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Chapter 5: Problem 81

Find angles \(u\) and \(v\) such that \(\cos (2 u)=\cos (2 v)\) but \(\cos u \neq \cos v\).

Short Answer

Expert verified
A pair of suitable angles \(u\) and \(v\) that satisfy the given conditions are \(u = 60^{\circ}\) and \(v = 120^{\circ}\).

Step by step solution

01

Write the double-angle formula for cosine

The double-angle formula for cosine is given by: \[\cos(2\theta) = 2\cos^2(\theta) - 1\] Now, apply this formula to both expressions: \[\cos(2u) = 2\cos^2(u) - 1\] \[\cos(2v) = 2\cos^2(v) - 1\]
02

Set expressions equal

Since \(\cos(2u) = \cos(2v)\), we can set the two expressions from Step 1 equal to each other: \[2\cos^2(u) - 1 = 2\cos^2(v) - 1\]
03

Isolate terms for \(\cos(u)\) and \(\cos(v)\)

Add 1 to both sides of the equation: \[2\cos^2(u)= 2\cos^2(v)\] Divide both sides by 2: \[\cos^2(u) = \cos^2(v)\] Take the square root of both sides. Keep in mind that when we take the square root of both sides, we have to consider both positive and negative roots: \[\cos(u) = \pm \cos(v)\]
04

Find suitable values for \(u\) and \(v\)

We need to find values for \(u\) and \(v\) that satisfy the given conditions: \(\cos(u) \neq \cos(v)\) and \(\cos(2u) = \cos(2v)\). From Step 3, we can see that: \[\cos(u) = \cos(v)\] or \[\cos(u) = -\cos(v)\] To satisfy the condition \(\cos(u) \neq \cos(v)\), we must select values such that \[\cos(u) = -\cos(v)\]. For a simple example, let's choose \(u=60^{\circ}\) and \(v=120^{\circ}\): \[\cos(u) = \cos(60^{\circ}) = \frac{1}{2}\] \[\cos(v) = \cos(120^{\circ}) = -\frac{1}{2}\] For these values of \(u\) and \(v\), the given conditions are satisfied: \[\cos(u) = -\cos(v)\] and \[\cos(2u) = \cos(2v)\] So, \(u = 60^{\circ}\) and \(v = 120^{\circ}\) are suitable angles that satisfy the given conditions.

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Most popular questions from this chapter

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