Question

which shows the correct trig equation?
sin 12 = 20 / 5
cos 5 = 20 / 12
tan 20 = 5 / 12
cos 20 = 5 / 12

Explanation:

Step1: Recall trigonometric ratios

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.
For the given right - triangle with angle $\theta = 20^{\circ}$, the hypotenuse $c = 12$, the adjacent side $a=5$, and the opposite side $b$ (not given in the options).

Step2: Check each option

• Option 1 ($\sin12 = 20/5$):

The angle in a trigonometric ratio should be the angle of the right - triangle (not a side length). Also, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, and the values $20$ and $5$ do not match the ratio formula for sine.

• Option 2 ($\cos5 = 20/12$):

The angle in a trigonometric ratio should be the angle of the right - triangle (not a side length). Also, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, and the values $20$ and $12$ do not match the ratio formula for cosine.

• Option 3 ($\tan20=\frac{5}{12}$):

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, for $\theta = 20^{\circ}$, the opposite side should be the side opposite to the $20^{\circ}$ angle (not $5$) and the adjacent side is $5$. The formula is incorrect.

• Option 4 ($\cos20=\frac{5}{12}$):

Since $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, for $\theta = 20^{\circ}$, the adjacent side $a = 5$ and the hypotenuse $c=12$. So, $\cos20=\frac{5}{12}$ is in line with the cosine ratio formula.

Answer:

$\cos20=\frac{5}{12}$ (the option in the red box)