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Use trigonometric identities to transform the left side of the equation into the right side (0 <0< 9 <1/1). cot(0) sin(0) = sin(8) = cos(8) EIN

Use trigonometric identities to transform the left side of the equation into the right side (0 <0< 9 <1/1). cot(0) sin(0) = sin(8) = cos(8) EIN

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 59E
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### Using Trigonometric Identities to Simplify Equations #### Problem Statement Use trigonometric identities to transform the left side of the equation into the right side \(\left(0 < \theta < \frac{\pi}{2}\right)\). #### Given Equation \[ \cot(\theta) \sin(\theta) = \left( \boxed{} \right) \sin(\theta) \] The goal is to simplify the left side to be equal to \(\cos(\theta)\). #### Step-by-Step Solution 1. **Express \(\cot(\theta)\) in terms of sine and cosine:** \[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \] 2. **Substitute \(\cot(\theta)\) in the given equation:** \[ \frac{\cos(\theta)}{\sin(\theta)} \sin(\theta) \] 3. **Simplify the equation:** \[ \cos(\theta) \] Thus, the transformation shows: \[ \cot(\theta) \sin(\theta) = \cos(\theta) \] This verifies the equality under the condition \(0 < \theta < \frac{\pi}{2}\).
Transcribed Image Text:### Using Trigonometric Identities to Simplify Equations #### Problem Statement Use trigonometric identities to transform the left side of the equation into the right side \(\left(0 < \theta < \frac{\pi}{2}\right)\). #### Given Equation \[ \cot(\theta) \sin(\theta) = \left( \boxed{} \right) \sin(\theta) \] The goal is to simplify the left side to be equal to \(\cos(\theta)\). #### Step-by-Step Solution 1. **Express \(\cot(\theta)\) in terms of sine and cosine:** \[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \] 2. **Substitute \(\cot(\theta)\) in the given equation:** \[ \frac{\cos(\theta)}{\sin(\theta)} \sin(\theta) \] 3. **Simplify the equation:** \[ \cos(\theta) \] Thus, the transformation shows: \[ \cot(\theta) \sin(\theta) = \cos(\theta) \] This verifies the equality under the condition \(0 < \theta < \frac{\pi}{2}\).
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