Trigonometric Function Values of Special Angles

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  • 0:01 A Right Triangle
  • 0:51 Our Trigonometric Functions
  • 3:44 Our Special Angles
  • 4:39 Our Trigonometric Values
  • 6:21 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After watching this video lesson, you will better understand the six trigonometric functions. You will also know what special angles there are and what the trigonometric values for these special angles are.

A Right Triangle

You can't have trigonometry without a right triangle. Why? We need our right triangle because a right triangle helps us to understand the relationships between our trigonometric functions. Recall that a right triangle is a triangle with one right angle that measures 90 degrees. This triangle, for example, is a right triangle:

trig functions

Do you see the right angle marked by the square box? That is our one right angle.

The long side, the side opposite the right angle, is always our hypotenuse. The side opposite the marked angle is called the opposite side. The side next to the angle, but not the hypotenuse, is called the adjacent side. We have labeled the one angle. If we are working off the other angle, then we would label that other angle. All of our trigonometric functions are ratios of the sides of our right triangle.

Our Trigonometric Functions

Why do we need to understand our trigonometric functions? We need to because we will see these trigonometric functions time and again, not just in math, but also in real life. Being able to understand these will make our lives easier. What real life examples are there, you ask? Why, sound waves for one. If you look at your computer speakers, then you are looking at and listening to the output of trigonometric functions, particularly the sine wave.

We have six trigonometric functions in total. They are sine, cosine, tangent, cosecant, secant, and cotangent. Let's go over each one in detail. Our first trigonometric (or trig for short) function is the sine function. It is the ratio of the opposite side to the hypotenuse. We write the sine function like this, with just three letters:

trig functions

In fact, all of our trig functions are written with only three letters.

Our second trigonometric function is the cosine function. It is defined as the adjacent side divided by the hypotenuse. We write the cosine function like this:

trig functions

Next, we have the tangent function. We define the tangent as the opposite side divided by the adjacent side. We write it like this:

trig functions

Let's take a quick break here. How are we going to remember all these fractions? That's a good question, and I happen to have an answer for you.

When I was in school, my teacher told me about this one chief of a tribe back in the day. This chief was really good in math, and he could easily give you the above three trig functions off the top of his head. What was his name? His name was Chief Sohcahtoa.

Look at his name and now look at the first letter of every word we have for the above three definitions and you will see that they match. We have 'Sine is Opposite over Hypotenuse;' then we have 'Cosine is Adjacent over Hypotenuse;' and, finally, we have 'Tangent is Opposite over Adjacent' - SOH - CAH - TOA. If you can remember these three, then you can figure out the other three trig functions. Let's see how that works.

The next one we have is the cosecant. This is actually the reciprocal of the sine function. It is written like this:

trig functions

Because this is the reciprocal of the sine function, we have simply flipped the definition of the sine function for the cosecant definition. The sine function is opposite over hypotenuse. Flipping it, we get hypotenuse over opposite, our cosecant function.

If you just remember Chief SOH - CAH - TOA, and then remember that the cosecant is the reciprocal of the sine, then you will be able to figure out the cosecant. This rule applies to the other two trig functions that are left, as well, since they are also reciprocals of one of the first three.

We have secant, which is the reciprocal of the cosine function. We write it like this:

trig functions

Again, our definition is the flipped version of the cosine function.

Last, we have the cotangent function, which is the reciprocal of the tangent function. We write it like this:

trig functions

Our Special Angles

If you've played with any of these functions before, you will see that you get some crazy answers when you use a calculator to calculate various angles. There are, however, some special angles where the answers are nicer.

You will see these special angles listed in either degrees or radians. Both are ways of measuring angles. I will give both of them to you. When you do your calculations, make sure that your calculator is set to degrees if you are working with degrees or radians if you are working with radians. If it is set incorrectly, then your answers will be way off.

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