Question

without using a calculator, compute the sine and cosine of 315° by using the reference angle.
(type sqrt(2) for √2 and sqrt(3) for √3.)
what is the reference angle? degrees.
in what quadrant is this angle? (answer 1, 2, 3, or 4)
sin(315°) =
cos(315°) =

Explanation:

Step1: Find the reference angle

The reference angle formula for an angle \(\theta\) in the fourth quadrant (\(270^{\circ}<\theta < 360^{\circ}\)) is \(\theta_{ref}=360^{\circ}-\theta\).
For \(\theta = 315^{\circ}\), \(\theta_{ref}=360^{\circ}-315^{\circ}=45^{\circ}\)

Step2: Determine the quadrant

Since \(270^{\circ}<315^{\circ}<360^{\circ}\), the angle \(315^{\circ}\) is in the fourth quadrant.

Step3: Calculate \(\sin(315^{\circ})\)

In the fourth quadrant, \(\sin\theta<0\). Using the identity \(\sin\theta=-\sin\theta_{ref}\) (because \(\sin\) is negative in QIV), and \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\), so \(\sin315^{\circ}=-\frac{\sqrt{2}}{2}\)

Step4: Calculate \(\cos(315^{\circ})\)

In the fourth quadrant, \(\cos\theta>0\). Using the identity \(\cos\theta = \cos\theta_{ref}\) (because \(\cos\) is positive in QIV), and \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), so \(\cos315^{\circ}=\frac{\sqrt{2}}{2}\)

Answer:

Reference angle: \(45\) degrees.
Quadrant: \(4\)
\(\sin(315^{\circ})=-\frac{\sqrt{2}}{2}\)
\(\cos(315^{\circ})=\frac{\sqrt{2}}{2}\)