Introduction

Recall that two triangles are called similar if one of them can be obtained by rescaling and moving around the other. Here’s an example:

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The dotted square symbol is a shorthand for “ degrees”. Triangles which have a angle are called right triangles, and will be the main focus of our discussion. What do similar triangles actually have in common? Certainly angles, but not necessarily the lengths of the sides. However, the ratios between any two sides of a triangle will remain the same, no matter how the triangle gets rescaled. Such ratios ultimately give us so much information about the given right triangle that they deserve special names: sine, cosine, and tangent.

Definitions and examples

Often, one has information about the angles, but not about all the sides. Knowing , and helps us find out missing sides of a given right triangle. For that, the following fact is extremely important:

Why is this true? Consider again a right triangle like below:

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Then we know that and . But the Pythagorean theorem also says that . Putting all of this together, we have that as required.

Let’s see how to apply this.

Values of trig functions for standard angles

We know that the sum of the inner angles of a triangle is always . For right triangles, one of the angles is , which means that the sum of the remaining two angles must also be . Frequently we encounter triangles whose angles are , and , and also triangles whose angles are , and .

These triangles have a special type of symmetry, which we’ll exploit to find the values of sine, cosine, and tangent, for , and . Finding the values of these trig functions for arbitrary angles by hand is a very difficult task. We will see later some trigonometric identities that may help us find such values for other angles but, in general, using a calculator (paying close attention to whether it is set to the right “units”) is the way to go.

For and

Consider an equilateral triangle of side length . Equilateral means that all the sides have the same length. This implies that all the inner angles must be equal and, since they must add up to , each of them equals . But also draw a height :

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By the Pythagorean theorem, we know that and so we may compute:

Now, relative to the angle, we recognize

This means that as well as

To find the values of , , and , we can use the same triangle, noting that the opposite side to is the adjacent side to , and that the adjacent side to is the opposite side to . Since the hypotenuse is always the side opposite to the right angle, we conclude that and, finally, that

For

Consider a square of side length , and draw a diagonal .

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By the Pythagorean theorem, implies that . Relative to either of the angles, we have Hence

Standard values

We can summarize what we have discovered here in a table. Besides our standard angles of , , and , we can also consider and as extreme cases. Let’s do a quick thought experiment to understand this: if a right triangle had an angle of , this triangle would in fact collapse to a line segment, and we would have , while , suggesting we set and . Since and are complementary, we’re forced to set and . But while computing does not make sense, as we would have a division by . We say that is undefined, or that it does not exist (“DNE” for short, as usual). So, we have:

Those values should be committed to heart, but it’s easier than it seems. Here’s how you can think about it:

  • No need to memorize the values for tangent: if you know and , you can just compute .
  • No need to memorize the values for cosine: recall that the cosine of an angle is the sine of the complement. So if you know the values for sine, you’re in business.
  • How to memorize the values for sine? The one thing you should remember here is that the values , , , and will appear. What is their order? Simple: write them in increasing order, just like the angles from to . So