(This discussion applies to functions of both angles and real numbers.)
Consider the following.
$$\begin{aligned}\cos (&\left.180^{\circ}-\theta\right) \\\&=\cos 180^{\circ}
\cos \theta+\sin 180^{\circ} \sin \theta \\\&=(-1) \cos \theta+(0) \sin \theta
\\\&=-\cos \theta\end{aligned}$$
\(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\) is an example of a
reduction formula, which is an identity that reduces a function of a
quadrantal angle plus or minus \(\theta\) to a function of \(\theta\) alone.
Another example of a reduction formula is \(\cos
\left(270^{\circ}+\theta\right)=\sin \theta\) Here is an interesting method for
quickly determining a reduction formula for a trigonometric function \(f\) of
the form \(f(Q \pm \theta),\) where \(Q\) is a quadrantal angle. There are two
cases to consider, and in each case, think of \(\boldsymbol{\theta}\) as a small
positive angle in order to determine the quadrant in which \(Q \pm \theta\) will
lie.
Case 1 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along
the \(x\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a
small positive angle \(\theta .\) If the given function \(f\) is positive in that
quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that
quadrant, use a - sign. The reduced form will have that \(\operatorname{sign},
f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE
GRAPH
Case 2 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along
the \(y\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a
small positive angle \(\theta .\) If the given function \(f\) is positive in that
quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that
quadrant, use a - sign. The reduced form will have that sign, the cofunction
of \(f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY
THE GRAPH
Use these ideas to write reduction formulas for each of the following.
$$\sin \left(180^{\circ}+\theta\right)$$
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