Question

Express your answer in a simplest rational form. Find each ratio.tan0 cos0 csc0

Answer 1

It is given that the terminal side of angle θ passes through the point (-5/13, 12/13) in quadrant II on a unit circle.

`(x,y)=(-(5)/(13),(12)/(13))`

Let us find the given trigonometric ratios in the simplest rational form.

First, we need to find the hypotenuse side using the Pythagorean theorem.

`\begin{gathered} c^2=a^2+b^2 \\ c^2=(-(5)/(13))^2+((12)/(13))^2 \\ c^2=(25)/(169)+(144)/(169) \\ c^2=(169)/(169) \\ c^=\sqrt{(169)/(169)} \\ c=1 \end{gathered}`

So, the hypotenuse side is 1.

`\begin{gathered} \tan\theta=(y)/(x) \\ \cos\theta=(x)/(h) \\ \csc\theta=(h)/(y) \end{gathered}`

Where

x = -5/13

y = 12/13

h = 1

`\begin{gathered} \tan\theta=((12)/(13))/(-(5)/(13))=-(12)/(5) \\ \cos\theta=((-5)/(13))/(1)=-(5)/(13) \\ \csc\theta=(1)/((12)/(13))=(13)/(12) \end{gathered}`

Therefore, the trigonometric ratios in the simplest rational form are

`\begin{gathered} \tan\theta=-(12)/(5) \\ \cos\theta=-(5)/(13) \\ \csc\theta=(13)/(12) \end{gathered}`