Negative Angle Identities - At A Glance

Negative angles, like rebellious teenagers, are contrarian. "I don't have to listen to you, Dad. I don't have to go counterclockwise if I don't want to. Gawd!"

Not that we have anything against rebellious teenagers; we have a few stories from our younger days that would make your hair stand on end, ya whippersnapper. Being so contrary does tend to make negative angles predictable, though.

The Trick to Working with Rebellious Angles

Cosine plays it cool. Positive, negative: cosine doesn't care, he'll just keep on doing what he wants. Sine, though, can't stand being told what to do. If you try to give orders to sine while there's a negative angle in the mix, sine will do the exact opposite of what you said.

See? The original angle (A) is in red, and the negative angle (-A) is in blue. The relationships between A and -A are the Negative Angle Identities (catchy name, right?):

cos (-A) = cos A

sin (-A) = -(sin A)

Mathematicians call cosine an even function, and sine an odd function, based on these identities. We'd call them Joe and Larry, but that isn't very descriptive.

Even functions are such that f(-x) = f(x), which means that putting in a negative value returns the positive value instead. Odd functions, though, return the opposite of the positive value; f(-x) = -f(x).

If we want to know the sine or cosine of a negative angle, we can express it in terms of the positive angle. This works for any angle of any size in any quadrant, any time and any where. If we say "any" any more times, we might run out of "any"s. And that wouldn't be any fun.

Sample Problem

What are the sine and cosine of ?

Why do you always have to be so negative, ? Let's inverse that frown upside down.

Let's start Operation No More Sad Face by figuring out sine and cosine of positive

Phase 1 is now complete; scowl inversion is at 20%. is located in the fourth quadrant, so x is super smiling while y is down in the dumps. That means that:

We've achieved a 70% bad-mood reversal. Don't mistake this for true happiness, though. We've found sine and cosine for positive , but we wanted to find them for .

Now is the time to deploy our secret weapon, the Negative Angle Identities.

Remember that cosine is chill; he was while the angle was positive, so he stays that way while it's negative. Sine is another story.

We've done it. We have achieved maximum happy. S.o. h.a.p.p.y.

The easiest mistake to make with negative angles is to stop too early and not actually apply the Negative Angle Identities. You can tell if you forgot by the aching sadness in your heart.