Find sec0, cot0, and cos0, where 0 is the angle shown in the figure. Give exact values, not decimal approximations. sec 0 %3D 11 cot e cos 0

Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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**Problem Statement:**

**Objective:**  
Find \( \sec \theta \), \( \cot \theta \), and \( \cos \theta \), where \( \theta \) is the angle shown in the figure. Provide exact values, not decimal approximations.

**Diagram Description:**

The diagram depicts a right triangle with:

- A hypotenuse of length 11 units.
- An opposite side of length 6 units.
- The angle \( \theta \) at the bottom left of the triangle.

**Trigonometric Functions:**

1. **\( \sec \theta \) (Secant of theta):**  
   This is the reciprocal of the cosine function.  
   Formula: \( \sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} \)

2. **\( \cot \theta \) (Cotangent of theta):**  
   This is the reciprocal of the tangent function.  
   Formula: \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \)

3. **\( \cos \theta \) (Cosine of theta):**  
   Formula: \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

**Calculations:**

To use these formulas, the length of the adjacent side needs to be found using the Pythagorean theorem:

\[ a^2 + b^2 = c^2 \]
\[ a^2 + 6^2 = 11^2 \]
\[ a^2 + 36 = 121 \]
\[ a^2 = 85 \]
\[ a = \sqrt{85} \]

Now, substitute the values into the trigonometric formulas:

- \( \sec \theta = \frac{11}{\sqrt{85}} \)
- \( \cot \theta = \frac{\sqrt{85}}{6} \)
- \( \cos \theta = \frac{\sqrt{85}}{11} \)
Transcribed Image Text:**Problem Statement:** **Objective:** Find \( \sec \theta \), \( \cot \theta \), and \( \cos \theta \), where \( \theta \) is the angle shown in the figure. Provide exact values, not decimal approximations. **Diagram Description:** The diagram depicts a right triangle with: - A hypotenuse of length 11 units. - An opposite side of length 6 units. - The angle \( \theta \) at the bottom left of the triangle. **Trigonometric Functions:** 1. **\( \sec \theta \) (Secant of theta):** This is the reciprocal of the cosine function. Formula: \( \sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} \) 2. **\( \cot \theta \) (Cotangent of theta):** This is the reciprocal of the tangent function. Formula: \( \cot \theta = \frac{\text{adjacent}}{\text{opposite}} \) 3. **\( \cos \theta \) (Cosine of theta):** Formula: \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \) **Calculations:** To use these formulas, the length of the adjacent side needs to be found using the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] \[ a^2 + 6^2 = 11^2 \] \[ a^2 + 36 = 121 \] \[ a^2 = 85 \] \[ a = \sqrt{85} \] Now, substitute the values into the trigonometric formulas: - \( \sec \theta = \frac{11}{\sqrt{85}} \) - \( \cot \theta = \frac{\sqrt{85}}{6} \) - \( \cos \theta = \frac{\sqrt{85}}{11} \)
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