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# 14.4: Sum and Difference Formulas

Difficulty Level: At Grade Created by: CK-12

Objective

To use and derive the sum and difference formulas.

Review Queue

Using your calculator, find the value of each trig function below. Round your answer to 4 decimal places.

1. sin75\begin{align*}\sin 75^\circ\end{align*}

2. cos15\begin{align*}\cos 15^\circ\end{align*}

3. tan105\begin{align*}\tan 105^\circ\end{align*}

4. cos255\begin{align*}\cos 255^\circ\end{align*}

## Finding Exact Trig Values using Sum and Difference Formulas

Objective

Here you'll use the sum and difference formulas to find exact values of angles other than the critical angles.

Guidance

You know that sin30=12,cos135=22,tan300=3,\begin{align*}\sin 30^\circ=\frac{1}{2}, \cos 135^\circ=-\frac{\sqrt{2}}{2}, \tan 300 ^\circ = -\sqrt{3},\end{align*} etc... from the special right triangles. In this concept, we will learn how to find the exact values of the trig functions for angles other than these multiples of 30,45,\begin{align*}30^\circ, 45^\circ,\end{align*} and 60\begin{align*}60^\circ\end{align*}. Using the Sum and Difference Formulas, we can find these exact trig values.

Sum and Difference Formulas

sin(a±b)cos(a±b)tan(a±b)=sinacosb±cosasinb=cosacosbsinasinb=tana±tanb1tanatanb

Example A

Find the exact value of sin75\begin{align*}\sin 75^\circ\end{align*}.

Solution: This is an example of where we can use the sine sum formula from above, sin(a+b)=sinacosb+cosasinb\begin{align*}\sin(a+ b)=\sin a \cos b+\cos a \sin b\end{align*}, where a=45\begin{align*}a = 45^\circ\end{align*} and b=30\begin{align*}b = 30^\circ\end{align*}.

sin75=sin(45+30)=sin45cos30+cos45sin30=2232+2212=6+24

In general, sin(a+b)sina+sinb\begin{align*}\sin (a+b)\ne \sin a+\sin b\end{align*} and similar statements can be made for the other sum and difference formulas.

Example B

Find the exact value of cos11π12\begin{align*}\cos \frac{11 \pi}{12}\end{align*}.

Solution: For this example, we could use either the sum or difference cosine formula, 11π12=2π3+π4\begin{align*}\frac{11\pi}{12}=\frac{2\pi}{3}+\frac{\pi}{4}\end{align*} or 11π12=7π6π4\begin{align*}\frac{11\pi}{12}=\frac{7\pi}{6}-\frac{\pi}{4}\end{align*}. Let’s use the sum formula.

cos11π12=cos(2π3+π4)=cos2π3cosπ4sin2π3sinπ4=12223222=2+64

Example C

Find the exact value of tan(π12)\begin{align*}\tan \left(-\frac{\pi}{12}\right)\end{align*}.

Solution: This angle is the difference between π4\begin{align*}\frac{\pi}{4}\end{align*} and π3\begin{align*}\frac{\pi}{3}\end{align*}.

tan(π4π3)=tanπ4tanπ31+tanπ4tanπ3=131+3

This angle is also the same as 23π12\begin{align*}\frac{23 \pi}{12}\end{align*}. You could have also used this value and done tan(π4+5π3)\begin{align*}\tan\left(\frac{\pi}{4}+\frac{5 \pi}{3}\right)\end{align*} and arrived at the same answer.

Guided Practice

Find the exact values of:

1. cos15\begin{align*}\cos 15^\circ\end{align*}

2. tan255\begin{align*}\tan 255^\circ\end{align*}

1.

cos15=cos(4530)=cos45cos30+sin45sin30=2232+2212=6+24

2.

tan(210+45)=tan210+tan451tan210tan45=33+1133=3+33333=3+333

Vocabulary

Sum and Difference Formulas
sin(a±b)cos(a±b)tan(a±b)=sinacosb±cosasinb=cosacosbsinasinb=tana±tanb1tanatanb

Problem Set

Find the exact value of the following trig functions.

