circ$, $67^\circ$. Wait, let's check the angles. The right angle is $90^\circ$, one angle is $23^\circ$, so the other is $67^\circ$ (since $90 + 23 + 67 = 180$). Now, for angle $67^\circ$: the side opposite to $67^\circ$ is the side opposite, which is the side of length $12$ (since opposite to $67^\circ$: let's see, the angle $23^\circ$ has opposite side $5$, adjacent $12$. Wait, no—wait, angle $67^\circ$: let's label the triangle. The right triangle has vertices: let's say, angle $23^\circ$ (let's call it angle A), right angle (angle C), and angle $67^\circ$ (angle B). So sides: opposite angle A ($23^\circ$) is $5$, adjacent is $12$, hypotenuse $13$. Then angle B is $67^\circ$, so opposite angle B is $12$, adjacent is $5$? Wait, no—wait, in a right triangle, for angle $\theta$, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$. So for angle $67^\circ$: what's opposite? The side opposite $67^\circ$ is the side opposite, which is the side that's not adjacent to $67^\circ$ and not the hypotenuse. Wait, angle $67^\circ$: the sides: the side adjacent to $67^\circ$ is the leg that forms the angle with the hypotenuse, and the opposite is the other leg. Wait, let's re-express: in triangle, angle at the top is $67^\circ$, right angle at the bottom right, angle at bottom left is $23^\circ$. So the sides: bottom leg (adjacent to $23^\circ$) is $12$, vertical leg (opposite to $23^\circ$) is $5$, hypotenuse $13$. Now, for angle $67^\circ$ (top angle): the adjacent side to $67^\circ$ is the vertical leg (length $5$), and the opposite side is the bottom leg (length $12$). So $\tan(67^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5}$. Let's check the options: option D is $\frac{12}{5}$.

Wait, let's confirm: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$. For angle $67^\circ$, the opposite side is the side opposite, which is the side of length $12$ (since angle $67^\circ$: the side opposite is the one that's across from it, which is the horizontal leg of length $12$), and the adjacent side is the vertical leg of length $5$ (since it's adjacent to the $67^\circ$ angle, forming the angle with the hypotenuse). So $\tan(67^\circ) = \frac{12}{5}$, which is option D.

Answer:

D. $\frac{12}{5}$