please simplify
sin (7pi/6 -x) + cos (2pi +x)
Use the formulas for sin (A - B) and cos (A + B)
I can tell you right away that cos (2 pi + x) = cos x
To simplify the expression sin(7π/6 - x) + cos(2π + x), let's break it down step by step:
1. Rewrite the angles using angle sum and difference formulas:
sin(7π/6 - x) = sin(7π/6)cos(x) - cos(7π/6)sin(x)
cos(2π + x) = cos(2π)cos(x) - sin(2π)sin(x)
2. Use the unit circle to determine the values of sine and cosine for the angles involved:
sin(7π/6) = -1/2
cos(7π/6) = -√3/2
cos(2π) = 1
sin(2π) = 0
3. Substitute the values obtained in step 2 into the expression in step 1:
sin(7π/6 - x) + cos(2π + x) = (-1/2)(cos(x)) - (-√3/2)(sin(x)) + 1(cos(x)) - 0(sin(x))
4. Simplify the expression:
sin(7π/6 - x) + cos(2π + x) = -1/2 cos(x) + √3/2 sin(x) + cos(x)
Therefore, the simplified form of sin(7π/6 - x) + cos(2π + x) is -1/2 cos(x) + √3/2 sin(x) + cos(x).