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Calculate b and 0. bl 8 22

Calculate b and 0. bl 8 22

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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### Problem: Calculate \( b \) and \( \theta \). #### Given Triangle: - Right-angled at one of its vertices. - One leg of the right triangle is labeled as \( 8 \) units. - The hypotenuse of the triangle is labeled as \( 22 \) units. - The other leg of the triangle is labeled as \( b \). ### Steps to Solve: 1. **Using Pythagorean Theorem to find \( b \):** The Pythagorean Theorem states: \[ a^2 + b^2 = c^2 \] where \( a \) and \( b \) are the legs of the right triangle and \( c \) is the hypotenuse. In this case: \[ 8^2 + b^2 = 22^2 \] Solving for \( b \): \[ 64 + b^2 = 484 \] \[ b^2 = 420 \] \[ b = \sqrt{420} \] \[ b \approx 20.49 \quad \text{(rounded to two decimal places)} \] 2. **Finding the angle \( \theta \):** To find \( \theta \), we can use trigonometric ratios. Here, we use the sine ratio: \[ \sin(\theta) = \frac{\text{opposite leg}}{\text{hypotenuse}} = \frac{8}{22} \] Solving for \( \theta \): \[ \theta = \sin^{-1}\left(\frac{8}{22}\right) \] \[ \theta \approx \sin^{-1}(0.3636) \] \[ \theta \approx 21.36^\circ \quad \text{(rounded to two decimal places)} \] ### Conclusion: - \( b \approx 20.49 \) - \( \theta \approx 21.36^\circ \)
Transcribed Image Text:### Problem: Calculate \( b \) and \( \theta \). #### Given Triangle: - Right-angled at one of its vertices. - One leg of the right triangle is labeled as \( 8 \) units. - The hypotenuse of the triangle is labeled as \( 22 \) units. - The other leg of the triangle is labeled as \( b \). ### Steps to Solve: 1. **Using Pythagorean Theorem to find \( b \):** The Pythagorean Theorem states: \[ a^2 + b^2 = c^2 \] where \( a \) and \( b \) are the legs of the right triangle and \( c \) is the hypotenuse. In this case: \[ 8^2 + b^2 = 22^2 \] Solving for \( b \): \[ 64 + b^2 = 484 \] \[ b^2 = 420 \] \[ b = \sqrt{420} \] \[ b \approx 20.49 \quad \text{(rounded to two decimal places)} \] 2. **Finding the angle \( \theta \):** To find \( \theta \), we can use trigonometric ratios. Here, we use the sine ratio: \[ \sin(\theta) = \frac{\text{opposite leg}}{\text{hypotenuse}} = \frac{8}{22} \] Solving for \( \theta \): \[ \theta = \sin^{-1}\left(\frac{8}{22}\right) \] \[ \theta \approx \sin^{-1}(0.3636) \] \[ \theta \approx 21.36^\circ \quad \text{(rounded to two decimal places)} \] ### Conclusion: - \( b \approx 20.49 \) - \( \theta \approx 21.36^\circ \)
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