the figure shows an angle θ in standard position with its terminal side intersecting the unit circle…

Explanation:

Step1: Recall circular - function definitions

For a point ((x,y)) on the unit - circle and an angle (\theta) in standard position whose terminal side intersects the unit - circle at ((x,y)), (\sin\theta=y), (\cos\theta = x), (\tan\theta=\frac{y}{x}(x\neq0)), (\csc\theta=\frac{1}{y}(y\neq0)), (\sec\theta=\frac{1}{x}(x\neq0)), (\cot\theta=\frac{x}{y}(y\neq0)).

Step2: Identify (x) and (y) values

The point on the unit - circle is ((x,y)=\left(-\frac{4}{5},-\frac{3}{5}\right)).

Step3: Calculate (\cos\theta)

Since (\cos\theta=x), then (\cos\theta =-\frac{4}{5}).

Step4: Calculate (\tan\theta)

(\tan\theta=\frac{y}{x}=\frac{-\frac{3}{5}}{-\frac{4}{5}}=\frac{3}{4}).

Step5: Calculate (\csc\theta)

(\csc\theta=\frac{1}{y}=\frac{1}{-\frac{3}{5}}=-\frac{5}{3}).

Step6: Calculate (\sec\theta)

(\sec\theta=\frac{1}{x}=\frac{1}{-\frac{4}{5}}=-\frac{5}{4}).

Step7: Calculate (\cot\theta)

(\cot\theta=\frac{x}{y}=\frac{-\frac{4}{5}}{-\frac{3}{5}}=\frac{4}{3}).

Answer:

(\cos\theta =-\frac{4}{5}), (\tan\theta=\frac{3}{4}), (\csc\theta =-\frac{5}{3}), (\sec\theta =-\frac{5}{4}), (\cot\theta=\frac{4}{3})