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Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

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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.

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:

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.

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:

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:

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

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:

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:

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:

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.

There are only five special angles that we need to concern ourselves with. They are 0 degrees or 0 radians, 30 degrees or pi/6 radians, 45 degrees or pi/4 radians, 60 degrees or pi/3 radians, and 90 degrees or pi/2 radians. In table form, we have this:

To find our trig values, we plug these measurements into each of our trig functions to see what we get. We can fill in a table with the values that we find:

This may seem like a lot of values to remember, but there's a pattern that can help you. In fact, you really only need to know the values of the sine and cosine functions for these special angles to be able to fill in the rest of the table. To use this pattern, you only need to know two things:

- Tangent = sine/cosine. We only need to look at the basic formulas to know that this is true: tangent = sine/cosine = (opposite/hypotenuse) / (adjacent/hypotenuse) = opposite/adjacent!
- To find the values for the reciprocal functions (secant, cosecant, and cotangent), just take the reciprocal of the sine, cosine, and tangent values.

Let's test out this pattern for the first row of the table. At 0 degrees or 0 radians, sine is 0 and cosine is 1. Since tangent is sine/cosine, this means that tangent is 0/1, or just 0. Okay, now we just have to take the reciprocals of these values to find the values of the remaining functions.

Cosecant = 1/sine = 1/0. We can't divide by 0, so you'll see that this value is undefined in the table. Similarly, cotangent = 1/tangent = 1/0 is also undefined.

Finally, secant = 1/cosine = 1/1 = 1. Feel free to test this pattern out on the other rows, too. Just remember that as long as you know the values for sine and cosine, you can quickly find the values of many other functions, even without a calculator!

Let's review everything we've learned now. We learned that trig functions are based on the **right triangle**, a triangle with one right angle that measures 90 degrees. The side opposite the right angle is always the **hypotenuse**. If we label one angle, the side opposite the angle is called the **opposite** side, and the side next to the angle, but not the hypotenuse, is called the **adjacent** side. Based on this triangle, we have our definitions of our trig functions.

The **sine** function is opposite over hypotenuse. The **cosine** function is adjacent over hypotenuse. The **tangent** function is opposite over adjacent. These three are the basic trig functions. The next three are reciprocals of one of these. The **cosecant** is the reciprocal of the sine function. The **secant** is the reciprocal of the cosine function. The **cotangent** is the reciprocal of the tangent function. We can summarize the special angles and their values in this table:

Following this lesson, you'll have the ability to:

- Describe the sine, cosine and tangent functions based on the right triangle
- Identify the reciprocals of the sine, cosine and tangent functions
- Explain how to easily recall the trig functions
- Recall the special angles and their values

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Precalculus: High School27 chapters | 212 lessons | 1 flashcard set

- Trigonometry: Sine and Cosine 7:26
- Other Trigonometric Functions: Cotangent, Secant & Cosecant 4:21
- Unit Circle: Memorizing the First Quadrant 5:15
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- Practice Problems with Circular Trigonometric Functions 7:54
- Special Right Triangles: Types and Properties 6:12
- Trigonometric Function Values of Special Angles 7:23
- Law of Cosines: Definition and Application 8:16
- The Double Angle Formula 9:44
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