Question

Solve the following equation for over the interval [0, 2π), giving exact answers in radian units. If an

equation has no solution, enter DNE. Multiple solutions should be entered as a comma-separated list.
cos(2b)-4=-5sin(b)
b=

Answer 1

The solutions for b over the interval [0, 2π) are b = 7π/6 or b = π/6, respectively.

Step-by-step explanation:

The given equation is cos(2b)-4=-5sin(b). To solve for b, we can use the trigonometric identity cos(2x) = 1-2sin²(x).

Substituting this identity into the equation, we get 1-2sin²(b) - 4 = -5sin(b).

Rearranging the terms, we obtain 2sin²(b) - 5sin(b) - 3 = 0.

This is a quadratic equation in sin(b), which can be factored as (2sin(b) + 1)(sin(b) - 3) = 0.

Setting each factor equal to zero, we find sin(b) = -1/2 or sin(b) = 3.

Therefore, the solutions for b over the interval [0, 2π) are b = 7π/6 or b = π/6, respectively.