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Lloyd's work to find the missing angle measure (to the nearest degree) is shown. What mistake did Lloyd make? 9 units Lloyd's work: = X COS = 1° COS ܙܩ ܩ ܙܩ 5 units X

Lloyd's work to find the missing angle measure (to the nearest degree) is shown. What mistake did Lloyd make? 9 units Lloyd's work: = X COS = 1° COS ܙܩ ܩ ܙܩ 5 units X

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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### Finding the Missing Angle of a Right Triangle (Example Problem) Lloyd's attempt to find the missing angle measure (to the nearest degree) is shown. Review his work and identify the mistake he made. #### Lloyd's Work: ``` cos (5/9) = x cos (5/9) = 1° ``` #### Diagram: We are given a right triangle where: - The hypotenuse is 9 units. - The adjacent side to the angle \( x \) is 5 units. #### Analysis: Lloyd is attempting to use the cosine function, which is defined as: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \] Given: \[ \cos(x) = \frac{5}{9} \] The correct method to find the angle \( x \) is to use the inverse cosine (arccos) function: \[ x = \cos^{-1}\left(\frac{5}{9}\right) \] #### Mistake Identified: Lloyd did not actually use the inverse cosine function (cosine inverse or arccos). Instead, he incorrectly interpreted \( \cos (5/9) \) as an angle measure, which is not mathematically appropriate. Moreover, cosine values must always be between -1 and 1, and thus the cosine of a ratio like 5/9 should not yield an angle directly. #### Corrected Work: Apply the inverse cosine to find the angle \( x \): \[ x = \cos^{-1}\left(\frac{5}{9}\right) \] Using a calculator: \[ x \approx \cos^{-1}(0.5556) \approx 56 \text{ degrees} \] #### Conclusion: Lloyd's mistake was not applying the inverse cosine function to find the angle. The correct measure of angle \( x \) is approximately 56 degrees.
Transcribed Image Text:### Finding the Missing Angle of a Right Triangle (Example Problem) Lloyd's attempt to find the missing angle measure (to the nearest degree) is shown. Review his work and identify the mistake he made. #### Lloyd's Work: ``` cos (5/9) = x cos (5/9) = 1° ``` #### Diagram: We are given a right triangle where: - The hypotenuse is 9 units. - The adjacent side to the angle \( x \) is 5 units. #### Analysis: Lloyd is attempting to use the cosine function, which is defined as: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \] Given: \[ \cos(x) = \frac{5}{9} \] The correct method to find the angle \( x \) is to use the inverse cosine (arccos) function: \[ x = \cos^{-1}\left(\frac{5}{9}\right) \] #### Mistake Identified: Lloyd did not actually use the inverse cosine function (cosine inverse or arccos). Instead, he incorrectly interpreted \( \cos (5/9) \) as an angle measure, which is not mathematically appropriate. Moreover, cosine values must always be between -1 and 1, and thus the cosine of a ratio like 5/9 should not yield an angle directly. #### Corrected Work: Apply the inverse cosine to find the angle \( x \): \[ x = \cos^{-1}\left(\frac{5}{9}\right) \] Using a calculator: \[ x \approx \cos^{-1}(0.5556) \approx 56 \text{ degrees} \] #### Conclusion: Lloyd's mistake was not applying the inverse cosine function to find the angle. The correct measure of angle \( x \) is approximately 56 degrees.
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