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=-(√(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=-√(15) \end{gathered}`

Then, the remaining trigonometric functions are:

`\sin\theta=-(√(15))/(5)`
`\begin{gathered} \tan\theta=(-√(15))/(-√(10))=(√(15)\cdot√(10))/(10) \\ \\ \Rightarrow\tan\theta=(√(6))/(2) \end{gathered}`
`\begin{gathered} \sec\theta=(1)/(\cos\theta)=-(5)/(√(10))\cdot(√(10))/(√(10)) \\ \\ \Rightarrow\sec\theta=-(√(10))/(2) \end{gathered}`
`\begin{gathered} \csc\theta=(1)/(\sin\theta)=-(5)/(√(15))\cdot(√(15))/(√(15)) \\ \\ \Rightarrow\csc\theta=-(√(15))/(3) \end{gathered}`
`\begin{gathered} \ctg\theta=(1)/(\tan\theta)=(2)/(√(6))\cdot(√(6))/(√(6)) \\ \\ \Rightarrow\ctg\theta=(√(6))/(3) \end{gathered}`