Question

mozzarella sticks
directions: complete the first task, and then complete the explanation.
if u and v are the measures of complementary angles such $\sin u = \frac{2}{5}$ $\tan v = \frac{\sqrt{21}}{2}$ , 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 angle $u$

For angle $u$, $\sin u = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2}{5}$. So the side opposite $u$ is $2$, hypotenuse is $5$.

Step2: Verify with Pythagorean theorem

Calculate adjacent side to $u$:
$$\sqrt{5^2 - 2^2} = \sqrt{25-4} = \sqrt{21}$$

Step3: Check tangent for angle $v$

Angle $v$ is complementary to $u$, so $\tan v = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{21}}{2}$, which matches the given value.

Step4: Label the triangle

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

of side values:

1. From $\sin u = \frac{2}{5}$, we identify the side opposite $u$ as $2$ and the hypotenuse as $5$ (since sine = opposite/hypotenuse).
2. Use the Pythagorean theorem $a^2 + b^2 = c^2$ to find the remaining side: $\sqrt{5^2 - 2^2} = \sqrt{21}$.
3. Confirm with $\tan v = \frac{\sqrt{21}}{2}$: since $v$ is complementary to $u$, the side opposite $v$ is $\sqrt{21}$ and adjacent to $v$ is $2$, which matches the given tangent value.

Answer:

Labeled Triangle Sides:

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