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Find the cosine of ZT.

Find the cosine of ZT.

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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### Example Problem: Finding the Cosine of an Angle in a Right Triangle Consider the following right triangle \( \Delta UST \) where: - \( \angle U \) is a right angle (90 degrees). - \( UT = \sqrt{15} \) (opposite side to angle \( T \)). - \( US = 4 \) (adjacent side to angle \( T \)). - \( TS \) is the hypotenuse. **Objective:** Find the cosine of \( \angle T \). **Solution:** The cosine of an angle in a right triangle is given by the ratio of the length of the adjacent side to the hypotenuse. 1. First, calculate the length of the hypotenuse \( TS \) using the Pythagorean theorem: \[ TS = \sqrt{(UT)^2 + (US)^2} \] Substituting the given values: \[ TS = \sqrt{(\sqrt{15})^2 + 4^2} \] \[ TS = \sqrt{15 + 16} \] \[ TS = \sqrt{31} \] 2. Next, find \( \cos(T) \): \[ \cos(T) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{US}{TS} \] \[ \cos(T) = \frac{4}{\sqrt{31}} \] 3. To write the answer in simplified, rationalized form: \[ \cos(T) = \frac{4}{\sqrt{31}} \times \frac{\sqrt{31}}{\sqrt{31}} = \frac{4\sqrt{31}}{31} \] So, \[ \cos(T) = \frac{4\sqrt{31}}{31} \] ### Answer: \[ \cos(T) = \frac{4\sqrt{31}}{31} \] **Note:** For the given answer entry box, ensure that expressions are entered in the rationalized form as shown above.
Transcribed Image Text:### Example Problem: Finding the Cosine of an Angle in a Right Triangle Consider the following right triangle \( \Delta UST \) where: - \( \angle U \) is a right angle (90 degrees). - \( UT = \sqrt{15} \) (opposite side to angle \( T \)). - \( US = 4 \) (adjacent side to angle \( T \)). - \( TS \) is the hypotenuse. **Objective:** Find the cosine of \( \angle T \). **Solution:** The cosine of an angle in a right triangle is given by the ratio of the length of the adjacent side to the hypotenuse. 1. First, calculate the length of the hypotenuse \( TS \) using the Pythagorean theorem: \[ TS = \sqrt{(UT)^2 + (US)^2} \] Substituting the given values: \[ TS = \sqrt{(\sqrt{15})^2 + 4^2} \] \[ TS = \sqrt{15 + 16} \] \[ TS = \sqrt{31} \] 2. Next, find \( \cos(T) \): \[ \cos(T) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{US}{TS} \] \[ \cos(T) = \frac{4}{\sqrt{31}} \] 3. To write the answer in simplified, rationalized form: \[ \cos(T) = \frac{4}{\sqrt{31}} \times \frac{\sqrt{31}}{\sqrt{31}} = \frac{4\sqrt{31}}{31} \] So, \[ \cos(T) = \frac{4\sqrt{31}}{31} \] ### Answer: \[ \cos(T) = \frac{4\sqrt{31}}{31} \] **Note:** For the given answer entry box, ensure that expressions are entered in the rationalized form as shown above.
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