Question

find the reference angle, the quadrant of the terminal side, and the sine and cosine of the angle. answer exactly, do not round. 135° reference angle: quadrant: sin(135°)= cos(135°)= question help: message instructor submit question

Explanation:

Step1: Determine the quadrant

Since \(90^{\circ}<135^{\circ}<180^{\circ}\), the terminal side of \(135^{\circ}\) is in Quadrant II.

Step2: Calculate the reference angle

For an angle \(\theta\) in Quadrant II, the reference angle \(\theta_{r}=180^{\circ}-\theta\). So, \(\theta_{r}=180^{\circ} - 135^{\circ}=45^{\circ}\)

Step3: Find \(\sin(135^{\circ})\)

We know that \(\sin(\theta)=\sin(180^{\circ}-\alpha)=\sin\alpha\) for \(\theta = 135^{\circ}\) and \(\alpha = 45^{\circ}\). So, \(\sin(135^{\circ})=\sin(45^{\circ})=\frac{\sqrt{2}}{2}\)

Step4: Find \(\cos(135^{\circ})\)

We know that \(\cos(\theta)=-\cos(180^{\circ}-\alpha)\) for \(\theta = 135^{\circ}\) and \(\alpha = 45^{\circ}\) (because in Quadrant II, \(\cos\) is negative). So, \(\cos(135^{\circ})=-\cos(45^{\circ})=-\frac{\sqrt{2}}{2}\)

Answer:

Reference angle: \(45^{\circ}\)
Quadrant: II
\(\sin(135^{\circ})=\frac{\sqrt{2}}{2}\)
\(\cos(135^{\circ})=-\frac{\sqrt{2}}{2}\)