Let’s solve the table row by row using trigonometric ratios.

a) Angle = $37^\circ$, Opposite = 7, Hypotenuse = ?

We use the sine function: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\sin(37^\circ) = \frac{7}{\text{Hypotenuse}}$ $\text{Hypotenuse} = \frac{7}{\sin(37^\circ)} \approx \frac{7}{0.6018} \approx 11.63$

Answer for a): Hypotenuse = 11.63

b) Opposite = 6, Hypotenuse = 8, Angle = ?

We use the sine function again: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\sin(\theta) = \frac{6}{8} = 0.75$ $\theta = \sin^{-1}(0.75) \approx 48.59^\circ$

Answer for b): Angle = $48.59^\circ$

c) Angle = $20^\circ$, Hypotenuse = 15, Adjacent = ?

We use the cosine function: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ $\cos(20^\circ) = \frac{\text{Adjacent}}{15}$ $\text{Adjacent} = 15 \cdot \cos(20^\circ) \approx 15 \cdot 0.9397 \approx 14.10$

Answer for c): Adjacent = 14.10

d) Angle = $45^\circ$, Hypotenuse = 20, Opposite = ?

We use the sine function: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\sin(45^\circ) = \frac{\text{Opposite}}{20}$ $\text{Opposite} = 20 \cdot \sin(45^\circ) \approx 20 \cdot 0.7071 \approx 14.14$

Answer for d): Opposite = 14.14

Final Table

Let me know if you'd like a further breakdown of any calculation or additional details!

Relative Questions:

1. How are trigonometric functions applied in practical scenarios?
2. Can you solve similar problems involving different triangles?
3. What is the significance of rounding to the nearest hundredth in calculations?
4. How does the Pythagorean theorem relate to trigonometric ratios?
5. How can you verify trigonometric calculations using a calculator?

Tip: Always double-check your trigonometric ratio and angle mode (degrees/radians) when solving problems.