1. sin15\begin{align*}\sin 15^\circ\end{align*}
2. cos5π12\begin{align*}\cos \frac{5\pi}{12}\end{align*}
3. tan345\begin{align*}\tan 345^\circ\end{align*}
4. cos(255)\begin{align*}\cos (-255^\circ)\end{align*}
5. sin13π12\begin{align*}\sin \frac{13 \pi}{12}\end{align*}
6. sin17π12\begin{align*}\sin \frac{17\pi}{12}\end{align*}
7. cos15\begin{align*}\cos 15^\circ\end{align*}
8. tan(15)\begin{align*}\tan (-15^\circ)\end{align*}
9. sin345\begin{align*}\sin 345^\circ\end{align*}
10. Now, use sin15\begin{align*}\sin 15^\circ\end{align*} from #1, and find sin345\begin{align*}\sin 345^\circ\end{align*}. Do you arrive at the same answer? Why or why not?
11. Using cos15\begin{align*}\cos 15^\circ\end{align*} from #7, find cos165\begin{align*}\cos 165^\circ\end{align*}. What is another way you could find cos165\begin{align*}\cos 165^\circ\end{align*}?
12. Describe any patterns you see between the sine, cosine, and tangent of these “new” angles.
13. Using your calculator, find the sin142\begin{align*}\sin 142^\circ\end{align*}. Now, use the sum formula and your calculator to find the sin142\begin{align*}\sin 142^\circ\end{align*} using 83\begin{align*}83^\circ\end{align*} and 59\begin{align*}59^\circ\end{align*}.
14. Use the sine difference formula to find sin142\begin{align*}\sin 142^\circ\end{align*} with any two angles you choose. Do you arrive at the same answer? Why or why not?
15. Challenge Using sin(a+b)=sinacosb+cosasinb\begin{align*}\sin (a+b)=\sin a \cos b +\cos a \sin b\end{align*} and cos(a+b)=cosacosbsinasinb\begin{align*}\cos (a+b)=\cos a \cos b - \sin a \sin b\end{align*}, show that tan(a+b)=tana+tanb1tanatanb\begin{align*}\tan (a+b)=\frac{\tan a + \tan b}{1-\tan a \tan b}\end{align*}.

## Simplifying Trig Expressions using Sum and Difference Formulas

Objective

Here you'll use the sum and difference formulas to simplify expressions.

Guidance

We can also use the sum and difference formulas to simplify trigonometric expressions.

Example A

The sina=35\begin{align*}\sin a = -\frac{3}{5}\end{align*} and cosb=1213\begin{align*}\cos b =\frac{12}{13}\end{align*}. a\begin{align*}a\end{align*} is in the 3rd\begin{align*}3^{rd}\end{align*} quadrant and b\begin{align*}b\end{align*} is in the 1st\begin{align*}1^{st}\end{align*}. Find sin(a+b)\begin{align*}\sin(a+b)\end{align*}.

Solution: First, we need to find cosa\begin{align*}\cos a\end{align*} and sinb\begin{align*}\sin b\end{align*}. Using the Pythagorean Theorem, missing lengths are 4 and 5, respectively. So, cosa=45\begin{align*}\cos a=-\frac{4}{5}\end{align*} because it is in the 3rd\begin{align*}3^{rd}\end{align*} quadrant and sinb=513\begin{align*}\sin b = \frac{5}{13}\end{align*}. Now, use the appropriate formulas.

sin(a+b)=sinacosb+cosasinb=351213+45513=5665

Example B

Using the information from Example A, find tan(a+b)\begin{align*}\tan (a+b)\end{align*}.

Solution: From the cosine and sine of a\begin{align*}a\end{align*} and b\begin{align*}b\end{align*}, we know that tana=34\begin{align*}\tan a=\frac{3}{4}\end{align*} and tanb=512\begin{align*}\tan b=\frac{5}{12}\end{align*}.

tan(a+b)=tana+tanb1tanatanb=34+512134512=14121116=5633

Example C

Simplify cos(πx)\begin{align*}\cos (\pi - x)\end{align*}.

Solution: Expand this using the difference formula and then simplify.

cos(πx)=cosπcosx+sinπsinx=1cosx+0sinx=cosx

Guided Practice

1. Using the information from Example A, find cos(ab)\begin{align*}\cos(a-b)\end{align*}.

2. Simplify tan(x+π)\begin{align*}\tan (x+\pi)\end{align*}.

1.

cos(ab)=cosacosb+sinasinb=451213+35513=6365

2.

tan(x+π)=tanx+tanπ1tanxtanπ=tanx+01tan0=tanx

Problem Set

sina=817,πa<3π2\begin{align*}\sin a =-\frac{8}{17}, \pi \le a < \frac{3\pi}{2}\end{align*} and sinb=12,3π2b<2π\begin{align*}\sin b =-\frac{1}{2}, \frac{3\pi}{2}\le b <2\pi\end{align*}. Find the exact trig values of:

