To find the angle of a right triangle when the hypotenuse and one side length (height) are given, we can use trigonometric functions rather than the Pythagorean theorem alone. Here, we have:

• Hypotenuse $c = 30 \, \text{cm}$
• Opposite side (height) $a = 9.2 \, \text{cm}$

Step 1: Choose the Right Trigonometric Function

To find the angle opposite the height $a$, we can use the sine function, defined as: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

Step 2: Substitute the Values

$\sin(\theta) = \frac{9.2}{30}$ Calculating $\frac{9.2}{30}$: $\sin(\theta) \approx 0.3067$

Step 3: Use the Inverse Sine to Find $\theta$

$\theta = \sin^{-1}(0.3067)$ Using a calculator, we find: $\theta \approx 17.86^\circ$

So, the angle opposite the height (9.2 cm) is approximately 17.86 degrees.

Would you like further details on using trigonometric functions to solve for angles?

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1. How would you find the other angle in the triangle?
2. How can the Pythagorean theorem be used here if two side lengths were given instead?
3. What happens if the height is the longer leg in the triangle?
4. How would you calculate the angle if the base and hypotenuse were given?
5. How can trigonometric functions help in solving non-right triangles?

Tip: The inverse functions (e.g., $\sin^{-1}, \cos^{-1}, \tan^{-1}$) are essential for finding angles when the side ratios are known.