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Chapter 5: Problem 6

Verify that the \(x\) -values are solutions of the equation. \(\sec x-2=0\) (a) \(x=\frac{\pi}{3}\) (b) \(x=\frac{5 \pi}{3}\)

Short Answer

Expert verified
Both \(x=\frac{\pi}{3}\) and \(x=\frac{5 \pi}{3}\) are solutions of the equation \(\sec x-2=0\).

Step by step solution

01

Substitution of \(x=\frac{\pi}{3}\)

Replace \(x\) with \(\frac{\pi}{3}\) in the given equation, which gives \(\sec(\frac{\pi}{3})-2\). To evaluate \(\sec(\frac{\pi}{3})\), we need to find the cosine of \(\frac{\pi}{3}\) first, which is \(0.5\). Hence, \(\sec(\frac{\pi}{3}) = \frac{1}{\cos(\frac{\pi}{3})} = 2\). Substituting back into the equation, we have \(2-2=0\). Hence, \(x=\frac{\pi}{3}\) is a solution to the given equation.
02

Substitution of \(x=\frac{5 \pi}{3}\)

Replace \(x\) with \(\frac{5 \pi}{3}\) in the given equation, which gives \(\sec(\frac{5 \pi}{3})-2\). To evaluate \(\sec(\frac{5 \pi}{3})\), we need to find the cosine of \(\frac{5 \pi}{3}\) first, which is \(0.5\). Hence, \(\sec(\frac{5 \pi}{3}) = \frac{1}{\cos(\frac{5 \pi}{3})} = 2\). Substituting back into the equation, we have \(2-2=0\). Hence, \(x=\frac{5 \pi}{3}\) is also a solution to the given equation.

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