TI - Addition and Subtraction Identities Lesson

Addition and Subtraction Identities

Now that you have reviewed the identities you knew, let's explore some new identities!

Is the following statement true? $\sin\left(30\right)°+\sin\left(60\right)°=\sin\left(90\right)°$

NO! Recall from your unit circle: $\textcolor{blue}\sin\left(\textcolor{blue}{30}\right)°+\textcolor{green}\sin\left(\textcolor{green}{60}\right)°=\sin\left(90\right)°\\ \frac{\textcolor{blue}1}{\textcolor{blue}2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\frac{\sqrt[]{\textcolor{green}3}}{\textcolor{green}2}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\ne1$

Addition and Subtraction Identities

But, at times, we do want to know the trig ratio of a sum or difference of angles so we have these identities:$\sin\left(x+y\right)=\sin x\cos y+\cos x\sin y;\:\cos\left(x+y\right)=\cos x\cos y-\sin x\sin y;\:\tan\left(x+y\right)=\frac{\tan x+\tan y}{1-\tan x\tan y} \\ \sin\left(x-y\right)=\sin x\cos y-\cos x\sin y;\:\cos\left(x-y\right)=\cos x\cos y+\sin x\sin y;\:\tan\left(x-y\right)=\frac{\tan x-\tan y}{1+\tan x\tan y}$

Let's see if you can use the identity rules to match the following statements.

Example

Find the exact value of $\frac{\tan\left(32°\right)+\tan\left(13°\right)}{1-\tan\left(32°\right)\cdot\tan\left(13°\right)}$

1. First, notice that this is the tangent addition identity, so: $\frac{\tan\left(32°\right)+\tan\left(13°\right)}{1-\tan\left(32°\right)\cdot\tan\left(13°\right)}=\tan\left(32+13\right)$
2. So, now we know we are looking for $\tan\left(32+13\right)=\tan\left(45\right)$
3. Using our unit circle, we know that $\tan\left(45\right)=1$

Try this problem on your own to see if you've got it: 

1. Problem: Find the exact value of $\frac{\tan\left(78°\right)+\tan\left(18°\right)}{1+\tan\left(78°\right)\cdot\tan\left(18°\right)}$

• Solution: $\sqrt[]{3}$

2. Problem: Find the exact value of $\cos\left(\frac{7\Pi}{8}\right)\cos\left(\frac{5\Pi}{24}\right)+\sin\left(\frac{7\Pi}{8}\right)\sin\left(\frac{5\Pi}{24}\right)$

• Solution: $-\frac{1}{2}$

Example

Find the exact value of $\cos\left(15°\right)$

First, we need to think about angles that we know the trigonometric identities and see if we can add or subtract those values to get 15. The angles that we know are 30, 45, and 60 - and their multiples. So, let's utilize the fact that 60 - 45 = 15.

Let's rewrite the given expression: 

1. So, now let's use the subtraction identity above: $\cos\left(15°\right)=\cos\left(60-45\right)\\ =\cos60\cos45+\sin60\sin45$
2. We know all the values above - use your Unit Circle to substitute: $=\cos60\cos45+\sin60\sin45\\ \left(\frac{1}{2}\right)\left(\frac{\sqrt[]{2}}{2}\right)+\left(\frac{\sqrt[]{3}}{2}\right)\left(\frac{\sqrt[]{2}}{2}\right)$
3. Now, we can simplify this expression: $\left(\frac{1}{2}\right)\left(\frac{\sqrt[]{2}}{2}\right)+\left(\frac{\sqrt[]{3}}{2}\right)\left(\frac{\sqrt[]{2}}{2}\right)\\ =\frac{\:\sqrt[]{2}}{4}+\frac{\sqrt[]{6}}{4}\\ =\frac{\sqrt[]{2}+\sqrt[]{6}}{4}$

Watch this video to try another expression:

 IMAGES CREATED BY GAVS