Question

for the right triangles below, find the exact values of the side lengths d and a. the figures are not drawn to scale. (a) d = (b) a =

Explanation:

Step1: Use cosine in first triangle

In the first right - triangle with hypotenuse 1, angle 30°, and adjacent side $d$. The cosine formula is $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. So, $\cos30^{\circ}=\frac{d}{1}$.
Since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, then $d = \frac{\sqrt{3}}{2}$.

Step2: Use sine in second triangle

In the second right - triangle (a 45 - 45 - 90 triangle) with hypotenuse 1 and opposite side $a$. The sine formula is $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. For a 45° angle in a right - triangle, $\sin45^{\circ}=\frac{a}{1}$.
Since $\sin45^{\circ}=\frac{\sqrt{2}}{2}$, then $a=\frac{\sqrt{2}}{2}$.

Answer:

(a) $d=\frac{\sqrt{3}}{2}$
(b) $a=\frac{\sqrt{2}}{2}$