rewrite the following ratio in another form: sin 7pi/3 in terms of cosine
sin (7π/3) can be rewritten using the identity sin^2θ + cos^2θ = 1 as follows:
sin (7π/3) = sin [(6π/3) + (π/3)] = sin (2π + π/3)
Using the sine of a sum identity, sin (a + b) = sin(a)cos(b) + cos(a)sin(b):
sin (2π + π/3) = sin(2π)cos(π/3) + cos(2π)sin(π/3)
Since sin(2π) = sin(0) = 0 and cos(2π) = cos(0) = 1:
sin(2π)cos(π/3) + cos(2π)sin(π/3) = 0 * cos(π/3) + 1 * sin(π/3)
This simplifies to:
0 + sin(π/3) = sin(π/3)
Therefore, sin (7π/3) in terms of cosine is equivalent to sin(π/3).