Solve for x 3csc(x)^2=4

Problem

3*csc^2(x)=4

Solution

1. Isolate the trigonometric term by dividing both sides of the equation by 3

csc^2(x)=4/3

2. Take the square root of both sides to solve for csc(x) remembering to include both the positive and negative roots.

csc(x)=±√(,4/3)

3. Simplify the radical expression on the right side.

csc(x)=±2/√(,3)

4. Convert the cosecant function to the sine function using the reciprocal identity sin(x)=1/csc(x)

sin(x)=±√(,3)/2

5. Identify the reference angle in the first quadrant where sin(x)=√(,3)/2 which is x=π/3 (or 60^∘.

(x_ref)=π/3

6. Determine all solutions within one period [0,2*π) by considering all four quadrants where the sine is either positive or negative.

x=π/3,(2*π)/3,(4*π)/3,(5*π)/3

7. Generalize the solution by adding multiples of the period π since the values are spaced exactly π apart.

x=π/3+n*π

x=(2*π)/3+n*π

Final Answer

x=π/3+n*π,(2*π)/3+n*π

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