List of the Basic Trig Identities

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  • 0:05 List of the Basic Trig…
  • 0:55 Sin and Cos
  • 2:26 Cot and Csc
  • 3:21 Tan and Sec
  • 4:13 Identities in Action
  • 6:28 Lesson Summary
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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.

List of the Basic Trig Identities

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.

Sin and Cos

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.

example of triangle

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(θ), and to get the y value of an angle, we take sin(θ). 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 (θ) + cos2 (θ) = 1.

example of triangle with x and y renamed

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.

Cot and Csc

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 (θ) + cos2 (θ) = 1.

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

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

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

Tan and Sec

Let's go back to the original equation, sin2 (θ) + cos2 (θ) = 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 (θ) / cos2 (θ) gives us tan2 (θ). We keep the plus sign. Cos2 (θ) / cos2 (θ) = 1. We keep the equals sign. And we'll move to the last term: 1 / cos2 (θ) = sec2 (θ).

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

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.

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