Solve for x sin(x)^2-2sin(x)-3=0

Problem

sin(x)−2*sin(x)−3=0

Solution

1. Substitute a variable to simplify the quadratic form by letting u=sin(x)

u2−2*u−3=0

2. Factor the quadratic equation by finding two numbers that multiply to −3 and add to −2

(u−3)*(u+1)=0

3. Solve for u by setting each factor to zero.

u−3=0⇒u=3

u+1=0⇒u=−1

4. Back-substitute the original expression sin(x) for u

sin(x)=3

sin(x)=−1

5. Evaluate the solutions for x Since the range of the sine function is [−1,1] the equation sin(x)=3 has no real solutions.

sin(x)=3⇒No solution

6. Determine the general solution for sin(x)=−1 This occurs when x is at the bottom of the unit circle.

x=(3*π)/2+2*n*π

Final Answer

sin(x)−2*sin(x)−3=0⇒x=(3*π)/2+2*n*π,n∈ℤ

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