The Negative Angle Identities in Trigonometry

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Trigonometric identities are excellent tools in trigonometry. In this lesson, we are going to look at the trigonometric identities that are negative angle identities. We will look at these identities and apply them to various examples.

Trigonometric Identities

Picture this. You have a successful career as a builder, and you and a coworker are working together to figure out the angle at which you should position two beams to create in order to secure a ceiling. You need an angle, x, such that cos(x) = √3 / 2.

Creating an Angle

As you are working on the problem and considering different angles, you begin working with the angle with measure 30 degrees. You calculate cos(30) to get the following.


Great! You think you've found your angle, but then your coworker says they've found the angle and it's -30 degrees. You see, while you were working with the 30 degree angle, your coworker was working with the angle with measure -30 degrees. They calculated cos(-30) to get the following.


Wait a minute?! How is this possible? You were calculating cosine of two different angles with different measures, so how did you both get the same result?

The answer to this lies in trigonometric identities. Trigonometric identities are relationships between different trigonometric functions. In this lesson, we are interested in some specific trigonometric identities called negative angle identities.

Negative Angle Identities

Negative angle identities are trigonometric identities that show the relationships between trigonometric functions when we take the trigonometric function of a negative angle. These identities are as follows.

  • sin(-x) = -sin(x)
  • cos(-x) = cos(x)
  • tan(-x) = -tan(x)
  • csc(-x) = -csc(x)
  • sec(-x) = sec(x)
  • cot(-x) = -cot(x)

The good news is these negative angle identities are fairly easy to remember. You can see that the cosine function and the secant function are the only ones in the list that don't change sign. For all of the other trignometric functions in the list, a negative angle will change the function's sign. Pretty straightforward, right?

Okay, now that we know these identities, we can see what happened with you and your coworker. Notice the negative angle identity involving the cosine function. We have that cos(-x) = cos(x). This tells us that cos(-30) = cos(30). Ah-ha! It's no wonder that you and your coworker ended up with the same result. You will need to decide which angle to use when connecting your beams using some other sort of criteria.

We see that the use of negative angle identities can occur in the world around us. Let's look at a few more examples to help familiarize ourselves with these identities.


Example 1

Let's consider another scenario. This time, suppose you are scrapbooking, and you want a photo to fit into a page in a certain way. You are going to be creating an angle, x, with the photo and some construction paper, such that sec(x) = 2. You find that when you work with the angle that has measure 60 degrees, you have the following.


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