Question

use the diagram shown to answer the question. right triangle with right angle, 33° angle, hypotenuse 19, side ( y ) which expression represents the value of ( y )? (\boldsymbol{\text{option a: } \frac{19}{sin 33^circ}}) (\boldsymbol{\text{option b: } \frac{19}{cos 33^circ}}) (\boldsymbol{\text{option c: } 19 sin 33^circ}) (\boldsymbol{\text{option d: } 19 cos 33^circ})

Explanation:

Step1: Recall trigonometric ratios

In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse, and the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse. The formulas are:
$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$.

Step2: Identify sides relative to $33^\circ$

In the given right triangle, the hypotenuse is $19$, the angle is $33^\circ$, and $y$ is the side opposite to the $33^\circ$ angle.

Step3: Apply the sine ratio

Using the sine ratio $\sin(33^\circ)=\frac{y}{19}$, we can solve for $y$ by multiplying both sides by $19$:
$y = 19\sin(33^\circ)$.

Answer:

C. $19\sin33^\circ$