Solve for x sec((3x)/2)=-2

Problem

sec((3*x)/2)=−2

Solution

1. Convert the secant function to its reciprocal cosine form.

cos((3*x)/2)=1/(−2)

2. Identify the reference angle for the cosine function.

cos(θ)=−1/2

θ=(2*π)/3+2*n*π

θ=(4*π)/3+2*n*π

3. Substitute the argument back into the general solution equations.

(3*x)/2=(2*π)/3+2*n*π

(3*x)/2=(4*π)/3+2*n*π

4. Isolate x by multiplying both sides of each equation by 2/3

x=2/3*((2*π)/3+2*n*π)

x=2/3*((4*π)/3+2*n*π)

5. Simplify the expressions to find the general solution for x where n is any integer.

x=(4*π)/9+(4*n*π)/3

x=(8*π)/9+(4*n*π)/3

Final Answer

sec((3*x)/2)=−2⇒x=(4*π)/9+(4*n*π)/3,(8*π)/9+(4*n*π)/3

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