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### Course: High school geometry>Unit 5

Lesson 9: Modeling with right triangles

# Right triangles and trigonometry FAQ

## What are the special right triangles?

Special right triangles are two specific types of right triangles that have unique ratios between their side lengths. We can use the Pythagorean theorem to convince ourselves that these ratios are true.
The first type is the $45\mathrm{°}$-$45\mathrm{°}$-$90\mathrm{°}$ triangle, where the two legs are the same length and the hypotenuse is $\sqrt{2}$ times the leg length. The second type is the $30\mathrm{°}$-$60\mathrm{°}$-$90\mathrm{°}$ triangle, where the side lengths are in a ratio of $1:\sqrt{3}:2$.
One way we use the special right triangles is to help us find the lengths of unknown sides. For example, if we know that we have a $45\mathrm{°}$-$45\mathrm{°}$-$90\mathrm{°}$ triangle and we know one of the side lengths, we can use the ratios to solve for the other two sides. If one of the legs is $5$ units long, then the other leg is also $5$ units long, and the hypotenuse is $5\sqrt{2}$ units long.
Try it yourself with our Special right triangles exercise.

## What are the trigonometric ratios?

Trigonometric ratios are ratios that compare the lengths of the sides of a right triangle. There are three primary trigonometric ratios: sine, cosine, and tangent.
We always calculate trigonometric ratios based on an angle measure. In right triangles, we use one of the acute angle measures. Then we call the longest side the "hypotenuse," the other side forming the angle the "adjacent" side, and the last side (which does not form one of the rays of the angle) the "opposite" side.
Sine is defined as the ratio of the opposite side of a right triangle to the hypotenuse. So, if we have a right triangle with an opposite side of length $a$ and a hypotenuse of length $c$, the sine of the angle is $\frac{a}{c}$.
Cosine is defined as the ratio of the adjacent side of a right triangle to the hypotenuse. So, if we have a right triangle with an adjacent side of length $b$ and a hypotenuse of length $c$, the cosine of the angle is $\frac{b}{c}$.
Tangent is defined as the ratio of the opposite side of a right triangle to the adjacent side. So, if we have a right triangle with an opposite side of length $a$ and an adjacent side of length $b$, the tangent of the angle is $\frac{a}{b}$.
We abbreviate sine as "sin", cosine as "cos", and tangent as "tan".
If we know the lengths of the sides of a right triangle, we can use these definitions to calculate the trigonometric ratios.

## How do we use the trigonometric ratios to solve for side lengths in a right triangle?

First, we need to identify which trigonometric ratio we need to use. Remember that the three primary ratios are sine, cosine, and tangent.
If we want to find the length of the opposite side and we know the hypotenuse, we can use the sine ratio. For example, if we know that $\mathrm{sin}\left(15\mathrm{°}\right)\approx 0.259$, we can set up an equation like this:
$\mathrm{sin}\left(15\mathrm{°}\right)=\frac{\text{opposite length}}{\text{hypotenuse length}}\approx 0.259$
If we know the hypotenuse is, say, $10$, we can solve for the opposite side:
$\begin{array}{rl}\frac{\text{opposite length}}{10}& \approx 0.259\\ \\ \text{opposite length}& \approx 2.59\end{array}$
In general, we can use the same process for any of the three ratios. If we know two of the three side lengths and the angle measure, we can use the appropriate trigonometric ratio to solve for the missing side length.

## How do we use the trigonometric ratios to solve for an angle in a right triangle?

To use the trigonometric ratios, we can set up an equation using the information we have. For example, if we know the length of the hypotenuse and the length of the opposite side, we can use the sine ratio to find an angle measure.
Let's say the hypotenuse is $10$ and the opposite side is $5$. We can set up an equation using the sine ratio:
$\mathrm{sin}\left(\theta \right)=\frac{5}{10}=\frac{1}{2}$
To solve for the angle measure $\theta$, we can use the inverse sine function on our calculator.
$\theta \approx {\mathrm{sin}}^{-1}\left(\frac{1}{2}\right)\approx 30\mathrm{°}$

## How do we use right triangles to model real-world situations?

There are many real-world applications of right triangle trigonometry. For example, we can use the trigonometric ratios to solve problems involving height and distance, such as determining the height of a building by measuring the shadow it casts. We can also use right triangle trigonometry in fields such as surveying, construction, and engineering.

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