Solve for x 4sec(x)+6=-2

Problem

4*sec(x)+6=−2

Solution

1. Isolate the trigonometric term by subtracting 6 from both sides of the equation.

4*sec(x)=−8

2. Divide by the coefficient 4 to solve for the secant function.

sec(x)=−2

3. Use the reciprocal identity sec(x)=1/cos(x) to rewrite the equation in terms of cosine.

1/cos(x)=−2

4. Solve for cosine by taking the reciprocal of both sides.

cos(x)=−1/2

5. Identify the reference angle in the first quadrant where cos(θ)=1/2 which is π/3 (or 60.

(θ_ref)=π/3

6. Determine the quadrants where cosine is negative, which are Quadrant II and Quadrant III.

x=π−π/3=(2*π)/3

x=π+π/3=(4*π)/3

7. Write the general solution by adding multiples of the period 2*π where n is any integer.

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

Final Answer

x=(2*π)/3+2*n*π,(4*π)/3+2*n*π

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