1. sin(a+b)\begin{align*}\sin (a+b)\end{align*}
2. cos(a+b)\begin{align*}\cos (a+b)\end{align*}
3. sin(ab)\begin{align*}\sin (a-b)\end{align*}
4. tan(a+b)\begin{align*}\tan (a+b)\end{align*}
5. cos(ab)\begin{align*}\cos (a-b)\end{align*}
6. tan(ab)\begin{align*}\tan (a-b)\end{align*}

Simplify the following expressions.

1. sin(2πx)\begin{align*}\sin (2\pi-x)\end{align*}
2. sin(π2+x)\begin{align*}\sin \left(\frac{\pi}{2}+x\right)\end{align*}
3. cos(x+π)\begin{align*}\cos (x+\pi)\end{align*}
4. cos(3π2x)\begin{align*}\cos \left(\frac{3\pi}{2}-x\right)\end{align*}
5. tan(x+2π)\begin{align*}\tan(x+2\pi)\end{align*}
6. tan(xπ)\begin{align*}\tan(x-\pi)\end{align*}
7. sin(π6x)\begin{align*}\sin \left(\frac{\pi}{6}-x\right)\end{align*}
8. tan(π4+x)\begin{align*}\tan \left(\frac{\pi}{4}+x\right)\end{align*}
9. cos(xπ3)\begin{align*}\cos \left(x-\frac{\pi}{3}\right)\end{align*}

Determine if the following trig statements are true or false.

1. sin(πx)=sin(xπ)\begin{align*}\sin(\pi - x)=\sin (x-\pi)\end{align*}
2. cos(πx)=cos(xπ)\begin{align*}\cos(\pi - x)=\cos (x-\pi)\end{align*}
3. tan(πx)=tan(xπ)\begin{align*}\tan(\pi - x)=\tan (x-\pi)\end{align*}

## Solving Trig Equations using Sum and Difference Formulas

Objective

Here you'll solve trig equations using the sum and difference formulas.

Guidance

Lastly, we can use the sum and difference formulas to solve trigonometric equations. For this concept, we will only find solutions in the interval 0x<2π\begin{align*}0\le x <2\pi\end{align*}.

Example A

Solve cos(xπ)=22\begin{align*}\cos (x-\pi)=\frac{\sqrt{2}}{2}\end{align*}.

Solution: Use the formula to simplify the left-hand side and then solve for x\begin{align*}x\end{align*}.

cos(xπ)cosxcosπ+sinxsinπcosxcosx=22=22=22=22

The cosine negative in the 2nd\begin{align*}2^{nd}\end{align*} and 3rd\begin{align*}3^{rd}\end{align*} quadrants. x=3π4\begin{align*}x=\frac{3\pi}{4}\end{align*} and 5π4\begin{align*}\frac{5\pi}{4}\end{align*}.

Example B

Solve sin(x+π4)+1=sin(π4x)\begin{align*}\sin \left(x+\frac{\pi}{4}\right)+1=\sin \left(\frac{\pi}{4}-x\right)\end{align*}.

Solution:

sin(x+π4)+1sinxcosπ4+cosxsinπ4+1sinx22+cosx22+12sinxsinx=sin(π4x)=sinπ4cosxcosπ4sinx=22.cosx22sinx=1=12=22

In the interval, x=5π4\begin{align*}x=\frac{5\pi}{4}\end{align*} and 7π4\begin{align*}\frac{7\pi}{4}\end{align*}.

Example C

Solve 2sin(x+π3)=tanπ3\begin{align*}2\sin \left(x+\frac{\pi}{3}\right)=\tan \frac{\pi}{3}\end{align*}.

Solution:

2sin(x+π3)2(sinxcosπ3+cosxsinπ3)2sinx12+2cosx32sinx+3cosxsinxsin2x1cos2x00=tanπ3=3=3=3=3(1cosx)=3(12cosx+cos2x)square both sides=36cosx+3cos2x  substitute sin2x=1cos2x=4cos2x6cosx+2=2cos2x3cosx+1

At this point, we can factor the equation to be (2cosx1)(cosx1)=0\begin{align*}(2\cos x -1)(\cos x -1)=0\end{align*}. cosx=12\begin{align*}\cos x =\frac{1}{2}\end{align*}, and 1, so x=0,π3,5π3\begin{align*}x=0, \frac{\pi}{3}, \frac{5 \pi}{3}\end{align*}. Be careful with these answers. When we check these solutions it turns out that 5π3\begin{align*}\frac{5\pi}{3}\end{align*} does not work.

