6) sketch the angle in standard position and find the exact value for each trigonometric function given the…

Explanation:

Step1: Calculate the radius (r)

For a point ((x,y)=(-3,-4)) on the terminal side of an angle (\theta) in standard position, use the formula (r = \sqrt{x^{2}+y^{2}}). [ \begin{align*} r&=\sqrt{(-3)^{2}+(-4)^{2}}\ &=\sqrt{9 + 16}\ &=\sqrt{25}\ &=5 \end{align*} ]

Step2: Calculate the trigonometric functions

• Sine function: (\sin\theta=\frac{y}{r}), with (y=-4) and (r = 5), so (\sin\theta=\frac{-4}{5})
• Cosecant function: (\csc\theta=\frac{r}{y}), with (r = 5) and (y=-4), so (\csc\theta=\frac{5}{-4}=-\frac{5}{4})
• Cosine function: (\cos\theta=\frac{x}{r}), with (x=-3) and (r = 5), so (\cos\theta=\frac{-3}{5})
• Secant function: (\sec\theta=\frac{r}{x}), with (r = 5) and (x=-3), so (\sec\theta=\frac{5}{-3}=-\frac{5}{3})
• Tangent function: (\tan\theta=\frac{y}{x}), with (y=-4) and (x=-3), so (\tan\theta=\frac{-4}{-3}=\frac{4}{3})
• Cotangent function: (\cot\theta=\frac{x}{y}), with (x=-3) and (y=-4), so (\cot\theta=\frac{-3}{-4}=\frac{3}{4})

Answer:

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