Question

Solve the equation f(x) = 6 for the function f(x) = 5 - 2 sin 2x.

Answer 1

To solve f(x) = 5 - 2 sin 2x for f(x) = 6, the equation becomes sin 2x = -1/2 and then we find x = π/12 or x = 11π/12, plus integer multiples of π, because the sine function has a periodic nature.

Step-by-step explanation:

To solve the equation f(x) = 6 for the function f(x) = 5 - 2 sin 2x, we set the function equal to 6:

1. 5 - 2 sin 2x = 6
2. -2 sin 2x = 1 (after subtracting 5 from both sides)
3. sin 2x = -1/2 (after dividing both sides by -2)
4. 2x = arcsin(-1/2)

Since the sin function is periodic, there are multiple solutions for x within one period (0 to 2π radians). The sine of an angle equals -1/2 at π/6 and 11π/6 radians in the unit circle. Therefore, the solutions within one period are:

1. x = π/12 + kπ, for integers k
2. x = 11π/12 + kπ, for integers k

To ensure our solutions make sense, we can plug them back into the original equation and verify that it yields a value close to 6. Since the values lie within the range of the sine function and we used correct algebraic steps, the answer is reasonable.