Solve for x 16sin(x)^2-12=0

Problem

16*sin(x)−12=0

Solution

1. Isolate the squared trigonometric term by adding 12 to both sides of the equation.

16*sin(x)=12

2. Divide both sides by 16 to solve for the squared sine term.

sin(x)=12/16

3. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 4.

sin(x)=3/4

4. Take the square root of both sides, remembering to include both the positive and negative roots.

sin(x)=±√(,3/4)

5. Simplify the radical expression.

sin(x)=±√(,3)/2

6. Identify the angles on the unit circle where the sine value is √(,3)/2 or −√(,3)/2

x=π/3,(2*π)/3,(4*π)/3,(5*π)/3

7. Generalize the solution to include all possible rotations by adding multiples of π since the solutions are separated by (2*π)/3 and π/3 intervals (specifically, they are ±π/3+n*π.

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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