Question

Find exact tan0, and csc0, where is the angle shown in the figure. Give exact values not decimals.

Answer 1

Given the triangle below:

To find the exact values of tan θ, cos θ, and csc θ.

`\begin{gathered} adj=√((hyp)^2-(opp)^2) \\ =√(8^2-7^2) \\ =√(15) \end{gathered}`

A) Tan θ:

`\begin{gathered} tan\theta=(opposite)/(adjacent) \\ =(7)/(√(15))*(√(15))/(√(15)) \\ \Rightarrow\tan\theta=(7√(15))/(15) \end{gathered}`

B) cos θ:

`\begin{gathered} \cos\theta=(adjavent)/(hypotenuse) \\ =(√(15))/(8) \\ \Rightarrow\cos\theta=(√(15))/(8) \end{gathered}`

C) csc θ:

`\begin{gathered} csc\theta=(1)/(\sin\theta)=(1)/((opposite)/(hypotenuse)) \\ =(1)/((7)/(8)) \\ \\ \Rightarrow csc\theta=(8)/(7) \end{gathered}`

Hence, we have

`\begin{gathered} \tan\theta=(7√(15))/(15) \\ \\ \cos\theta=(√(15))/(8) \\ \\ csc\theta=(8)/(7) \\ \end{gathered}`