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6. Use the triangle to find sin(20), cos(20), sin(), cos 7. Write the exact answer for cos (2) si sin ( 8. Write the exact answer for sin () - sin e 17 15 R\ L 8 00 09

6. Use the triangle to find sin(20), cos(20), sin(), cos 7. Write the exact answer for cos (2) si sin ( 8. Write the exact answer for sin () - sin e 17 15 R\ L 8 00 09

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.2: Graphs Of Equations
Problem 40E
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## Mathematical Problems and Solutions ### Problem 6: **Use the triangle to find \(\sin(2\theta), \cos(2\theta), \sin\left(\frac{\pi}{4}\right), \cos\left(\frac{\pi}{2}\right)** Below is a right triangle with the following properties: - Hypotenuse \( AC = 17 \) - Opposite side to angle \( \theta \), \( AB = 15 \) - Adjacent side to angle \( \theta \), \( BC = 8 \) ``` C /| / | / | / | / | A-----B ``` #### Solution Details: - **\(\sin(\theta)\)** is found by the ratio of the opposite side to the hypotenuse: \[\sin(\theta) = \frac{AB}{AC} = \frac{15}{17}\] - **\(\cos(\theta)\)** is found by the ratio of the adjacent side to the hypotenuse: \[\cos(\theta) = \frac{BC}{AC} = \frac{8}{17}\] Using double-angle identities to find \(\sin(2\theta)\) and \(\cos(2\theta)\): - **\(\sin(2\theta)\)** \[\sin(2\theta) = 2 \sin(\theta) \cos(\theta) = 2 \left(\frac{15}{17}\right) \left(\frac{8}{17}\right) = 2 \left(\frac{120}{289}\right) = \frac{240}{289}\] - **\(\cos(2\theta)\)** \[\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = \left(\frac{8}{17}\right)^2 - \left(\frac{15}{17}\right)^2 = \frac{64}{289} - \frac{225}{289} = \frac{64 - 225}{289} = -\frac{161}{289}\] - **\(\sin\left(\frac{\pi}{4}\right)\)** \[\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\
Transcribed Image Text:## Mathematical Problems and Solutions ### Problem 6: **Use the triangle to find \(\sin(2\theta), \cos(2\theta), \sin\left(\frac{\pi}{4}\right), \cos\left(\frac{\pi}{2}\right)** Below is a right triangle with the following properties: - Hypotenuse \( AC = 17 \) - Opposite side to angle \( \theta \), \( AB = 15 \) - Adjacent side to angle \( \theta \), \( BC = 8 \) ``` C /| / | / | / | / | A-----B ``` #### Solution Details: - **\(\sin(\theta)\)** is found by the ratio of the opposite side to the hypotenuse: \[\sin(\theta) = \frac{AB}{AC} = \frac{15}{17}\] - **\(\cos(\theta)\)** is found by the ratio of the adjacent side to the hypotenuse: \[\cos(\theta) = \frac{BC}{AC} = \frac{8}{17}\] Using double-angle identities to find \(\sin(2\theta)\) and \(\cos(2\theta)\): - **\(\sin(2\theta)\)** \[\sin(2\theta) = 2 \sin(\theta) \cos(\theta) = 2 \left(\frac{15}{17}\right) \left(\frac{8}{17}\right) = 2 \left(\frac{120}{289}\right) = \frac{240}{289}\] - **\(\cos(2\theta)\)** \[\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = \left(\frac{8}{17}\right)^2 - \left(\frac{15}{17}\right)^2 = \frac{64}{289} - \frac{225}{289} = \frac{64 - 225}{289} = -\frac{161}{289}\] - **\(\sin\left(\frac{\pi}{4}\right)\)** \[\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\
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