RTT - Solving Right Triangles Lesson

Solving Right Triangles

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Armed with the Pythagorean Theorem, trigonometric functions, and special right triangles, you are now almost ready to solve right triangles! Every right triangle has a right angle, two acute angles, a hypotenuse and two legs. Solving a right triangle means that you will be able to find ALL 3 angles and ALL 3 sides when you are given just a few select parts of the triangle.

How do we solve the right triangle when we only know one side length and one acute angle?

First, we can find the missing acute angle by using the Triangle Sum Theorem, which states that the angles of any triangle will always sum to 180 $LaTeX: ^\circ$ $^\circ$ .

180 $LaTeX: ^\circ$ $^\circ$ -25 $LaTeX: ^\circ$ $^\circ$ -90 $LaTeX: ^\circ$ $^\circ$ = 65 $LaTeX: ^\circ$ $^\circ$

To find the side lengths, we will use trigonometric ratios.

Since x is opposite of angle 25 $LaTeX: ^\circ$ $^\circ$ and we know the hypotenuse is 13, we will use the sine function. $LaTeX: \sin\theta=\frac{opp}{hyp}$ $\sin\theta=\frac{opp}{hyp}$

$LaTeX: \sin25^{\circ}=\frac{x}{13}$ $\sin25^{\circ}=\frac{x}{13}$

$LaTeX: x=13\sin25^{\circ}$ $x=13\sin25^{\circ}$

$LaTeX: x\approx5.5$ $x\approx5.5$

Since y is adjacent to angle 25 $LaTeX: ^\circ$ $^\circ$ , we will use the cosine function. $LaTeX: \cos\theta=\frac{adj}{hyp}$ $\cos\theta=\frac{adj}{hyp}$

$LaTeX: Cos25^{\circ}=\frac{y}{13}$ $Cos25^{\circ}=\frac{y}{13}$

$LaTeX: y=13\cos25^{\circ}$ $y=13\cos25^{\circ}$

$LaTeX: y\approx11.8$ $y\approx11.8$

Just to make sure we have it, let's try another practice problem together...

Try some for yourself!

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