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.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. For an angle θ, sin(θ) = opposite/hypotenuse. Understanding this ratio is crucial for solving problems involving right triangles, especially when given a specific sine value, such as sin θ = ¾.

Right Triangle Properties

A right triangle is characterized by one angle measuring 90 degrees. The relationships between the angles and sides of a right triangle are governed by trigonometric ratios, including sine, cosine, and tangent. Recognizing these properties allows for the construction of right triangle problems and the application of trigonometric functions to find unknown angles or side lengths.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, are used to determine the angle when the value of a trigonometric function is known. For example, if sin θ = ¾, then θ can be found using θ = arcsin(¾). This concept is essential for solving problems where the angle needs to be calculated from a given sine value.