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# 14.5: Double and Half Angle Formulas

Difficulty Level: At Grade Created by: CK-12

Objective

You’ll learn how to use the double and half angle formulas.

Review Queue

Use your calculator to find the value of the trig functions below. Round your answers to 4 decimal places.

1. sin22.5\begin{align*}\sin 22.5^\circ\end{align*}

2. tan157.5\begin{align*}\tan 157.5\end{align*}

Find the exact values of the trig expressions below.

3. cos(π4+π4)\begin{align*}\cos \left(\frac{\pi}{4}+\frac{\pi}{4}\right)\end{align*}

4. sin(2π3π4)\begin{align*}\sin \left(\frac{2 \pi}{3} - \frac{\pi}{4}\right)\end{align*}

## Finding Exact Trig Values using Double and Half Angle Formulas

Objective

Here you'll use the half and double angle formulas to find exact values of angles other than the critical angles.

Guidance

In the previous concept, we added two different angles together to find the exact values of trig functions. In this concept, we will learn how to find the exact values of the trig functions for angles that are half or double of other angles. Here we will introduce the Double-Angle (2a)\begin{align*}(2a)\end{align*} and Half-Angle (a2)\begin{align*}\left(\frac{a}{2}\right)\end{align*} Formulas.

Double-Angle and Half-Angle Formulas

cos2asina2cosa2=cos2asin2a=2cos2a1=1sin2a=±1cosa2=±1+cosa2sin2a=2sinacosatan2a=2tana1tan2atana2=1cosasina  =sina1+cosa

The signs of sina2\begin{align*}\sin \frac{a}{2}\end{align*} and cosa2\begin{align*}\cos \frac{a}{2}\end{align*} depend on which quadrant a2\begin{align*}\frac{a}{2}\end{align*} lies in. For cos2a\begin{align*}\cos 2a\end{align*} and tana2\begin{align*}\tan \frac{a}{2}\end{align*} any formula can be used to solve for the exact value.

Example A

Find the exact value of cosπ8\begin{align*}\cos \frac{\pi}{8}\end{align*}.

Solution: π8\begin{align*}\frac{\pi}{8}\end{align*} is half of π4\begin{align*}\frac{\pi}{4}\end{align*} and in the first quadrant.

cos(12π4)=1+cosπ42=1+222=122+22=2+22

Example B

Find the exact value of sin2a\begin{align*}\sin 2a\end{align*} if cosa=45\begin{align*}\cos a=- \frac{4}{5}\end{align*} and 3π2a<2π\begin{align*}\frac{3 \pi}{2} \le a < 2 \pi\end{align*}.

Solution: To use the sine double-angle formula, we also need to find sina\begin{align*}\sin a\end{align*}, which would be 35\begin{align*}\frac{3}{5}\end{align*} because a\begin{align*}a\end{align*} is in the 4th\begin{align*}4^{th}\end{align*} quadrant.

sin2a=2sinacosa=23545=2425

Example C

Find the exact value of tan2a\begin{align*}\tan 2a\end{align*} for a\begin{align*}a\end{align*} from Example B.

Solution: Use tana=sinacosa=3545=34\begin{align*}\tan a=\frac{\sin a}{\cos a}=\frac{\frac{3}{5}}{- \frac{4}{5}}=- \frac{3}{4}\end{align*} to solve for tan2a\begin{align*}\tan 2a\end{align*}.

tan2a=2341(34)2=32716=32167=247

Guided Practice

1. Find the exact value of cos(5π8)\begin{align*}\cos \left(-\frac{5 \pi}{8}\right)\end{align*}.

2. cosa=47\begin{align*}\cos a=\frac{4}{7}\end{align*} and 0a<π2\begin{align*}0 \le a < \frac{\pi}{2}\end{align*}. Find:

a) sin2a\begin{align*}\sin 2a\end{align*}

b) tana2\begin{align*}\tan \frac{a}{2}\end{align*}

1. 5π8\begin{align*}- \frac{5 \pi}{8}\end{align*} is in the 3rd\begin{align*}3^{rd}\end{align*} quadrant.

5π8=12(5π4)cos12(5π4)=1+cos(5π4)2=1222=12222=222\begin{align*}- \frac{5 \pi}{8}=\frac{1}{2} \left(- \frac{5 \pi}{4}\right) \rightarrow \cos \frac{1}{2} \left(- \frac{5 \pi}{4}\right)=- \sqrt{\frac{1+ \cos \left(- \frac{5 \pi}{4}\right)}{2}}=-\sqrt{\frac{1- \frac{\sqrt{2}}{2}}{2}}=\sqrt{\frac{1}{2} \cdot \frac{2-\sqrt{2}}{2}}=\frac{\sqrt{2- \sqrt{2}}}{2}\end{align*}

2. First, find sina\begin{align*}\sin a\end{align*}. 42+y2=72y=33\begin{align*}4^2+y^2=7^2\rightarrow y=\sqrt{33}\end{align*}, so sina=337\begin{align*}\sin a=\frac{\sqrt{33}}{7}\end{align*}

a) sin2a=233747=83349\begin{align*}\sin 2a=2 \cdot \frac{\sqrt{33}}{7} \cdot \frac{4}{7}=\frac{8 \sqrt{33}}{49}\end{align*}

b) You can use either tana2\begin{align*}\tan \frac{a}{2}\end{align*} formula.

tana2=147337=37733=333=3311

Vocabulary

Double-Angle and Half-Angle Formulas
cos2asina2cosa2=cos2asin2a=2cos2a1=1sin2a=±1cosa2=±1+cosa2sin2a=2sinacosatan2a=2tana1tan2atana2=1cosasina  =sina1+cosa

Problem Set

Find the exact value of the following angles.

