Question

find the exact value of the sine function of the given angle.
2115°

sin 2115° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Reduce the angle by full rotations

A full rotation is \(360^\circ\). We divide \(2115^\circ\) by \(360^\circ\) to find the equivalent angle within \(0^\circ\) to \(360^\circ\).
\(2115\div360 = 5\) with a remainder. \(5\times360 = 1800\), so \(2115 - 1800=315^\circ\). So \(\sin2115^\circ=\sin315^\circ\).

Step2: Determine the reference angle and sign

\(315^\circ\) is in the fourth quadrant where sine is negative. The reference angle for \(315^\circ\) is \(360^\circ - 315^\circ = 45^\circ\).
We know that \(\sin45^\circ=\frac{\sqrt{2}}{2}\), so \(\sin315^\circ=-\sin45^\circ\).

Step3: Substitute the value of \(\sin45^\circ\)

Substituting \(\sin45^\circ=\frac{\sqrt{2}}{2}\) into \(-\sin45^\circ\), we get \(-\frac{\sqrt{2}}{2}\).

Answer:

\(-\frac{\sqrt{2}}{2}\)