Solve for x sec((5x)/4)=2

Problem

$\sec \frac{5x}{4}=2$

Solution

1. Rewrite the equation in terms of the cosine function using the reciprocal identity $\sec \theta =\frac{1}{\cos \theta }$

$\cos \frac{5x}{4}=\frac{1}{2}$

2. Identify the general solutions for the cosine equation $\cos \theta =\frac{1}{2}$ on the unit circle.

$\frac{5x}{4}=\frac{\pi }{3}+2\pi n$

$\frac{5x}{4}=-\frac{\pi }{3}+2\pi n$

3. Isolate $x$ in the first general solution by multiplying both sides by $\frac{4}{5}$

$x=\frac{4}{5}(\frac{\pi }{3}+2\pi n)$

$x=\frac{4\pi }{15}+\frac{8\pi n}{5}$

4. Isolate $x$ in the second general solution by multiplying both sides by $\frac{4}{5}$

$x=\frac{4}{5}(-\frac{\pi }{3}+2\pi n)$

$x=-\frac{4\pi }{15}+\frac{8\pi n}{5}$

Final Answer

$\sec \frac{5x}{4}=2\Rightarrow x=\pm \frac{4\pi }{15}+\frac{8\pi n}{5}$

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