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Answer 1

Trigonometric Ratios

A right triangle has one angle of 90° and two acute angles. There are special proportions between the side lengths called the trigonometric ratios.

All the trigonometric ratios in a right triangle are positive.

We need to use some trigonometric identities to find the other five ratios, given one of them:

`\begin{gathered} \sin ^2\theta+\cos ^2\theta=1 \\ \tan \theta=(\sin \theta)/(\cos \theta) \\ \csc \theta=(1)/(\sin \theta) \\ \sec \theta=(1)/(\cos \theta) \\ \cot \theta=(\cos\theta)/(\sin\theta)=(1)/(\tan \theta) \end{gathered}`

We are given:

`\sin \theta=(6)/(11)`

From the first identity, solve for the cosine:

`\cos \theta=√(1-\sin^2\theta)`

Substituting:

`\begin{gathered} \cos \theta=\sqrt[]{1-((6)/(11))^2} \\ \cos \theta=\sqrt[]{1-(36)/(121)} \\ \cos \theta=\sqrt[]{(121-36)/(121)} \\ \cos \theta=\sqrt[]{(85)/(121)} \\ \cos \theta=\frac{\sqrt[]{85}}{11} \end{gathered}`

Calculate the tangent:

`\begin{gathered} \tan \theta=\frac{(6)/(11)}{\frac{\sqrt[]{85}}{11}} \\ \tan \theta=\frac{6}{\sqrt[]{85}} \\ \text{Rationalizing:} \\ \tan \theta=\frac{6}{\sqrt[]{85}}\cdot\frac{\sqrt[]{85}}{\sqrt[]{85}} \\ \tan \theta=\frac{6\sqrt[]{85}}{85} \end{gathered}`

Calculate the cosecant:

`\begin{gathered} \csc \theta=(1)/((6)/(11)) \\ \csc \theta=(11)/(6) \end{gathered}`

Calculate the secant:

`\begin{gathered} \sec \theta=\frac{1}{\frac{\sqrt[]{85}}{11}} \\ \sec \theta=\frac{11}{\sqrt[]{85}} \\ \text{Rationalizing:} \\ \sec \theta=\frac{11}{\sqrt[]{85}}\cdot\frac{\sqrt[]{85}}{\sqrt[]{85}} \\ \sec \theta=\frac{11\sqrt[]{85}}{85} \end{gathered}`

Finally, calculate the cotangent:

`\begin{gathered} \cot \theta=\frac{\frac{\sqrt[]{85}}{11}}{(6)/(11)} \\ \cot \theta=\frac{\sqrt[]{85}}{6} \end{gathered}`