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. insert step 1> Identify that you are given \( \sin \theta = \frac{3}{4} \). This means in a right triangle, the opposite side to angle \( \theta \) is 3 units, and the hypotenuse is 4 units.
2. insert step 2> Use the Pythagorean theorem to find the length of the adjacent side. The theorem states \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.
3. insert step 3> Substitute the known values into the Pythagorean theorem: \( a^2 + 3^2 = 4^2 \).
4. insert step 4> Solve for \( a \) by simplifying the equation: \( a^2 + 9 = 16 \), then \( a^2 = 7 \), and finally \( a = \sqrt{7} \).
5. insert step 5> Now, you can find \( \theta \) using the inverse sine function: \( \theta = \sin^{-1}(\frac{3}{4}) \).

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.