Find all solutions in the interval [0,2pi].
sin 2x + sin x = 0
We should also include 360° , (from sinx = 0)
To find all solutions in the interval [0, 2pi] for the equation sin 2x + sin x = 0, we can use some trigonometric identities.
Step 1: Rewrite the equation using a trigonometric identity.
sin 2x + sin x = 0
Using the double angle formula for sine, we have:
2sin x cos x + sin x = 0
Step 2: Factor out sin x.
sin x(2cos x + 1) = 0
Step 3: Set each factor equal to 0 and find the solutions.
First, let's consider sin x = 0:
sin x = 0
x = 0, pi, 2pi
Now, let's consider 2cos x + 1 = 0:
2cos x + 1 = 0
2cos x = -1
cos x = -1/2
Using the unit circle or trigonometric ratios, we know that the angles where cos x = -1/2 are x = 2pi/3 and x = 4pi/3.
Therefore, the solutions in the interval [0, 2pi] for the equation sin 2x + sin x = 0 are:
x = 0, pi, 2pi, 2pi/3, 4pi/3.
To find all solutions in the interval [0, 2π] for the equation sin(2x) + sin(x) = 0, we can use some trigonometric identities and solve step by step.
Let's start by using the trigonometric identity sin(2x) = 2sin(x)cos(x). We can rewrite the equation as:
2sin(x)cos(x) + sin(x) = 0
Now, we can factor out sin(x):
sin(x)(2cos(x) + 1) = 0
To find the solutions, we set each factor equal to zero:
sin(x) = 0 (1)
or
2cos(x) + 1 = 0 (2)
Let's solve equation (1) first:
sin(x) = 0
In the interval [0, 2π], the solutions for sin(x) = 0 are x = 0 and x = π.
Now, let's solve equation (2):
2cos(x) + 1 = 0
Subtract 1 from both sides:
2cos(x) = -1
Divide both sides by 2:
cos(x) = -1/2
In the interval [0, 2π], the solutions for cos(x) = -1/2 are x = 2π/3 and x = 4π/3.
Therefore, the solutions for the given equation sin(2x) + sin(x) = 0 in the interval [0, 2π] are x = 0, x = π, x = 2π/3, and x = 4π/3.
sin 2 x = 2 sin x cos x
2 sin x cos x + sin x = 0
sin x (2 cos x + 1) = 0
so
x = 0 is a solution and x = 180 is a solution
then
cos x = -1/2
that is at x = 120 and x = 240