In right triangle , angle is the right angle. If , what is ?

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SAT Right Triangles & Trigonometry Question 44 | Free SAT Prep | Aniko

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Question 44·Medium·Right Triangles and Trigonometry

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In right triangle $XYZ$ , angle $Z$ is the right angle. If $\sin X = \dfrac{4}{7}$ , what is $\tan Y$ ?

When a sine value is given in a right triangle, treat it as opposite over hypotenuse. Assign proportional side lengths, find the third side with the Pythagorean theorem, then form the requested tangent ratio from opposite over adjacent.

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Use the sine ratio

Since sin X is opposite over hypotenuse, label the side opposite X as 4 parts and the hypotenuse as 7 parts.

Find the third side

Use the Pythagorean theorem with one leg 4 and hypotenuse 7.

Use tangent for angle Y

For angle Y, tangent is the side opposite Y divided by the side adjacent to Y.

Step-by-step Explanation

Label sides from the sine ratio

Angle $Z$ is the right angle, so $XY$ is the hypotenuse. Since $\sin X=\dfrac{4}{7}$ , the side opposite angle $X$ can be represented by $4$ and the hypotenuse by $7$ . Thus let $YZ=4$ and $XY=7$ .

Find the missing leg

Use the Pythagorean theorem:

XZ^2+YZ^2=XY^2.

Substitute $YZ=4$ and $XY=7$ :

XZ^2+4^2=7^2 \Rightarrow XZ^2=33 \Rightarrow XZ=\sqrt{33}.

Compute tangent for angle Y

For angle $Y$ , the opposite side is $XZ=\sqrt{33}$ and the adjacent side is $YZ=4$ . Therefore

\tan Y=\frac{\sqrt{33}}{4}.

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