Trigonometry and the Pythagorean Theorem

New York State Common Core Math Geometry, Module 2, Lesson 30

Worksheets for Geometry, Module 2, Lesson 30

Student Outcomes

• Students rewrite the Pythagorean Theorem in terms of sine and cosine ratios, and use it in this form to solve problems.
• Students write tangent as an identity in terms of sine and cosine, and use it in this form to solve problems.

Trigonometry and the Pythagorean Theorem

Classwork

Exercises 1–2

1. In a right triangle with acute angle of measure 𝜃, sin 𝜃 = 1/2. What is the value of cos 𝜃? Draw a diagram as part of your response.
2. In a right triangle with acute angle of measure 𝜃, sin 𝜃 = 7/9. What is the value of tan 𝜃? Draw a diagram as part of your response.

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Example 1

a. What common right triangle was probably modeled in the construction of the triangle in Figure 2? Use sin 53° ≈ 0.8. b. The actual angle between the base and lateral faces of the pyramid is actually closer to 52°. Considering the age of the pyramid, what could account for the difference between the angle measure in part (a) and the actual measure? c. Why do you think the architects chose to use a 3–4–5 as a model for the triangle?

Example 2

Show why tan 𝜃 = sinθ/cosθ

Exercises 3–4

3. In a right triangle with acute angle of measure 𝜃, sin 𝜃 = 1/2, use the Pythagorean identity to determine the value of cos 𝜃.
4. Given a right triangle with acute angle of measure 𝜃, sin 𝜃 = 7/9, use the Pythagorean identity to determine the value of tan 𝜃.

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