Solve each problem. (Source for Exercises 49 and 50: Parker, M., Editor, She Does Math, Mathematical Association of America.) Create a right triangle problem whose solution can be found by evaluating θ if sin θ = ¾.

Verified step by step guidance

1. Understand that the problem asks to create a right triangle where the angle \( \theta \) satisfies \( \sin \theta = \frac{3}{4} \). Recall that \( \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \) in a right triangle.
2. Assign the opposite side length as 3 units and the hypotenuse as 4 units, based on the sine ratio \( \sin \theta = \frac{3}{4} \).
3. Use the Pythagorean theorem to find the length of the adjacent side: \( \text{adjacent} = \sqrt{\text{hypotenuse}^2 - \text{opposite}^2} = \sqrt{4^2 - 3^2} \).
4. Express the problem: "In a right triangle, the side opposite angle \( \theta \) is 3 units, and the hypotenuse is 4 units. Find \( \theta \) by evaluating \( \sin^{-1} \left( \frac{3}{4} \right) \)."
5. To solve for \( \theta \), use the inverse sine function: \( \theta = \sin^{-1} \left( \frac{3}{4} \right) \). This step completes the setup for finding the angle \( \theta \).

Key Concepts

Right Triangle and Trigonometric Ratios

A right triangle has one 90-degree angle, and its sides relate through trigonometric ratios like sine, cosine, and tangent. The sine of an angle θ is the ratio of the length of the side opposite θ to the hypotenuse. Understanding this relationship allows solving for unknown sides or angles.

Inverse Sine Function (sin⁻¹ or arcsin)

The inverse sine function is used to find the angle θ when the sine value is known. Given sin θ = 3/4, θ can be found by calculating θ = sin⁻¹(3/4). This function returns an angle in the range of -90° to 90°, which corresponds to the angle in a right triangle.

Constructing a Right Triangle from a Given Ratio

To create a right triangle problem from sin θ = 3/4, assign side lengths consistent with this ratio, such as opposite side = 3 units and hypotenuse = 4 units. Using the Pythagorean theorem, the adjacent side can be found, enabling full characterization of the triangle and solving related problems.