Question

1. (36 points) for each of the following angles, determine the following

(a) (3 points) \\( \theta=\frac{-\pi}{4} \\)

• the quadrant the angle lies in
• the reference angle
• the values of \\( \sin (\theta), \cos (\theta) \\), and \\( \tan (\theta) \\)

Explanation:

Step1: Determine the quadrant

Since \(\theta =-\frac{\pi}{4}\), we can rewrite it as \(\theta = 2\pi-\frac{\pi}{4}=\frac{7\pi}{4}\). \(\frac{3\pi}{2}<\frac{7\pi}{4}<2\pi\), so the angle \(\theta =-\frac{\pi}{4}\) lies in the fourth quadrant.

Step2: Find the reference angle

For an angle \(\theta\) in the fourth quadrant, the reference angle \(\theta_{r}\) is given by \(\theta_{r} = 2\pi-\theta\) (when \(\theta\) is in standard position). If \(\theta =-\frac{\pi}{4}\), in standard position \(\theta = 2\pi-\frac{\pi}{4}\), and the reference angle \(\theta_{r}=\frac{\pi}{4}\)

Step3: Calculate \(\sin\theta\), \(\cos\theta\) and \(\tan\theta\)

We know that:

• \(\sin\theta=-\sin\theta_{r}\) (in the fourth quadrant, sine is negative). Since \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), then \(\sin(-\frac{\pi}{4})=-\frac{\sqrt{2}}{2}\)
• \(\cos\theta=\cos\theta_{r}\) (in the fourth quadrant, cosine is positive). Since \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), then \(\cos(-\frac{\pi}{4})=\frac{\sqrt{2}}{2}\)
• \(\tan\theta=-\tan\theta_{r}\) (in the fourth quadrant, tangent is negative). Since \(\tan\frac{\pi}{4} = 1\), then \(\tan(-\frac{\pi}{4})=- 1\)

Answer:

• The quadrant: Fourth quadrant
• The reference angle: \(\frac{\pi}{4}\)
• \(\sin(-\frac{\pi}{4})=-\frac{\sqrt{2}}{2}\), \(\cos(-\frac{\pi}{4})=\frac{\sqrt{2}}{2}\), \(\tan(-\frac{\pi}{4})=-1\)