Question

unit 2: full review (triangles) | wayground
12.
solve for x:
a) 19 cos (40)=14.55
b) 19 sin (40)=12.21
c) 19 tan (40)=15.94
13.
solve for a:
a) 11 sin (43)=7.50
b) 11 cos (43)=8.04
c) 11 tan (43)=10.26

Explanation:

Step1: Recall sine - cosine - tangent relations

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Solve for $x$ in the first triangle

In the first right - triangle with hypotenuse $19$ and an angle of $40^{\circ}$, and $x$ is the side opposite to the $40^{\circ}$ angle. Using the sine relation $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, we have $\sin(40^{\circ})=\frac{x}{19}$, so $x = 19\sin(40^{\circ})$.

Step3: Solve for $a$ in the second triangle

In the second right - triangle with adjacent side to the $43^{\circ}$ angle equal to $11$ and $a$ is the opposite side to the $43^{\circ}$ angle. Using the tangent relation $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, we have $\tan(43^{\circ})=\frac{a}{11}$, so $a = 11\tan(43^{\circ})$.

Answer:

For the first problem, the correct option is b) $19\sin(40)=12.21$.
For the second problem, the correct option is c) $11\tan(43)=10.26$.