Solve the following interval
0<_ (less than or equal to)0(Theta)<2pi:
1. sin0(theta) + sin2 0(theta) = 0
2. 2cos^20 (theta) + cos0 (theta) -1 = 0
sinӨ + sin2Ө = 0
recall that sin2Ө = 2sinӨcosӨ
so sinӨ + sin2Ө = 0
sinӨ + 2sinӨcosӨ = 0
sinӨ(1+2cosӨ)=0
sinӨ = 0 or 2cosӨ = -1
sinӨ = 0 or or cosӨ = -1/2
Ө = 0,pi,2pi or Ө=2pi/3,4pi/3
in degrees: 0,180,360,120,240
To solve the given intervals, we need to find the values of theta that satisfy the equations. Let's solve each equation separately:
1. sin(theta) + sin(2 * theta) = 0
Step 1: Rewrite the equation using trigonometric formulas.
sin(theta) + 2 * sin(theta) * cos(theta) = 0
Step 2: Simplify the equation.
sin(theta) * (1 + 2 * cos(theta)) = 0
Step 3: Set each factor equal to zero and solve.
sin(theta) = 0 or 1 + 2 * cos(theta) = 0
For sin(theta) = 0, the theta values are 0, pi, 2pi, etc., since sin(0) = sin(pi) = sin(2pi) = 0. These solutions lie in the given interval.
For 1 + 2 * cos(theta) = 0, rearrange the equation:
cos(theta) = -1/2
We know that cos(theta) is equal to -1/2 at two points on the unit circle: 2pi/3 and 4pi/3. However, we need to consider only the solutions within the given interval.
Within the interval 0 ≤ theta ≤ 2pi, the values of theta that satisfy the equation sin(theta) + sin(2 * theta) = 0 are:
- theta = 0, pi, 2pi
- theta ≈ 2pi/3, 4pi/3
2. 2 * cos^2(theta) + cos(theta) - 1 = 0
Step 1: Let's make a substitution to simplify the equation.
Let x = cos(theta). Now the equation becomes:
2 * x^2 + x - 1 = 0
Step 2: Solve the quadratic equation.
Factor the equation:
(2 * x - 1)(x + 1) = 0
Setting each factor equal to zero:
2 * x - 1 = 0 or x + 1 = 0
Solving these equations, we find:
x = 1/2 or x = -1
Step 3: Convert back to theta.
For x = 1/2, we find theta using the inverse cosine function:
cos(theta) = 1/2
theta = ±π/3, ±5π/3
For x = -1, we have:
cos(theta) = -1
Within the given interval 0 ≤ theta ≤ 2pi, the values of theta that satisfy the equation 2 * cos^2(theta) + cos(theta) - 1 = 0 are:
- theta = π/3, 5π/3
In summary:
For the equation sin(theta) + sin(2 * theta) = 0:
- theta = 0, pi, 2pi
- theta ≈ 2pi/3, 4pi/3
For the equation 2 * cos^2(theta) + cos(theta) - 1 = 0:
- theta = π/3, 5π/3