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A right triangle has side lengths 8, 15, and 17 as shown below. Use these lengths to find sin X, tan X, and cosX. X 15 za 8 17 sin X = tan X = cos X= 0 8 08 EEE D U Aal Aa

A right triangle has side lengths 8, 15, and 17 as shown below. Use these lengths to find sin X, tan X, and cosX. X 15 za 8 17 sin X = tan X = cos X= 0 8 08 EEE D U Aal Aa

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.
Chapter1: Line And Angle Relationships
Section1.2: Angles And Their Relationships
Problem 37E: Draw a triangle with three acute angles. Construct angle bisectors for each of the three angles. On...
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## Understanding Trigonometric Ratios in a Right Triangle A **right triangle** has side lengths \(8, 15, \) and \(17\) as shown below. Use these lengths to find \(\sin X\), \(\tan X\), and \(\cos X\). ### Diagram Explanation The right triangle is labeled with vertices \(X\), \(Y\), and \(Z\), where \(\angle Z\) is the right angle. - The side opposite \(\angle X\) (denoted as \(YZ\)) is \(8\) units. - The side adjacent to \(\angle X\) (denoted as \(ZX\)) is \(15\) units. - The hypotenuse \(XY\) is \(17\) units. ### Calculation of Trigonometric Ratios To find the trigonometric ratios (sine, tangent, and cosine) for \(\angle X\), we use the following definitions: - \(\sin X = \frac{\text{opposite}}{\text{hypotenuse}}\) - \(\tan X = \frac{\text{opposite}}{\text{adjacent}}\) - \(\cos X = \frac{\text{adjacent}}{\text{hypotenuse}}\) 1. **Sine (\(\sin X\))**: \[ \sin X = \frac{8}{17} \] 2. **Tangent (\(\tan X\))**: \[ \tan X = \frac{8}{15} \] 3. **Cosine (\(\cos X\))**: \[ \cos X = \frac{15}{17} \] ### Summary This problem demonstrates how to use the side lengths of a right triangle to calculate the primary trigonometric ratios for one of its angles.
Transcribed Image Text:## Understanding Trigonometric Ratios in a Right Triangle A **right triangle** has side lengths \(8, 15, \) and \(17\) as shown below. Use these lengths to find \(\sin X\), \(\tan X\), and \(\cos X\). ### Diagram Explanation The right triangle is labeled with vertices \(X\), \(Y\), and \(Z\), where \(\angle Z\) is the right angle. - The side opposite \(\angle X\) (denoted as \(YZ\)) is \(8\) units. - The side adjacent to \(\angle X\) (denoted as \(ZX\)) is \(15\) units. - The hypotenuse \(XY\) is \(17\) units. ### Calculation of Trigonometric Ratios To find the trigonometric ratios (sine, tangent, and cosine) for \(\angle X\), we use the following definitions: - \(\sin X = \frac{\text{opposite}}{\text{hypotenuse}}\) - \(\tan X = \frac{\text{opposite}}{\text{adjacent}}\) - \(\cos X = \frac{\text{adjacent}}{\text{hypotenuse}}\) 1. **Sine (\(\sin X\))**: \[ \sin X = \frac{8}{17} \] 2. **Tangent (\(\tan X\))**: \[ \tan X = \frac{8}{15} \] 3. **Cosine (\(\cos X\))**: \[ \cos X = \frac{15}{17} \] ### Summary This problem demonstrates how to use the side lengths of a right triangle to calculate the primary trigonometric ratios for one of its angles.
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