To solve the first equation, we need to isolate the cosine term:
1.5x - 1/2 = 01.5x = 1/2x = 1/3
Now, to solve the second equation, we need to use the identity cos(π/6 - θ) = cos(π/6)cos(θ) + sin(π/6)sin(θ)
2cos(π/6 - 3x) = 32(cos(π/6)cos(3x) + sin(π/6)sin(3x)) = 32(√3/2cos(3x) + 1/2sin(3x)) = 3√3cos(3x) + sin(3x) = 3/2
Now, we can use the identity cos(3x) = 4cos^3(x) - 3cos(x) and sin(3x) = 3sin(x) - 4sin^3(x)
√3(4cos^3(x) - 3cos(x)) + 3sin(x) - 4sin^3(x) = 3/24√3cos^3(x) - 3√3cos(x) + 3sin(x) - 4√3sin^3(x) = 3/2
At this point, it seems difficult to solve the equation algebraically. You might want to use a numerical method or graphing to find the solution for x.
To solve the first equation, we need to isolate the cosine term:
1.5x - 1/2 = 0
1.5x = 1/2
x = 1/3
Now, to solve the second equation, we need to use the identity cos(π/6 - θ) = cos(π/6)cos(θ) + sin(π/6)sin(θ)
2cos(π/6 - 3x) = 3
2(cos(π/6)cos(3x) + sin(π/6)sin(3x)) = 3
2(√3/2cos(3x) + 1/2sin(3x)) = 3
√3cos(3x) + sin(3x) = 3/2
Now, we can use the identity cos(3x) = 4cos^3(x) - 3cos(x) and sin(3x) = 3sin(x) - 4sin^3(x)
√3(4cos^3(x) - 3cos(x)) + 3sin(x) - 4sin^3(x) = 3/2
4√3cos^3(x) - 3√3cos(x) + 3sin(x) - 4√3sin^3(x) = 3/2
At this point, it seems difficult to solve the equation algebraically. You might want to use a numerical method or graphing to find the solution for x.