Question

the figure shows an angle θ in standard position with its terminal side intersecting the unit - circle. evaluate the six circular function values of θ.
sin θ = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
cos θ = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
tan θ = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
sec θ = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
csc θ = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
cot θ = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall unit - circle definitions

For a point \((x,y)\) on the unit - circle corresponding to an angle \(\theta\), \(\sin\theta=y\), \(\cos\theta = x\), \(\tan\theta=\frac{y}{x}(x
eq0)\), \(\sec\theta=\frac{1}{x}(x
eq0)\), \(\csc\theta=\frac{1}{y}(y
eq0)\), \(\cot\theta=\frac{x}{y}(y
eq0)\). From the given point on the unit - circle \((x = \frac{3}{5},y=-\frac{4}{5})\).

Step2: Calculate \(\sin\theta\)

\(\sin\theta=y=-\frac{4}{5}\)

Step3: Calculate \(\cos\theta\)

\(\cos\theta=x = \frac{3}{5}\)

Step4: Calculate \(\tan\theta\)

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

Step5: Calculate \(\sec\theta\)

\(\sec\theta=\frac{1}{x}=\frac{5}{3}\)

Step6: Calculate \(\csc\theta\)

\(\csc\theta=\frac{1}{y}=-\frac{5}{4}\)

Step7: Calculate \(\cot\theta\)

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

Answer:

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