angle $\theta$ is in standard position. if $(...

# Explanation: ## Step1: Calculate the radius $r$ Given the point $(x = 8,y=- 15)$ on the terminal - ray of the angle $\theta$ in standard position. Use the formula $r=\sqrt{x^{2}+y^{2}}$. So, $r=\sqrt{8^{2}+(-15)^{2}}=\sqrt{64 + 225}=\sqrt{289}=17$. ## Step2: Calculate $\sin(\theta)$ By the definition of sine function $\sin(\theta)=\frac{y}{r}$. Substitute $y=-15$ and $r = 17$, we get $\sin(\theta)=-\frac{15}{17}$. ## Step3: Calculate $\cos(\theta)$ By the definition of cosine function $\cos(\theta)=\frac{x}{r}$. Substitute $x = 8$ and $r = 17$, we get $\cos(\theta)=\frac{8}{17}$. ## Step4: Calculate $\tan(\theta)$ By the definition of tangent function $\tan(\theta)=\frac{y}{x}$. Substitute $x = 8$ and $y=-15$, we get $\tan(\theta)=-\frac{15}{8}$. ## Step5: Calculate $\csc(\theta)$ Since $\csc(\theta)=\frac{1}{\sin(\theta)}$, and $\sin(\theta)=-\frac{15}{17}$, then $\csc(\theta)=-\frac{17}{15}$. ## Step6: Calculate $\sec(\theta)$ Since $\sec(\theta)=\frac{1}{\cos(\theta)}$, and $\cos(\theta)=\frac{8}{17}$, then $\sec(\theta)=\frac{17}{8}$. ## Step7: Calculate $\cot(\theta)$ Since $\cot(\theta)=\frac{1}{\tan(\theta)}$, and $\tan(\theta)=-\frac{15}{8}$, then $\cot(\theta)=-\frac{8}{15}$. # Answer: $\sin(\theta)=-\frac{15}{17}$ $\cos(\theta)=\frac{8}{17}$ $\tan(\theta)=-\frac{15}{8}$ $\csc(\theta)=-\frac{17}{15}$ $\sec(\theta)=\frac{17}{8}$ $\cot(\theta)=-\frac{8}{15}$