2sin(5π3+π3)2sin2π0=tanπ3=33

Therefore, 5π3\begin{align*}\frac{5\pi}{3}\end{align*} is an extraneous solution.

Guided Practice

Solve the following equations in the interval 0x<2π\begin{align*}0\le x<2\pi\end{align*}.

1. cos(2πx)=12\begin{align*}\cos(2\pi - x)=\frac{1}{2}\end{align*}

2. sin(π6x)+1=sin(x+π6)\begin{align*}\sin \left(\frac{\pi}{6}-x\right)+1 = \sin \left(x+\frac{\pi}{6}\right)\end{align*}

3. cos(π2+x)=tanπ4\begin{align*}\cos \left(\frac{\pi}{2}+x\right)=\tan \frac{\pi}{4}\end{align*}

1.

cos(2πx)cos2πcosx+sin2πsinxcosxx=12=12=12=π3 and 5π3

2.

3.

cos(π2+x)cosπ2cosxsinπ2sinxsinxsinxx=tanπ4=1=1=1=3π2

Problem Set

Solve the following trig equations in the interval 0x<2π\begin{align*}0\le x < 2\pi\end{align*}.

1. sin(xπ)=22\begin{align*}\sin (x-\pi)=-\frac{\sqrt{2}}{2}\end{align*}
2. cos(2π+x)=1\begin{align*}\cos(2\pi +x)=-1\end{align*}
3. tan(x+π4)=1\begin{align*}\tan \left(x+\frac{\pi}{4}\right)=1\end{align*}
4. sin(π2x)=12\begin{align*}\sin \left(\frac{\pi}{2}-x\right)=\frac{1}{2}\end{align*}
5. sin(x+3π4)+sin(x3π4)=1\begin{align*}\sin \left(x+\frac{3\pi}{4}\right)+\sin \left(x-\frac{3\pi}{4}\right)=1\end{align*}
6. sin(x+π6)=sin(xπ6)\begin{align*}\sin \left(x+\frac{\pi}{6}\right)=-\sin \left(x-\frac{\pi}{6}\right)\end{align*}
7. cos(x+π6)=cos(xπ6)+1\begin{align*}\cos \left(x+\frac{\pi}{6}\right)=\cos \left(x-\frac{\pi}{6}\right)+1\end{align*}
8. cos(x+π3)+cos(xπ3)=1\begin{align*}\cos \left(x+\frac{\pi}{3}\right)+\cos \left(x-\frac{\pi}{3}\right)=1\end{align*}
9. tan(x+π)+2sin(x+π)=0\begin{align*}\tan(x+\pi)+2\sin (x+\pi)=0\end{align*}
10. tan(x+π)+cos(x+π2)=0\begin{align*}\tan (x+\pi)+\cos \left(x+\frac{\pi}{2}\right)=0\end{align*}
11. tan(x+π6)=tan(x+π4)\begin{align*}\tan \left(x+\frac{\pi}{6}\right)=\tan \left(x+\frac{\pi}{4}\right)\end{align*}
12. sin(x5π3)2sin(x2π3)=0\begin{align*}\sin \left(x-\frac{5\pi}{3}\right)-2\sin \left(x-\frac{2\pi}{3}\right)=0\end{align*}
13. 4sin(x+π)2=2cos(x+π2)\begin{align*}4\sin (x+\pi)-2=2\cos\left(x+\frac{\pi}{2}\right)\end{align*}
14. 1+2cos(xπ)+cosx=0\begin{align*}1+2\cos(x-\pi)+\cos x =0\end{align*}
15. Real Life Application The height, h\begin{align*}h\end{align*} (in feet), of two people in different seats on a Ferris wheel can be modeled by h1=50cos6t+46\begin{align*}h_1=50\cos 6t+46\end{align*} and h2=50cos6(tπ3)+46\begin{align*}h_2=50\cos 6\left(t-\frac{\pi}{3}\right)+46\end{align*} where t\begin{align*}t\end{align*} is the time (in minutes). When are the two people at the same height?

## Date Created:

Apr 23, 2013

Jun 11, 2015
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