Question

1. illustrate each of the following trigonometric ratios using a sketch on the provided unit circle with a properly labeled triangle, if possible. make sure to show the standard angle and the reference angle, if possible. illustrations without the unit circle will not be accepted! k/u - 10

a) ( cosleft(\frac{3pi}{4}
ight)=-\frac{sqrt{2}}{2} ) b) ( cscleft(\frac{7pi}{6}
ight)=-2 )

c) ( cotleft(\frac{5pi}{3}
ight)=-\frac{sqrt{3}}{3} ) d) ( \tanleft(\frac{pi}{2}
ight) ) or ( \tanleft(\frac{3pi}{2}
ight) dne )

e) using what you learned in class, write one/two sentences on how the unit circle is useful.

Explanation:

Step1: Analyze the unit - circle and trigonometric ratios

The unit circle is a circle with a radius \(r = 1\). For a point \((x,y)\) on the unit circle corresponding to an angle \(\theta\), we have \(\cos\theta=x\), \(\sin\theta = y\), \(\tan\theta=\frac{y}{x}(x
eq0)\), \(\cot\theta=\frac{x}{y}(y
eq0)\), \(\csc\theta=\frac{1}{y}(y
eq0)\), \(\sec\theta=\frac{1}{x}(x
eq0)\)

Step2: For part (a) \(\cos(\frac{3\pi}{4})\)

The standard angle \(\theta=\frac{3\pi}{4}\). The reference angle \(\theta_{r}=\pi-\frac{3\pi}{4}=\frac{\pi}{4}\). On the unit circle, the \(x\) - coordinate (since \(\cos\theta=x\)) of the point corresponding to \(\theta = \frac{3\pi}{4}\) is \(-\frac{\sqrt{2}}{2}\). The terminal side of the angle \(\frac{3\pi}{4}\) lies in the second quadrant.

Step3: For part (b) \(\csc(\frac{7\pi}{6})\)

First, \(\csc\theta=\frac{1}{\sin\theta}\). If \(\csc(\frac{7\pi}{6})=- 2\), then \(\sin(\frac{7\pi}{6})=-\frac{1}{2}\). The standard angle \(\theta=\frac{7\pi}{6}\). The reference angle \(\theta_{r}=\frac{7\pi}{6}-\pi=\frac{\pi}{6}\). The terminal side of the angle \(\frac{7\pi}{6}\) lies in the third quadrant.

Step4: For part (c) \(\cot(\frac{5\pi}{3})\)

Since \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), if \(\cot(\frac{5\pi}{3})=-\frac{\sqrt{3}}{3}\), and \(\cos(\frac{5\pi}{3})=\frac{1}{2}\), \(\sin(\frac{5\pi}{3})=-\frac{\sqrt{3}}{2}\), \(\cot(\frac{5\pi}{3})=\frac{\cos(\frac{5\pi}{3})}{\sin(\frac{5\pi}{3})}=\frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=-\frac{\sqrt{3}}{3}\). The standard angle \(\theta=\frac{5\pi}{3}\). The reference angle \(\theta_{r}=2\pi-\frac{5\pi}{3}=\frac{\pi}{3}\). The terminal side of the angle \(\frac{5\pi}{3}\) lies in the fourth quadrant.

Step5: For part (d) \(\tan(\frac{\pi}{2})\) and \(\tan(\frac{3\pi}{2})\)

Since \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), for \(\theta=\frac{\pi}{2}\), \(\cos(\frac{\pi}{2}) = 0\) and \(\sin(\frac{\pi}{2})=1\); for \(\theta=\frac{3\pi}{2}\), \(\cos(\frac{3\pi}{2}) = 0\) and \(\sin(\frac{3\pi}{2})=-1\). Division by zero is undefined, so \(\tan(\frac{\pi}{2})\) and \(\tan(\frac{3\pi}{2})\) do not exist (DNE). The terminal side of \(\frac{\pi}{2}\) is the positive \(y\) - axis and the terminal side of \(\frac{3\pi}{2}\) is the negative \(y\) - axis.

Step6: For part (e) Importance of the unit circle

The unit circle is useful because it provides a geometric interpretation of trigonometric functions. For any angle \(\theta\), the coordinates of the point on the unit circle corresponding to \(\theta\) give the values of \(\cos\theta\) ( \(x\) - coordinate) and \(\sin\theta\) ( \(y\) - coordinate). It helps in visualizing the signs of trigonometric functions in different quadrants and in finding reference angles.

Answer:

a) Standard angle \(\frac{3\pi}{4}\), reference angle \(\frac{\pi}{4}\), terminal side in the second quadrant with \(x =-\frac{\sqrt{2}}{2}\)
b) Standard angle \(\frac{7\pi}{6}\), reference angle \(\frac{\pi}{6}\), terminal side in the third quadrant with \(\sin\theta=-\frac{1}{2}\)
c) Standard angle \(\frac{5\pi}{3}\), reference angle \(\frac{\pi}{3}\), terminal side in the fourth quadrant with \(\cot\theta =-\frac{\sqrt{3}}{3}\)
d) \(\tan(\frac{\pi}{2})\) and \(\tan(\frac{3\pi}{2})\) are undefined (DNE) as \(\cos(\frac{\pi}{2})=\cos(\frac{3\pi}{2}) = 0\)
e) The unit circle is useful as it gives a geometric way to understand trigonometric functions ( \(\cos\theta=x\), \(\sin\theta = y\) for a point \((x,y)\) on the unit - circle corresponding to angle \(\theta\)) and helps in determining signs and reference angles of trigonometric ratios.