Solve for x 2sin(x)-3=-csc(x)

Problem

2*sin(x)−3=−csc(x)

Solution

1. Use the reciprocal identity to rewrite the cosecant function in terms of sine.

csc(x)=1/sin(x)

2. Substitute the identity into the original equation.

2*sin(x)−3=−1/sin(x)

3. Multiply the entire equation by sin(x) to eliminate the fraction, noting that sin(x)≠0

2*sin2(x)−3*sin(x)=−1

4. Rearrange the equation into standard quadratic form by adding 1 to both sides.

2*sin2(x)−3*sin(x)+1=0

5. Factor the quadratic expression by looking for two numbers that multiply to 2 and add to −3

(2*sin(x)−1)*(sin(x)−1)=0

6. Set each factor to zero to find the possible values for sin(x)

2*sin(x)−1=0⇒sin(x)=1/2

sin(x)−1=0⇒sin(x)=1

7. Solve for x using the unit circle values for each case, where n is any integer.

sin(x)=1/2⇒x=π/6+2*n*π

sin(x)=1/2⇒x=(5*π)/6+2*n*π

sin(x)=1⇒x=π/2+2*n*π

Final Answer

x=π/6+2*n*π,(5*π)/6+2*n*π,π/2+2*n*π

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