Solve this equation on the interval 0 θ < 2π. Round your answer(s) to two decimal places.
2 sin θ + 3 = 2
(smaller value)
(larger value)
2 sin θ + 3 = 2
2 sin θ = -1
sin θ = -1/2
θ = ?
2sin@ + 3 = 2
2sin@ = -1
sin@ = -1/2
@ = 210, 330 convert to radians if desired
To solve the equation 2 sin θ + 3 = 2 on the interval 0 θ < 2π, we can follow these steps:
Step 1: Subtract 3 from both sides of the equation:
2 sin θ = 2 - 3
2 sin θ = -1
Step 2: Divide both sides of the equation by 2:
sin θ = -1/2
Step 3: Find the solution(s) to the equation sin θ = -1/2 on the interval 0 θ < 2π.
To find the solution(s), we can refer to the unit circle or the values of sine in the four quadrants.
On the unit circle, the y-coordinate (sine value) is positive in the first and second quadrants, and negative in the third and fourth quadrants.
Since we are looking for sin θ = -1/2, which is negative, we can focus on the third and fourth quadrants.
In the third quadrant (π < θ < 3π/2), sin θ is negative. One of the angles in this quadrant where sin θ = -1/2 is π + π/6, which is π/2.
In the fourth quadrant (3π/2 < θ < 2π), sin θ is also negative. One of the angles in this quadrant where sin θ = -1/2 is 2π - π/6, which is 11π/6.
So, the solutions to the equation 2 sin θ + 3 = 2 on the interval 0 θ < 2π are:
- θ = π/2 (approx. 1.57) [smaller value]
- θ = 11π/6 (approx. 5.76) [larger value]
Remember to round the answers to two decimal places as requested.