To start, a **ratio** is a mathematical *comparison*. In other words, the *comparison* of any two quantities is called a *ratio*. Any time you work with fractions, for example, you’re technically comparing the numerator quantity to the denominator:

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## The Tangent Ratio

If you have a right triangle, and you chose one of the acute angles in it (acute meaning less than 90°) as your **reference angle**, the **opposite length** from that angle compared to the **adjacent length** is called the **tangent ratio**. Notice in the animation that the **acute angles** are 30° and 60º. The side *opposite* of 30° is y and *adjacent* is x. Whereas if we chose 60º as our acute angle, the *opposite* length is x and *adjacent* is y. That’s summarized below:

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In case you forget, the tangent ratio can be remembered as **T**_{angent}**O**_{pposite}**A**_{djacent} (TOA).

## The Sine Ratio

What if we wanted to compare the ** opposite **to

**length? In that case, we’d use another trigonometric ratio known as**

*hypotenuse***Sine**. Let’s investigate sine using the generic right triangle below:

Notice the symbol (θ, “theta”). That will be considered our **reference angle **here. As mentioned before, the reference angle can be any of the acute angles (less than 90°) in the right triangle – never choose the 90° angle as your reference. Therefore, the sine ratio applied to θ is summarized below:

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The sine ratio can be remembered as **S**_{ine}**O**_{pposite}**H**_{potenuse} (SOH).

## The Cosine Ratio

The last comparison we’ll make using the same right triangle shown above is the **cosine ratio**. The cosine ratio is a comparison of the **adjacent length** to the **hypotenuse** relative to the angle θ:

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The sine ratio can be remembered as **C**_{osine}**A**_{djacent}**H**_{potenuse} (CAH). Collectively, the **sine**, **cosine**, and **tangent ratios** are referred to as **trigonometric ratios**. Using all three abbreviations stated above, the best way to remember the ratios is through the mnemonic:

**SOH CAH** **TOA**

Trigonometric ratios can be used to find two main things:

**Missing sides**in a right triangle**Missing angles**in a right triangle

To find the length of an unknown side, you must be given **one acute reference angle** and **one known side**. Then you have a straight forward calculation involving your calculator and some algebraic manipulation. Keep in mind that most modern day calculators come **pre-programmed** with all possible ratios of sine, cosine, and tangent for any angle. So if you wanted to find the ratio for **sin(50°)**, you’d click the sine button (sine function), then your angle (shown in yellow):

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Try it yourself, and make sure your calculator is in **degree mode** (shown in red). Notice that four numbers after the decimal place were kept for good measure. This will ensure that if you do the opposite – where you want to find an angle given the ratio – you get an angle that’s accurate to 50° (this will be discussed further below). Also, by writing **0.7660 over 1**, it illustrates how sine of 50 degrees is comparing the opposite length (0.7660) to the hypotenuse (1).

This leads us to the second reason we use trigonometric ratios. As mentioned in point (2), if we have **2 known sides**, the unknown acute angle of any of the vertices can be found using **inverse trigonometric functions** that are also found on your calculator (blue arrow). You’ll have to click “shift” or “2nd” first to access these functions. The two videos below will walk you through four different examples related to finding the angle when only the sides are known.

Of course, this lesson wouldn’t be complete if we didn’t see examples where we find an **unknown side** when given a known **acute angle** and a **known side**.

Summary:The three primary trigonometric ratios are sine, cosine, and tangent. They are defined as follows: null

You can find any side length or angle measure of a right triangle if you know

two pieces of informationin addition to the right angle.