Question

Trig help me find the other five trig values deep explanatory needed pls #5

Answer 1

As we can see, the sine and the cosine are negative:

$\begin{gathered} \cos\theta=-\frac{\sqrt{10}}{5} \\ \\ \sin\theta\lt0 \end{gathered}$

This is true for angles in the third quadrant:

$180\degree\lt\theta\lt270\degree$

Then, using the cosine value, we can find the remaining side. From the Pythagorean Theorem:

$\begin{gathered} CO^2+CA^2=H^2 \\ \\ CO^2+10=25 \\ \\ \Rightarrow CO=-\sqrt{15} \end{gathered}$

Then, the remaining trigonometric functions are:

$\sin\theta=-\frac{\sqrt{15}}{5}$$\begin{gathered} \tan\theta=\frac{-\sqrt{15}}{-\sqrt{10}}=\frac{\sqrt{15}\cdot\sqrt{10}}{10} \\ \\ \Rightarrow\tan\theta=\frac{\sqrt{6}}{2} \end{gathered}$$\begin{gathered} \sec\theta=\frac{1}{\cos\theta}=-\frac{5}{\sqrt{10}}\cdot\frac{\sqrt{10}}{\sqrt{10}} \\ \\ \Rightarrow\sec\theta=-\frac{\sqrt{10}}{2} \end{gathered}$$\begin{gathered} \csc\theta=\frac{1}{\sin\theta}=-\frac{5}{\sqrt{15}}\cdot\frac{\sqrt{15}}{\sqrt{15}} \\ \\ \Rightarrow\csc\theta=-\frac{\sqrt{15}}{3} \end{gathered}$$\begin{gathered} \ctg\theta=\frac{1}{\tan\theta}=\frac{2}{\sqrt{6}}\cdot\frac{\sqrt{6}}{\sqrt{6}} \\ \\ \Rightarrow\ctg\theta=\frac{\sqrt{6}}{3} \end{gathered}$