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Chapter 5: Problem 67

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
The simplified form of the expression is - cos(胃)

Step by step solution

01

Simplify trigonometric expression

The expression \(\sin \left(\frac{3 \pi}{2}+\theta\right)\) can be simplified using the addition formula in trigonometry, which states that \(\sin(a + b) = \sin(a) * \cos(b) + \cos(a) * \sin(b)\). Applied to the given expression, it transforms into: \(-\cos(\theta)\)
02

Use a graphing utility to confirm

Once simplified, the expression should be plotted using a graphing tool. On the same graph, plot both the original equation \(\sin \left(\frac{3 \pi}{2}+\theta\right)\) and the simplified one \(-\cos(\theta)\). If both plots overlap on the graphing tool, this confirms the validity of the simplification.

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Most popular questions from this chapter

Verify the identity. $$\cos \left(\frac{\pi}{3}+x\right)+\cos \left(\frac{\pi}{3}-x\right)=\cos x$$

Verify the identity. $$a \sin B \theta+b \cos B \theta=\sqrt{a^{2}+b^{2}} \sin (B \theta+C)\( where \)C=\arctan (b / a)\( and \)a>0$$

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$75^{\circ}$$

Determine whether the statement is true or false. Justify your answer. \(\sin \frac{u}{2}=-\sqrt{\frac{1-\cos u}{2}}\) when \(u\) is in the second quadrant.

Proof (a) Write a proof of the formula for \(\sin (u+v)\) (b) Write a proof of the formula for \(\sin (u-v)\)
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