Find the exact solutions of the equation In The interval.. Sin 2x -sin x=0
sin 2 x = sin x when x = 0, 180, 360 etc
sin 2x - sinx =0
2sinx cosx - sinx = 0
sinx(2cosx - 1) = 0
sinx = 0 or cosx = 1/2
for sinx = 0
x = 0, 180°, 360°
for cosx = 1/2
x = 60° , 300°
so for 0 ≤ x ≤ 360°
x = 0, 60, 180, 300, 360
in radians:
x = 0 , π/3 , π , 5π/3 , 2π
To find the exact solutions, we need to solve the equation sin(2x) - sin(x) = 0 in the given interval.
First, let's rewrite the equation using a trigonometric identity:
sin(2x) - sin(x) = 2sin(x)cos(x) - sin(x) = sin(x)(2cos(x) - 1) = 0
Now, we have two cases to consider:
Case 1: sin(x) = 0
If sin(x) = 0, then x can be any multiple of π:
x = nπ, where n is an integer.
Case 2: 2cos(x) - 1 = 0
Solving this equation for cos(x):
2cos(x) - 1 = 0
2cos(x) = 1
cos(x) = 1/2
Using the unit circle or trigonometric ratios, we can find the angles in the interval [0, 2π] where cos(x) = 1/2:
x = π/3, 5π/3
So, the exact solutions in the interval [0, 2π] are:
x = nπ, π/3, 5π/3, where n is an integer.
To find the exact solutions of the equation sin(2x) - sin(x) = 0 in the given interval, we can use algebraic manipulations and trigonometric identities.
First, let's rewrite the equation as sin(2x) = sin(x).
We can use the identity sin(2θ) = 2sin(θ)cos(θ) to rewrite the equation as 2sin(x)cos(x) = sin(x).
Now, we have two cases to consider:
Case 1: sin(x) = 0
If sin(x) = 0, then x must be an integer multiple of π, because sin(0) = sin(π) = 0.
So, one set of solutions is x = nπ, where n is an integer.
Case 2: cos(x) = 1/2
If cos(x) = 1/2, then x must be an angle whose cosine value is 1/2.
To find these angles, we recall the unit circle and the values of cos(x) for different angles. The angles where cos(x) = 1/2 are π/3 and 5π/3 (among others), because cos(π/3) = cos(2π/3) = cos(4π/3) = cos(5π/3) = 1/2.
For these values, sin(x) = √(1 - cos^2(x)) = √(1 - (1/2)^2) = √(1 - 1/4) = √(3/4) = √3/2.
So, another set of solutions is x = π/3 + 2nπ and x = 5π/3 + 2nπ, where n is an integer.
Combining both sets of solutions, the exact solutions of the equation sin(2x) - sin(x) = 0 in the given interval are:
x = nπ, π/3 + 2nπ, and 5π/3 + 2nπ, where n is an integer.