Mathematics
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• 7 week duration
• Unit 3: Geometric Properties

Most relations that you have studied in mathematics have been linear. However, many non-linear also exist in real life.

The Tangent, Sine, and Cosine Ratio

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:

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 ratioNotice 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:

In case you forget, the tangent ratio can be remembered as TangentOppositeAdjacent (TOA).

The Sine Ratio

What if we wanted to compare the opposite to hypotenuse length? In that case, we’d use another trigonometric ratio known as 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:

$\mathrm{sin}\left(\theta \right)=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}\phantom{\rule{0ex}{0ex}}$

The sine ratio can be remembered as SineOppositeHpotenuse (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:

$\mathrm{cos}\left(\theta \right)=\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}\phantom{\rule{0ex}{0ex}}$

The sine ratio can be remembered as CosineAdjacentHpotenuse (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   CAHTOA

Trigonometric ratios can be used to find two main things:

1. Missing sides in a right triangle
2. 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):

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:

You can find any side length or angle measure of a right triangle if you know two pieces of information in addition to the right angle.