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Chapter 5: Problem 73

Simplify the expression algebraically and use a graphing utility to confirm your answer graphically. $$\sin \left(\frac{3 \pi}{2}+\theta\right)$$

Short Answer

Expert verified
- cos(\theta)

Step by step solution

01

Apply the Trigonometric Addition Identity

The given expression is \( \sin \left(\frac{3 \pi}{2}+\theta\right) \). Use the trigonometric addition identity on this expression, which states \( \sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b) \). In our case, \( a = \frac{3\pi}{2} \) and \( b = \theta \). So we get: \( \sin \left(\frac{3 \pi}{2}\right)\cos(\theta) + \cos \left(\frac{3\pi}{2}\right)\sin(\theta) \).
02

Evaluate Trigonometric Functions

We know that \( \sin{\frac{3\pi}{2}} = -1 \) and \( \cos{\frac{3\pi}{2}} = 0 \). Substituting these values we get: \( -1 \cdot \cos(\theta) + 0 \cdot \sin(\theta) \).
03

Simplify the Expression

Simplifying this expression, we obtain - cos(\theta).
04

Verify Graphically

Graph the original expression \( \sin \left(\frac{3 \pi}{2}+\theta\right) \) and the simplified form - cos(\theta) on the same set of axes. The two plots should coincide, confirming that the simplification is correct. Trial values of theta may be chosen to test this.

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