Question

mozzarella sticks
directions: complete the first task, and then complete the explanation.
if u and v are the measures of complementary angles such is $sin u = \frac{2}{5}$, $\tan v = \frac{sqrt{21}}{2}$, label the sides of the right triangle in the diagram below with the possible side lengths.
explain how you decided the side values.

Explanation:

Step1: Use sine definition for $\sin u$

$\sin u = \frac{\text{opposite to } u}{\text{hypotenuse}} = \frac{2}{5}$
So, opposite side to $u$ = 2, hypotenuse $u$ = 5.

Step2: Calculate adjacent side to $u$

Use Pythagorean theorem: $\text{adjacent}^2 + 2^2 = 5^2$
$\text{adjacent} = \sqrt{5^2 - 2^2} = \sqrt{25-4} = \sqrt{21}$

Step3: Verify $\tan v$ (complementary angles)

For angle $v$, opposite side = $\sqrt{21}$, adjacent side = 2.
$\tan v = \frac{\text{opposite to } v}{\text{adjacent to } v} = \frac{\sqrt{21}}{2}$, which matches the given value.

1. Start with the sine of angle $u$: $\sin u = \frac{2}{5}$ tells us the side opposite $u$ is 2, and the hypotenuse (the side labeled $u$) is 5, since sine equals opposite over hypotenuse in a right triangle.
2. Use the Pythagorean theorem to find the remaining side: $\sqrt{5^2 - 2^2} = \sqrt{21}$. This side is adjacent to angle $u$ and opposite angle $v$.
3. Confirm with $\tan v$: since tangent is opposite over adjacent, $\tan v = \frac{\sqrt{21}}{2}$, which matches the given value, confirming the side lengths are correct.

Answer:

• Side opposite angle $u$: 2
• Side adjacent to angle $u$ (opposite angle $v$): $\sqrt{21}$
• Hypotenuse (side labeled $u$): 5