Let theta be an angle in the first quadrant, and suppose sin theta equals a.
Evaluate sin(180 degrees + theta) in terms of a
sin(180 degrees + theta) =
for any x, sin(x+180) = -sin(x)
check the sum formula for sines
To evaluate sin(180 degrees + theta) in terms of a, we can use the trigonometric identity:
sin(A + B) = sin A * cos B + cos A * sin B
In this case, we have A = 180 degrees and B = theta. Since sin 180 degrees is 0 and cos 180 degrees is -1, we can substitute these values into our identity:
sin(180 degrees + theta) = sin 180 degrees * cos theta + cos 180 degrees * sin theta
Since sin 180 degrees is 0 and cos 180 degrees is -1, the equation simplifies to:
sin(180 degrees + theta) = 0 * cos theta + (-1) * sin theta
Multiplying 0 by cos theta gives 0, so we have:
sin(180 degrees + theta) = -sin theta
Therefore, sin(180 degrees + theta) in terms of a is -a.