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56 ? 70 42

56 ? 70 42

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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Find angle? measure

**Right Triangle Analysis** In the diagram, we have a right triangle with the lengths of its sides provided: - The base of the triangle is 42 units. - The perpendicular height is 56 units. - The hypotenuse, which is the longest side opposite the right angle, is 70 units. The right angle is located at the bottom left corner of the triangle. The angle opposite the base labeled 42 is marked with a question mark, indicating that it is the angle we need to find or analyze. Here is a breakdown of how one might approach solving for the unknown angle if required: 1. **Identify Known Quantities:** - Adjacent Side (to the unknown angle): 42 units - Opposite Side (to the unknown angle): 56 units - Hypotenuse: 70 units 2. **Using Trigonometric Ratios:** - **Tangent:** \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{56}{42} = \frac{4}{3}\) - **Sine:** \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{56}{70} = \frac{4}{5}\) - **Cosine:** \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{42}{70} = \frac{3}{5}\) To find the exact value of the angle using a calculator or the inverse trigonometric functions: - \(\theta = \tan^{-1}( \frac{4}{3} )\) - \(\theta = \sin^{-1}( \frac{4}{5} )\) - \(\theta = \cos^{-1}( \frac{3}{5} )\) Given that this is an educational setting, these trigonometric principles and calculations can be used to derive further understanding of the properties and ratios within right-angled triangles. This diagram is a useful aid in visualizing and applying these fundamental principles in trigonometry.
Transcribed Image Text:**Right Triangle Analysis** In the diagram, we have a right triangle with the lengths of its sides provided: - The base of the triangle is 42 units. - The perpendicular height is 56 units. - The hypotenuse, which is the longest side opposite the right angle, is 70 units. The right angle is located at the bottom left corner of the triangle. The angle opposite the base labeled 42 is marked with a question mark, indicating that it is the angle we need to find or analyze. Here is a breakdown of how one might approach solving for the unknown angle if required: 1. **Identify Known Quantities:** - Adjacent Side (to the unknown angle): 42 units - Opposite Side (to the unknown angle): 56 units - Hypotenuse: 70 units 2. **Using Trigonometric Ratios:** - **Tangent:** \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{56}{42} = \frac{4}{3}\) - **Sine:** \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{56}{70} = \frac{4}{5}\) - **Cosine:** \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{42}{70} = \frac{3}{5}\) To find the exact value of the angle using a calculator or the inverse trigonometric functions: - \(\theta = \tan^{-1}( \frac{4}{3} )\) - \(\theta = \sin^{-1}( \frac{4}{5} )\) - \(\theta = \cos^{-1}( \frac{3}{5} )\) Given that this is an educational setting, these trigonometric principles and calculations can be used to derive further understanding of the properties and ratios within right-angled triangles. This diagram is a useful aid in visualizing and applying these fundamental principles in trigonometry.
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