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(θ), cos(θ), and tan(θ)

(a) (3 points) θ = \frac{-π}{4}

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 \(\theta\) 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 \(\theta_{r} = 2\pi-\theta\). For \(\theta =-\frac{\pi}{4}\) (or \(\theta=\frac{7\pi}{4}\)), \(\theta_{r}=\frac{\pi}{4}\)

Step3: Calculate the trigonometric functions

We know that:

• \(\sin(-\frac{\pi}{4})=-\sin(\frac{\pi}{4})\) (since \(\sin\theta\) is odd function and in the fourth quadrant \(\sin\theta<0\)). 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 \(\cos\theta\) is even function and in the fourth quadrant \(\cos\theta > 0\)). 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 \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) and \(\tan\theta\) is odd function, and in the fourth quadrant \(\tan\theta<0\)). Using the unit - circle value \(\tan(\frac{\pi}{4}) = 1\), so \(\tan(-\frac{\pi}{4})=- 1\)

Answer:

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