sin(x-6)=cos(3x-4)

Question

solve for x

Answer 1

sin(x-6) = cos(3x-4)

sin(x-6) = sin(pi/2-(3x-4)+n(2pi)), n integer

x-6 = pi/2-(3x-4)+n(2pi)

x - 6 + 3x - 4 = pi/2 + n2pi

4x - 10 = pi/2 + n2pi

4x = 10 + pi/2 + n2pi

x = 5/2 + pi/8 + npi/2

x = 5/2 + pi/8 + npi/2 radians, n integer

x = (180°/pi)(5/2 + pi/8 + npi/2)

x = 450°/pi + 180°/8 + n180°/2

x = 450°/pi + 22.5° + n90°, n integer

x ≈ 165.73944878270581° - 90° + (n+1)90°, n integer

x ≈ 75.73944878270581° + m90°, m integer [m = n+1]

See GeoGebra graph and check here:
http://www.wyzant.com/resources/files/262934/find_x_if_sin_x_6_cos_3x_4

Note: If you wanted to work in degrees from the beginning,
you would modify the original equations like this:

sin((x-6)*(180°/pi)) = cos((3x-4)*(180°/pi))

Answer 2

The first step to solving this is to make use of the fact that sin(θ) = cos(θ – 90?). You did not indicate the units of the arguments (degrees or radians), but the technique is the same either way. Based on the relationship between sin() and cos() just stated, and assuming degrees:

sin(x-6) = cos(3x-4) = cos((x-6)-90), where the angle θ in terms of this equation is x-6. From this,
cos(3x-4) = cos((x-6)-90), therefore,
3x-4 = (x-6)-90 = x-96
2x – 4 = -96
2x = -92
x = -46
As a check,
sin(x-6) = sin(-52) = -0,78801
cos(3x-4) = cos(-142) = -0.78801

Answer 3

Assuming degrees, remember that if a±b=90±360n then, sin a = cos b.

 

Hence you have (x - 6) + (3x - 4) = 90 + 360n ⇒ 4x - 10 = 90 + 360n ⇒ x = 25 + 90n

 

or (x - 6) - (3x - 4) = 90 - 360n ⇒ -2x - 2 = 90 - 360n ⇒ x = 180n - 46.