Find all solutions of the given equation in the interval
[0, 2pi)
cos x/2 - sin x = 0
Hi, I am struggling with this question. Can anybody help me please? Thanks!
don't like fractions, so how about Ø = x/2 or x = 2Ø
then your equation becomes
cosØ - sin 2Ø = 0
cosØ - 2sinØcosØ = 0
cosØ(1 - 2sinØ) = 0
cosØ = 0 or sinØ = 1/2
Ø = π/2, 3π/2 ---> x = π, 3π
or
Ø = π/6, 5π/6, ---> x = π/3, 5π/3
in your given interval:
x = π, π/3, 5π/3
or
recall cos 2A = 2cos^2 A - 1 or 1 - 2sin^2 x
so if
cos (x/2) = sinx
squaring both sides brings that equation into play. A messier way of doing it.
Also remember since squaring has taken place, all answers must be checked
Of course! I'd be happy to help you solve this equation.
First, let's rewrite the equation as:
cos(x/2) - sin(x) = 0
To find the solutions in the interval [0, 2pi), we need to determine the values of x that satisfy the equation.
Step 1: Solve for sin(x):
sin(x) = cos(x/2)
Step 2: Square both sides of the equation to eliminate the square root:
sin(x)^2 = cos(x/2)^2
Step 3: Use the identity sin^2(x) + cos^2(x) = 1 to rewrite the equation:
1 - cos^2(x) = cos(x/2)^2
Step 4: Simplify the equation:
cos^2(x) + cos(x/2)^2 = 1
Step 5: Use the double-angle identity for cosine to rewrite the equation:
cos^2(x) + 1/2 * (1 + cos(x)) = 1
Step 6: Combine like terms:
cos^2(x) + 1/2 * cos(x) + 1/2 = 1
Step 7: Rearrange the equation:
cos^2(x) + 1/2 * cos(x) - 1/2 = 0
Step 8: Factor the quadratic equation:
(cos(x) - 1)(cos(x) + 1/2) = 0
Step 9: Set each factor to zero and solve for x:
cos(x) - 1 = 0 --> cos(x) = 1
cos(x) + 1/2 = 0 --> cos(x) = -1/2
To find the solutions in the interval [0, 2pi), we need to consider all values of x that satisfy these equations.
For cos(x) = 1, the solutions are x = 0 and x = 2pi.
For cos(x) = -1/2, we can use the inverse cosine function to find the angles:
x = arccos(-1/2)
Using a calculator or a trigonometric table, we find two solutions in the interval [0, 2pi): x ≈ 2.094 and x ≈ 4.189.
Therefore, the solutions to the given equation in the interval [0, 2pi) are x = 0, x = 2pi, x ≈ 2.094, and x ≈ 4.189.
I hope this explanation helps! Let me know if you have any further questions.