1. illustrate each of the following trigonometric ratios using a sketch on the provided unit circle with a…

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\neq0)), (\cot\theta=\frac{x}{y}(y\neq0)), (\csc\theta=\frac{1}{y}(y\neq0)), (\sec\theta=\frac{1}{x}(x\neq0))

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.