Question

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

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

(a) (3 points) \\( \theta = \frac { - \pi } { 4 } \\)
(b) (3 points) \\( \theta = \frac { \pi } { 6 } \\)
(c) (3 points) \\( \theta = \frac { 5 \pi } { 6 } \\)

Explanation:

Step1: Determine the quadrant

Since \(\theta =-\frac{\pi}{4}\), we know that \(2\pi-\frac{\pi}{4}=\frac{7\pi}{4}\). Angles between \(\frac{3\pi}{2}\) and \(2\pi\) lie in the fourth quadrant. So \(\theta =-\frac{\pi}{4}\) (or \(\frac{7\pi}{4}\)) lies in the fourth quadrant.

Step2: Find the reference angle

The reference angle \(\theta_{r}\) for an angle \(\theta\) in the fourth quadrant is given by \(\theta_{r} = 2\pi-\theta\) (when \(\theta\) is positive equivalent). For \(\theta =-\frac{\pi}{4}\) (or \(\theta=\frac{7\pi}{4}\)), the reference angle \(\theta_{r}=\frac{\pi}{4}\)

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

We know that:

• \(\sin(-\frac{\pi}{4})=-\sin(\frac{\pi}{4})\) (since \(\sin\) is an odd function and in the fourth quadrant \(\sin\) is negative). Using the unit - circle value \(\sin(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\), so \(\sin(-\frac{\pi}{4})=-\frac{\sqrt{2}}{2}\)
• \(\cos(-\frac{\pi}{4})=\cos(\frac{\pi}{4})\) (since \(\cos\) is an even function and in the fourth quadrant \(\cos\) is positive). Using the unit - circle value \(\cos(\frac{\pi}{4})=\frac{\sqrt{2}}{2}\), so \(\cos(-\frac{\pi}{4})=\frac{\sqrt{2}}{2}\)
• \(\tan(-\frac{\pi}{4})=-\tan(\frac{\pi}{4})\) (since \(\tan\) is an odd function). Using the unit - circle value \(\tan(\frac{\pi}{4}) = 1\), so \(\tan(-\frac{\pi}{4})=- 1\)

Answer:

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