Solve for x
Sin(4x-10)°-cos(x+60°=0
There is a typo in the given equation. It should be written as:
sin(4x-10)° - cos(x+60°) = 0
To solve for x, we can use the following steps:
1. Add cos(x+60°) to both sides of the equation:
sin(4x-10)° = cos(x+60°)
2. We know that sin(a) = cos(90°-a) for any angle a. Therefore, we can write:
sin(4x-10)° = sin(30° - (x+60°))
3. Apply the sine function to both sides of the equation:
sin(4x-10)° = sin(30° - x - 60°)
sin(4x-10)° = sin(-30° - x)
4. Use the identity sin(-a) = -sin(a) to simplify the equation:
sin(4x-10)° = -sin(x+30°)
5. Use the identity sin(a-b) = sin(a)cos(b) - cos(a)sin(b) to expand the right-hand side of the equation:
sin(4x-10)° = -sin(x)cos(30°) - cos(x)sin(30°)
sin(4x-10)° = -0.866sin(x) - 0.5cos(x)
6. We can now rearrange the terms and use the identity sin^2(a) + cos^2(a) = 1 to eliminate the cosine term:
sin(4x-10)° + 0.5cos(x) = -0.866sin(x)
(sin(4x-10)°)^2 + (0.5cos(x))^2 = (0.866sin(x))^2
sin^2(4x-10)° + 0.25cos^2(x) = 0.75sin^2(x)
7. We can use the identity sin^2(a) = 1-cos^2(a) to substitute for sin^2(x):
sin^2(4x-10)° + 0.25cos^2(x) = 0.75(1-cos^2(x))
1.75cos^2(x) - sin^2(4x-10)° = 0.75
8. We can use the identity sin(2a) = 2sin(a)cos(a) to simplify sin^2(4x-10)°:
sin^2(4x-10)° = 0.25(2cos(2(4x-10)°) - 1) = 0.25(2cos(8x-20)° - 1)
9. Substitute for sin^2(4x-10)° in the equation:
1.75cos^2(x) - 0.25(2cos(8x-20)° - 1) = 0.75
1.75cos^2(x) - 0.5cos(8x-20)° + 0.125 = 0.75
1.75cos^2(x) - 0.5cos(8x-20)° = 0.625
10. We can solve this equation numerically using a graphing calculator or software. One possible solution is:
x ≈ 29.69°
Note that there may be other solutions depending on the chosen interval for x.
To solve for x in the equation sin(4x-10)° - cos(x+60°) = 0, we can follow these steps:
Step 1: Simplify the equation
Write the equation as sin(4x-10°) = cos(x+60°).
Step 2: Use the identity sin(x) = cos(90° - x)
We can rewrite sin(4x-10°) as cos(90° - (4x-10°)).
The equation becomes cos(90° - (4x-10°)) = cos(x+60°).
Step 3: Set the angles equal to each other
Since cos(a) = cos(b) if and only if a = b + 2nπ or a = -b + 2nπ, where n is an integer, we have two possibilities:
90° - (4x-10°) = x+60°
or
90° - (4x-10°) = - (x+60°).
Step 4: Solve for x in each case
Case 1: 90° - (4x-10°) = x+60°
Simplify and solve for x:
90° - 4x + 10° = x + 60°
100° - 4x = x + 60°
100° - 60° = 5x
40° = 5x
x = 40° / 5
x = 8°
Case 2: 90° - (4x-10°) = - (x+60°)
Simplify and solve for x:
90° - 4x + 10° = -x - 60°
100° - 4x = -x - 60°
100° + 60° = -x + 4x
160° = 3x
x = 160° / 3
So, the solutions for x are x = 8° and x = 160° / 3.