Problem

Solve the following trigonometric equation for x: 2sin⁡(x)cos⁡(x)+sin⁡(x)−1=0

Answer

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Answer

Finally, we solve for x by taking the inverse sine of each solution: x=sin−1⁡(−1+52),sin−1⁡(−1−52)

Steps

Step 1 ：First, we rewrite the equation using the double-angle identity sin⁡(2x)=2sin⁡(x)cos⁡(x): sin⁡(2x)+sin⁡(x)−1=0

Step 2 ：Next, we rearrange the equation in a standard quadratic form: sin⁡(2x)+sin⁡(x)−1=0⇒sin⁡(x)2+sin⁡(x)−1=0

Step 3 ：Now, we can apply the quadratic formula to solve for sin⁡(x): sin⁡(x)=−b±b2−4ac2a=−1±1+42=−1±52

Step 4 ：Finally, we solve for x by taking the inverse sine of each solution: x=sin−1⁡(−1+52),sin−1⁡(−1−52)

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