Question

If cos(theta) = 5/0 and is in the 1st quadrant, find the following:

Answer 1

step 1

Find out sine

Remember the identity

`cos^2\theta+sin^2\theta=1`

substitute the given value of cosine

`\begin{gathered} ((5)/(9))^2+s\imaginaryI n^2\theta=1 \\ \\ s\imaginaryI n^2\theta=1-(25)/(81) \\ \\ s\mathrm{i}n^2\theta=(56)/(81) \\ \\ sin\theta=(√(56))/(9) \\ \\ sin\theta=(2√(14))/(9) \end{gathered}`

step 2

Find out cosecant

`\begin{gathered} csc\theta=(1)/(sin\theta) \\ \\ csc\theta=(9)/(2√(14))*(√(14))/(√(14))=(9√(14))/(28) \\ \\ csc\theta=(9√(14))/(28) \end{gathered}`

step 3

Find out secant

`\begin{gathered} sec\theta=(1)/(cos\theta) \\ \\ sec\theta=(9)/(5) \end{gathered}`

step 4

Find out tangent

`\begin{gathered} tan\theta=(sin\theta)/(cos\theta) \\ \\ tan\theta=((2√(14))/(9))/((5)/(9))=(2√(14))/(5) \end{gathered}`

step 5

Find out cotangent

`\begin{gathered} cot\theta=(1)/(tan\theta) \\ \\ cot\theta=(5)/(2√(14))*(√(14))/(√(14))=(5√(14))/(28) \\ \\ cot\theta=(5√(14))/(28) \end{gathered}`