# Solving Problems Using Sine and Cosine

### New York State Common Core Math Geometry, Module 2, Lesson 28

Worksheets for Geometry, Module 2, Lesson 28

Student Outcomes

• Students use graphing calculator to find the values of sin ⁡θ and cos ⁡θ for θ between 0 and 90.
• Students solve for missing sides of a right triangle given the length of one side and the measure of one of the acute angles.
• Students find the length of the base of a triangle with acute base angles given the lengths of the other two sides and the measure of each of the base angles.

Solving Problems Using Sine and Cosine

Classwork

Exercises 1–4

1. a. The bus drops you off at the corner of H Street and 1st Street, approximately 300 ft. from school. You plan to walk to your friend Janneth’s house after school to work on a project. Approximately how many feet will you have to walk from school to Janneth’s house? Round your answer to the nearest foot. (Hint: Use the ratios you developed in Lesson 25.)
b. In real life, it is unlikely that you would calculate the distance between school and Janneth’s house in this manner. Describe a similar situation in which you might actually want to determine the distance between two points using a trigonometric ratio.
2. Use a calculator to find the sine and cosine of 𝜃. Give your answer rounded to the ten-thousandth place.
3. What do you notice about the numbers in the row sin 𝜃 compared with the numbers in the row cos 𝜃?
4. Provide an explanation for what you noticed in Exercise 2.

Example 1

Find the values of 𝑎 and 𝑏.

Exercise 5

1. A shipmate set a boat to sail exactly 27° NE from the dock. After traveling 120 miles, the shipmate realized he had misunderstood the instructions from the captain; he was supposed to set sail going directly east!
a. How many miles will the shipmate have to travel directly south before he is directly east of the dock? Round your answer to the nearest mile.
b. How many extra miles does the shipmate travel by going the wrong direction compared to going directly east? Round your answer to the nearest mile.

Example 2

Johanna borrowed some tools from a friend so that she could precisely, but not exactly, measure the corner space in her backyard to plant some vegetables. She wants to build a fence to prevent her dog from digging up the seeds that she plants. Johanna returned the tools to her friend before making the most important measurement: the one that would give the length of the fence! Johanna decided that she could just use the Pythagorean theorem to find the length of the fence she would need. Is the Pythagorean theorem applicable in this situation? Explain.

Exercise 6

1. The measurements of the triangle shown below are rounded to the nearest hundredth. Calculate the missing side length to the nearest hundredth.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 