Question

find the exact values below. if applicable, click on \undefined.\
\\(\sin 675^{\circ} = \square\\)
\\(\sec\left(-4\pi\
ight) = \square\\)

Explanation:

For \(\boldsymbol{\sin 675^\circ}\):

Step1: Reduce the angle

Subtract multiples of \(360^\circ\) from \(675^\circ\) to get an equivalent angle between \(0^\circ\) and \(360^\circ\).
\(675^\circ - 360^\circ = 315^\circ\), so \(\sin 675^\circ = \sin 315^\circ\).

Step2: Identify the reference angle

\(315^\circ\) is in the fourth quadrant, where sine is negative. The reference angle is \(360^\circ - 315^\circ = 45^\circ\).

Step3: Use the reference angle

\(\sin 315^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2}\).

For \(\boldsymbol{\sec(-4\pi)}\):

Step1: Use the even property of secant

\(\sec(-\theta) = \sec\theta\), so \(\sec(-4\pi) = \sec(4\pi)\).

Step2: Reduce the angle

The period of \(\sec\theta\) (and \(\cos\theta\)) is \(2\pi\), so \(\sec(4\pi) = \sec(2\pi \cdot 2) = \sec(0)\) (since \(\sec\theta\) has period \(2\pi\)).

Step3: Evaluate \(\sec(0)\)

\(\sec\theta = \frac{1}{\cos\theta}\), and \(\cos(0) = 1\), so \(\sec(0) = \frac{1}{1} = 1\).

Answer:

s:
\(\sin 675^\circ = \boldsymbol{-\frac{\sqrt{2}}{2}}\)
\(\sec(-4\pi) = \boldsymbol{1}\)