for the right triangles below, find the exact values of the side lengths $a$ and $b$.\nif necessary, write…

for the right triangles below, find the exact values of the side lengths $a$ and $b$.\nif necessary, write your responses in simplified radical form.\n$a = \\square$\n$b = \\square$

for the right triangles below, find the exact values of the side lengths $a$ and $b$.\nif necessary, write your responses in simplified radical form.\n$a = \\square$\n$b = \\square$

Answer

Explanation:

Step1: Find $a$ using cosine

$\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{2}{a}$ Since $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, substitute: $\frac{\sqrt{3}}{2} = \frac{2}{a}$ Solve for $a$: $a = \frac{2 \times 2}{\sqrt{3}} = \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3}$

Step2: Find $b$ using cosine

$\cos(45^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{b}$ Since $\cos(45^\circ) = \frac{\sqrt{2}}{2}$, substitute: $\frac{\sqrt{2}}{2} = \frac{5}{b}$ Solve for $b$: $b = \frac{5 \times 2}{\sqrt{2}} = \frac{10}{\sqrt{2}} = 5\sqrt{2}$

Answer:

$a = \frac{4\sqrt{3}}{3}$ $b = 5\sqrt{2}$