for the right triangles below, find the exact values of the side lengths h and d. if necessary, write your…

Explanation:

Step1: Find (h) using cosine function

In the first right - triangle with hypotenuse (2) and angle (30^{\circ}), we know that (\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}). Here (\theta = 30^{\circ}), and the adjacent side to the (30^{\circ}) angle is (h). So (h = 2\cos30^{\circ}). Since (\cos30^{\circ}=\frac{\sqrt{3}}{2}), then (h = 2\times\frac{\sqrt{3}}{2}=\sqrt{3}).

Step2: Find (d) using sine function

In the second right - triangle with hypotenuse (7) and angle (45^{\circ}), we know that (\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}). Here (\theta = 45^{\circ}), and the opposite side to the (45^{\circ}) angle is (d). So (d = 7\sin45^{\circ}). Since (\sin45^{\circ}=\frac{\sqrt{2}}{2}), then (d=\frac{7\sqrt{2}}{2}).

Answer:

(h = \sqrt{3}) (d=\frac{7\sqrt{2}}{2})