Question

w8 version (november 2025) name: 4. the point (-5,-6) lies on the terminal arm of an angle θ in standard position, where 0° ≤ θ ≤ 360°. determine the exact value of each of the following. do not find the value of the angle. 3 a) cscθ = b) tanθ = 5. determine the value of x. round to the nearest tenth. 5

Explanation:

Step1: Find the radius \( r \)

For a point \((x,y)\) on the terminal arm of an angle, \(r=\sqrt{x^{2}+y^{2}}\). Given \(x = - 5\) and \(y=-6\), then \(r=\sqrt{(-5)^{2}+(-6)^{2}}=\sqrt{25 + 36}=\sqrt{61}\).

Step2: Calculate \(\csc\theta\)

We know that \(\csc\theta=\frac{r}{y}\). Substituting \(r = \sqrt{61}\) and \(y=-6\), we get \(\csc\theta=\frac{\sqrt{61}}{-6}=-\frac{\sqrt{61}}{6}\).

Step3: Calculate \(\tan\theta\)

We know that \(\tan\theta=\frac{y}{x}\). Substituting \(x=-5\) and \(y = - 6\), we get \(\tan\theta=\frac{-6}{-5}=\frac{6}{5}\).

Answer:

a) \(-\frac{\sqrt{61}}{6}\)
b) \(\frac{6}{5}\)