If sin(x-20°)=cos(3x-10°),find x.
use the fact that
cos Ø = sin (90° - Ø)
cos(3x - 10) = sin(90 - (3x-10) )
cos(3x - 10) = sin(100 - 3x)
so sin(x-20) = sin(100 - 3x)
then:
x-20 = 100-3x
4x = 120
x = 30
check with your calculator
To find the value of x, we can start by expressing both sin(x-20°) and cos(3x-10°) in terms of sin and cos of x. Let's go through the steps:
1. Using the trigonometric identity sin(A-B) = sin(A)cos(B) - cos(A)sin(B), we can rewrite sin(x-20°) as sin(x)cos(20°) - cos(x)sin(20°).
2. Similarly, using the trigonometric identity cos(3A) = cos(A)^3 - 3cos(A)sin(A)^2, we can rewrite cos(3x-10°) as cos(x)^3cos(10°) - 3cos(x)sin(x)^2sin(10°).
3. Now, we can equate the two expressions: sin(x)cos(20°) - cos(x)sin(20°) = cos(x)^3cos(10°) - 3cos(x)sin(x)^2sin(10°).
4. Rearranging the equation, we get sin(x)cos(20°) + 3cos(x)sin(x)^2sin(10°) = cos(x)^3cos(10°) + cos(x)sin(20°).
5. We can now simplify further:
- Distribute sin(x)cos(20°) and cos(x)sin(20°) to get: sin(x)cos(20°) + 3cos(x)sin(x)^2sin(10°) = cos(x)^3cos(10°) + cos(x)sin(20°).
- Factor out cos(x) from the left side: cos(x)(sin(20°) + 3sin(x)^2sin(10°)) = cos(x)^3cos(10°) + cos(x)sin(20°).
- Divide both sides by cos(x): sin(20°) + 3sin(x)^2sin(10°) = cos(x)^2cos(10°) + sin(20°).
- Move all terms to one side of the equation: sin(x)^2sin(10°) + cos(x)^2cos(10°) = sin(20°) - sin(20°).
- Simplify the right side: sin(x)^2sin(10°) + cos(x)^2cos(10°) = 0.
6. Now, we have a trigonometric equation sin(x)^2sin(10°) + cos(x)^2cos(10°) = 0.
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as sin^2(x)(sin(10°)) + cos^2(x)(cos(10°)) = 0.
Since sin(10°) and cos(10°) are positive values, both sin^2(x) and cos^2(x) must be equal to zero. This happens only when sin(x) = 0 and cos(x) = 0.
7. Therefore, the solutions for x are x = 0°, 180°, and any angle that is an integer multiple of 90°.
So, x can be 0°, 180°, or any angle that is a multiple of 90°.