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Chapter 6: Problem 29

Find a formula for \(\cos \left(\theta+\frac{\pi}{2}\right)\).

Short Answer

Expert verified
The formula for \(\cos\left(\theta+\frac{\pi}{2}\right)\) is: \(\cos\left(\theta+\frac{\pi}{2}\right) = -\sin(\theta)\).

Step by step solution

01

Write down the angle addition formula for cosine

Our starting point is the angle addition formula for cosine: \[\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)\]
02

Replace \(\alpha\) with \(\theta\) and \(\beta\) with \(\frac{\pi}{2}\)

Now we replace \(\alpha\) with \(\theta\) and \(\beta\) with \(\frac{\pi}{2}\) in the formula: \[\cos(\theta + \frac{\pi}{2}) = \cos(\theta)\cos\left(\frac{\pi}{2}\right) - \sin(\theta)\sin\left(\frac{\pi}{2}\right)\]
03

Use trigonometric values of the angles

Recall that \(\cos\left(\frac{\pi}{2}\right) = 0\) and \(\sin\left(\frac{\pi}{2}\right) = 1\). Now we substitute these values into the formula: \[\cos(\theta + \frac{\pi}{2}) = \cos(\theta)(0) - \sin(\theta)(1)\]
04

Simplify

Now we simplify the formula: \[\cos(\theta + \frac{\pi}{2}) = - \sin(\theta)\] So, the formula for \(\cos \left(\theta+\frac{\pi}{2}\right)\) is: \[\cos \left(\theta+\frac{\pi}{2}\right) = - \sin(\theta)\]

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Most popular questions from this chapter

Convert the rectangular coordinates given for each point to polar coordinates \(r\) and \(\theta .\) Use radians, and always choose the angle to be in the interval \((-\pi, \pi)\). $$ (3,-7) $$

Convert the polar coordinates given for each point to rectangular coordinates in the \(x y\) -plane. $$ r=8, \theta=\frac{\pi}{3} $$

What is the relationship between the point with polar coordinates \(r=5, \theta=0.2\) and the point with polar coordinates \(r=5, \theta=-0.2 ?\)

Convert the polar coordinates given for each point to rectangular coordinates in the \(x y\) -plane. $$ r=5, \theta=-\frac{\pi}{2} $$

Convert the polar coordinates given for each point to rectangular coordinates in the \(x y\) -plane. $$ r=\sqrt{19}, \theta=5 \pi $$
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