Solve for x if sin2x=cos(x+60 and x€ [-90,180].show all work details
What does x€ [-90,180] mean?
The domain limits of x?
You seem to have omitted a parenth after x+60
cos(x+60) = sin(90-x-60)
= sin (30-x)
sin2x = sin(30-x)
Therefore 2x = 30 -x
x = 10 degrees is one solution
See if you can find others. Note that sin2x = sin(180 -2x); that will lead to another solution.
Yes it mean domaim limit of x that guied u when plot the graph
To solve for x, we will use the trigonometric identity for sine of double angle:
sin(2x) = 2sin(x)cos(x)
Given that sin(2x) = cos(x + 60), we can rewrite the equation as:
2sin(x)cos(x) = cos(x + 60)
Expanding the right side using the cosine of a sum identity:
2sin(x)cos(x) = cos(x)cos(60) - sin(x)sin(60)
Simplifying further:
2sin(x)cos(x) = (1/2)cos(x) - (√3/2)sin(x)
Now, let's bring all the terms to one side of the equation:
2sin(x)cos(x) - (1/2)cos(x) + (√3/2)sin(x) = 0
Next, let's factor out sin(x) and cos(x):
sin(x)(2cos(x) + (√3/2)) - (1/2)cos(x) = 0
Now we have two possibilities to consider:
1) sin(x) = 0
If sin(x) = 0, then x can be any integer multiple of 180 degrees within the given range [-90, 180]. Therefore:
x = 0, -180
2) 2cos(x) + (√3/2) = (1/2)
Let's solve for x using this equation. Subtracting (1/2)cos(x) from both sides:
sin(x)(2cos(x) + (√3/2) - (1/2)cos(x)) = 0 - (1/2)cos(x)
Combining like terms:
(√3/2)sin(x) + (1/2)sin(x) = (1/2)cos(x)
Multiplying both sides by 2:
√3sin(x) + sin(x) = cos(x)
Expanding the left side using the sine of a sum identity:
(√3 + 1)sin(x) = cos(x)
Rearranging the equation:
(√3 + 1)sin(x) - cos(x) = 0
Now, solving for x in the range [-90, 180], we can use a calculator or reference table to find the values of x that satisfy this equation. Some approximate solutions in degrees are:
x ≈ -64.32, 57.67, 155.66
Therefore, the solutions for x are:
x ≈ 0, -180, -64.32, 57.67, 155.66
To solve for x, we need to use trigonometric identities and equations:
1. Start with the given equation: sin(2x) = cos(x + 60).
2. We know that sin(2x) = 2sin(x)cos(x) and cos(x + 60) = cos(x)cos(60) - sin(x)sin(60).
So, the equation becomes: 2sin(x)cos(x) = cos(x)cos(60) - sin(x)sin(60).
3. Simplify cos(60) and sin(60). We know that cos(60) = 1/2 and sin(60) = √3/2. The equation now becomes:
2sin(x)cos(x) = (1/2)cos(x) - (√3/2)sin(x).
4. Multiply both sides of the equation by 2 to eliminate the fractions:
4sin(x)cos(x) = cos(x) - √3sin(x).
5. Rearrange the equation: 4sin(x)cos(x) - cos(x) + √3sin(x) = 0.
6. Factor out cos(x) and sin(x):
cos(x)(4sin(x) - 1) + √3sin(x) = 0.
7. Set each factor equal to zero and solve for x:
cos(x) = 0:
x = 90°, 270°.
4sin(x) - 1 = 0:
4sin(x) = 1,
sin(x) = 1/4.
Since x ∈ [-90, 180], we need to identify the corresponding quadrants where sin(x) = 1/4. These are the first and second quadrants.
Using the inverse sine function, we solve for x:
x = sin^(-1)(1/4) ≈ 14.475° or x ≈ 165.525°.
8. The solutions for x are: x = 90°, 270°, 14.475°, and 165.525°.