Question

Find the TANGENT X. Write your answers in simplified, rationalized form. DO NOT ROUND!

Answer 1

`\tan X=\frac{3\sqrt[]{65}}{13}`

Step-by-step explanation:

From the diagram:

• The side ,opposite, angle X is ZY

,

• The side ,adjacent to, angle X is XY

From trigonometric ratios, we know that:

`\begin{gathered} \tan \theta=\frac{Opposite}{\text{Adjacent}} \\ \implies\tan X=(ZY)/(XY) \end{gathered}`

Since we require the value of ZY, we find it using Pythagoras Theorem.

`\begin{gathered} XZ^2=XY^2+ZY^2 \\ \sqrt[]{58}^2=\sqrt[]{13}^2+ZY^2 \\ ZY^2=58-13 \\ ZY^2=45 \\ ZY=\sqrt[]{45} \end{gathered}`

Therefore:

`\begin{gathered} \tan X=(ZY)/(XY) \\ =\frac{\sqrt[]{45}}{\sqrt[]{13}} \end{gathered}`

We rationalize our result.

`\begin{gathered} =\frac{\sqrt[]{45}}{\sqrt[]{13}}*\frac{\sqrt[]{13}}{\sqrt[]{13}} \\ \tan X=\frac{3\sqrt[]{65}}{13} \end{gathered}`

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