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

Problem

sec((5*x)/4)=2

Solution

1. Rewrite the equation in terms of the cosine function using the reciprocal identity sec(θ)=1/cos(θ)

cos((5*x)/4)=1/2

2. Identify the general solutions for the cosine equation cos(θ)=1/2 on the unit circle.

(5*x)/4=π/3+2*π*n

(5*x)/4=−π/3+2*π*n

3. Isolate x in the first general solution by multiplying both sides by 4/5

x=4/5*(π/3+2*π*n)

x=(4*π)/15+(8*π*n)/5

4. Isolate x in the second general solution by multiplying both sides by 4/5

x=4/5*(−π/3+2*π*n)

x=−(4*π)/15+(8*π*n)/5

Final Answer

sec((5*x)/4)=2⇒x=±(4*π)/15+(8*π*n)/5

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