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Math 103: Precalculus12 chapters | 94 lessons | 10 flashcard sets

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Lesson Transcript

Instructor:
*Tyler Cantway*

Tyler has tutored math at two universities and has a master's degree in engineering.

There are specific trig functions that have very special and very simple relationships with each other. Learn the main one and how you can use it to understand the others.

In this lesson, we will learn and memorize the three basic **trig identities**. These are sometimes referred to as the **Pythagorean identities**. Sometimes we can be given equations and expressions that look like a jumbled mess. It's usually better if we can organize everything. Pythagorean identities help us group things together in specific ways that simplify them. Simplifying and organizing equations can be very helpful.

Think back and you'll remember the Pythagorean theorem states that for right triangles, there is a special relationship between the lengths of the legs and the hypotenuse. This is written in a formula as **a2 + b2 = c2**. We can actually use that formula to describe the trig identities.

If we took a look at the unit circle and chose a point, we could draw a right triangle from it. Let's begin by taking a look at a basic right triangle and see if we can relate it to a trig identity. The hypotenuse is 1, and we know that if we square and add the two legs, they will add to 1. Because we are on the unit circle, we can give the angle that we formed a name. That angle could vary depending on the triangle we drew, so we'll just call it theta.

If we took a look at this triangle, we could see that x2 + y2 = 12 or just 1. More importantly, because we are on the unit circle, we can rename the *x* and *y* values to their trigonometric values. To get the *x* value of an angle, we take **cos(Î¸)**, and to get the *y* value of an angle, we take **sin(Î¸)**. This allows us to simplify the Pythagorean theorem to just use one angle. Since we replaced the variables with their trig values, our formula simplifies into **sin2 (Î¸) + cos2 (Î¸) = 1**.

Now, if you look closely, you'll see that it is still very similar to the Pythagorean theorem. And because this is an identity, it means that no matter what that angle equals, if we take its sine and cosine, square them and add them, it will always equal 1.

Unfortunately, there are some other trig values that we can't forget. The good news is we can take this first identity, give it a little twist, and it'll show us how to come up with the other Pythagorean identities.

We start with **sin2 (Î¸) + cos2 (Î¸) = 1**.

Let's divide each part of this formula by the first term to see what happens: **sin2 (Î¸) / sin2 (Î¸) = 1**. Let's move on to the second term. We keep the plus sign. **Cos2 (Î¸) / sin2 (Î¸)** gives us **cot2 (Î¸)**. We keep the equals sign and move on to the last term: **1 / sin2 (Î¸)** gives us **csc2 (Î¸)**.

Just like that we have an identity with two of the other trig functions: **1 + cot2 (Î¸) = csc2 (Î¸)**.

Last but not least, we have two more trig functions we have to cover.

Let's go back to the original equation, **sin2 (Î¸) + cos2 (Î¸) = 1**, and see if we can come up with the other identity.

Last time we divided everything by the first term. This time we want to divide everything by the second term, which is cos2. **Sin2 (Î¸) / cos2 (Î¸)** gives us **tan2 (Î¸)**. We keep the plus sign. **Cos2 (Î¸) / cos2 (Î¸) = 1**. We keep the equals sign. And we'll move to the last term: **1 / cos2 (Î¸) = sec2 (Î¸)**.

That's the last of our Pythagorean identities: **tan2 (Î¸) + 1 = sec2 (Î¸)**.

Now that we know where these identities come from and how to remember them, let's see how we can use this to simplify equations.

Let's take a look at the quantity **(sin(Î¸) + cos(Î¸))2 = 3x + 2sin(Î¸)cos(Î¸)**.

Let's start on the left side. You'll see that we have a binomial squared. When you have a binomial squared, you can simplify this by squaring the first term, then adding 2 times the first term times the second term, and then adding the second term squared. In this case, that would give us **sin2 (Î¸) + 2sin(Î¸)cos(Î¸) + cos2 (Î¸)**. We keep the equal sign and we keep everything on the right side the same.

Moving back to the left side, let's rearrange it just a bit. If you'll see, we have sin2 (Î¸) and we have cos2 (Î¸), but they're not together. Let's switch the second term and the third term so that we can put them together.

After we rearrange terms on the left side, we have **sin2 (Î¸) plus cos2 (Î¸)**. And we remember that from our identity. That always equals 1. So, sin2 (Î¸) + cos2 (Î¸) - we can remove it and replace it with a simple 1.

After we use the Pythagorean identity to make a simplification, we're left with **1 + 2sin(Î¸)cos(Î¸) = 3x + 2(sin(Î¸)cos(Î¸))**.

We can simplify this further by subtracting off the **2sin(Î¸)cos(Î¸)** terms from each side. Since it's on both sides and we can subtract it, it will eliminate it from each side of the equal sign. We're left with **1 = 3x**.

Now things are starting to look a lot better. We want to solve for *x*, so we need to do the inverse operation of the 3. Since 3 is multiplied by *x*, we want to undo that. So we will divide by 3. And you can't just divide one side of the equation by 3; you must do it for both sides. The 3 is eliminated and we are left with **1/3 = x**

We can get the basic trigonometric Pythagorean identity by doing just a small tweak on the Pythagorean theorem. It is **sin2 (Î¸) + cos2 (Î¸) = 1** .

We can use this first identity and divide it by the first term and get **1 + cot2 (Î¸) = csc2 (Î¸)** .

And finally, we can go back to that original identity and divide everything by the second term to give us **tan2 (Î¸) + 1 = sec2 (Î¸)**.

This lesson should show you how to:

- Recognize what trig identities are and memorize the three basic ones
- Understand sin and cos
- Memorize cot and csc
- Recognize applications for tan and sec
- Understand how to use the identities in trig

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Math 103: Precalculus12 chapters | 94 lessons | 10 flashcard sets

- Go to Functions

- Graphing Sine and Cosine 7:50
- Graphing Sine and Cosine Transformations 8:39
- Graphing the Tangent Function: Amplitude, Period, Phase Shift & Vertical Shift 9:42
- Unit Circle: Memorizing the First Quadrant 5:15
- Using Unit Circles to Relate Right Triangles to Sine & Cosine 5:46
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- Law of Sines: Definition and Application 6:04
- Law of Cosines: Definition and Application 8:16
- The Double Angle Formula 9:44
- Converting Between Radians and Degrees 7:15
- How to Solve Trigonometric Equations for X 4:57
- List of the Basic Trig Identities 7:11
- Go to Trigonometry

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