Trigonometry: Negative Angle Identities

So far in trigonometry, we have covered many different ways to simplify trig functions, from using the unit circle to find sine, cosine, and tangent to using trig ratios to find other trig ratios. However, now we start learning about trig identities. Trig identities are very useful as we begin solving more complex trig functions. Let's learn about our first trig identity: negative angle identities!

##Negative Angle Identities Negative angle identities give us a method to transform negative angles into positive angles, which are often easier to work with. There is a negative angle identity for each trig function, but they are all very similar, except a few key differences, so they should be easy to remember. Let's take a look!

$$\sin(-\theta) = -\sin(\theta)$$ $$\cos(-\theta) = \cos(\theta)$$ $$\tan(-\theta) = -\tan(\theta)$$ $$\csc(-\theta) = -\csc(\theta)$$ $$\sec(-\theta) = \sec(\theta)$$ $$\cot(-\theta) = -\cot(\theta)$$

We have seen a similar concept before when learning about even and odd functions. We previously learned that a function is even when $f(x) = f(-x)$. A function is odd when $-f(x) = f(-x)$.

Looking at the identities written above, we can see that $\cos(\theta)$ and $\sec(\theta)$ are even functions, whereas $\sin(\theta)$, $\tan(\theta)$, $\csc(\theta)$, and $\cot(\theta)$ are odd functions.

Use the negative angle identities to help you simplify complex trig functions, like $f(x) = -42\sin(-2x+4) +2$.

(Video) Trigonometry - Negative angle identities

by MySecretMathTutor

So far, we have learned about finding trigonometric functions and reciprocal trig functions using the unit circle, but not all trigonometric functions are positive. Check out this video by MySecretMathTutor to learn about negative angle identities and how to use them!

As is said in the video, converting from a negative angle to a positive angle is actually pretty easy, you just need to remember the following relationships:

$$\sin{(-\theta)} = -\sin{(\theta)}$$ $$\cos{(-\theta)} = \cos{(\theta)}$$ $$\csc{(-\theta)} = -\csc{(\theta)}$$ $$\sec{(-\theta)} = \sec{(\theta)}$$ $$\tan{(-\theta)} = -\tan{(\theta)}$$ $$\cot{(-\theta)} = -\cot{(\theta)}$$

MySecretMathTutor does a great job explaining the sign relationships using graphs of the sine, cosine, and tangent functions, which I encourage you to focus on to better understand these relationships.

Let's try an example used in the video.

We are given $\tan{(-\theta)}=\frac{1}{3}$ and we want to solve for $\tan(\theta)$.

We know that $\tan{(-\theta)} = -\tan{(\theta)}$ from the known negative angle identities, so we can substitute $-\tan{(\theta)}$ into the given equation to have $-\tan{(\theta)}=\frac{1}{3}$.

Then, we can multiply each side by $-1$ to switch the side the negative is on, leaving us with $\tan{(\theta)}=-\frac{1}{3}$.

Keep watching to learn how to use these negative angle identities to combine like terms and simplify large expressions!

Negative Angle Identities

There are six negative angle identities you need to memorize!

• Negative angle identities (this lesson)
• Co-function identities
• Angle and sum differences identities
• Double angle identities
• Half angle identities

Negative angle identities help us understand what happens when an angle goes in the clockwise direction instead of the usual counterclockwise direction.

In trigonometry, positive angles move counterclockwise, and negative angles move clockwise around the unit circle.

Negative angle identities tell us how trig functions behave when the angle is negative. These identities show that trig functions fall into two groups:

Odd Functions - change signs

• Sine and tangent are odd, so the output flips from positive to negative (or vice versa)
• $\sin(-x) = -\sin(x)$
• $\tan(-x)=-\tan(x)$

Even Functions - stay the same

• Cosine is even, so the value does not change
• $\cos(-x) = \cos(x)$

Think about the unit circle: a positive angle and its negative version land at points that are mirrored across the $x$-axis. Here, the $x$-coordinate (cosine) stays the same, while the $y$-coordinate (sine) changes signs. Since tangent is sine over cosine, it also changes signs.

For the future, when you will need to use these negative angle identities, it is best to memorize these!