1. sin105\begin{align*}\sin 105^\circ\end{align*}
2. tanπ8\begin{align*}\tan \frac{\pi}{8}\end{align*}
3. cos5π12\begin{align*}\cos \frac{5 \pi}{12}\end{align*}
4. cos165\begin{align*}\cos 165^\circ\end{align*}
5. sin3π8\begin{align*}\sin \frac{3 \pi}{8}\end{align*}
6. tan(π12)\begin{align*}\tan \left(- \frac{\pi}{12}\right)\end{align*}
7. sin11π8\begin{align*}\sin \frac{11 \pi}{8}\end{align*}
8. cos19π12\begin{align*}\cos \frac{19 \pi}{12}\end{align*}

The cosa=513\begin{align*}\cos a=- \frac{5}{13}\end{align*} and 3π2a<2π\begin{align*}\frac{3 \pi}{2} \le a < 2 \pi\end{align*}. Find:

1. sin2a\begin{align*}\sin 2a\end{align*}
2. cosa2\begin{align*}\cos \frac{a}{2}\end{align*}
3. tana2\begin{align*}\tan \frac{a}{2}\end{align*}
4. cos2a\begin{align*}\cos 2a\end{align*}

The sina=811\begin{align*}\sin a=\frac{8}{11}\end{align*} and π2a<π\begin{align*}\frac{\pi}{2} \le a < \pi\end{align*}. Find:

1. tan2a\begin{align*}\tan 2a\end{align*}
2. sina2\begin{align*}\sin \frac{a}{2}\end{align*}
3. cosa2\begin{align*}\cos \frac{a}{2}\end{align*}
4. sin2a\begin{align*}\sin 2a\end{align*}

## Simplifying Trig Expressions using Double and Half Angle Formulas

Objective

Here you'll use the half and double angle formulas to simplify more complicated expressions.

Guidance

We can also use the double-angle and half-angle formulas to simplify trigonometric expressions.

Example A

Simplify cos2xsinxcosx\begin{align*}\frac{\cos 2x}{\sin x \cos x}\end{align*}.

Solution: Use cos2a=cos2asin2a\begin{align*}\cos 2a=\cos^2a-\sin^2a\end{align*} and then factor.

cos2xsinxcosx=cos2xsin2xsinx+cosx=(cosxsinx)(cosx+sinx)sinx+cosx=cosxsinx

Example B

Find the formula for sin3x\begin{align*}\sin 3x\end{align*}.

Solution: You will need to use the sum formula and the double-angle formula. sin3x=sin(2x+x)\begin{align*}\sin 3x=\sin(2x+x)\end{align*}

sin3x=sin(2x+x)=sin2xcosx+cos2xsinx=2sinxcosxcosx+sinx(2cos2x1)=2sinxcos2x+2sinxcos2xsinx=4sinxcos2xsinx=sinx(4cos2x1)

We will explore other possibilities for the sin3x\begin{align*}\sin 3x\end{align*} because of the different formulas for cos2a\begin{align*}\cos 2a\end{align*} in the Problem Set.

Example C

Verify the identity cosx+2sin2x2=1\begin{align*}\cos x+2 \sin^2 \frac{x}{2}=1\end{align*}.

Solution: Simplify the left-hand side use the half-angle formula.

cosx+2sin2x2cosx+2(1cosx2)2cosx+21cosx2cosx+1cosx1

Guided Practice

1. Simplify sin2xsinx\begin{align*}\frac{\sin 2x}{\sin x}\end{align*}.

2. Verify cosx+2cos2x2=1+2cosx\begin{align*}\cos x+2 \cos^2 \frac{x}{2}=1+ 2 \cos x\end{align*}.

1. sin2xsinx=2sinxcosxsinx=2cosx\begin{align*}\frac{\sin 2x}{\sin x}=\frac{2 \sin x \cos x}{\sin x}=2 \cos x\end{align*}

2.

cosx+2cos2x2cosx+21+cosx22cosx+1+cosx1+2cosx=1+2cosx===

Problem Set

Simplify the following expressions.

1. 2+2cosx(cosx2)\begin{align*}\sqrt{2+2 \cos x} \left(\cos \frac{x}{2}\right)\end{align*}
2. cos2xcos2x\begin{align*}\frac{\cos 2x}{\cos^2x}\end{align*}
3. tan2x(1+tanx)\begin{align*}\tan 2x(1+ \tan x)\end{align*}
4. cos2x3sin2x\begin{align*}\cos 2x- 3 \sin^2x\end{align*}
5. 1+cos2xcotx\begin{align*}\frac{1+\cos 2x}{\cot x}\end{align*}
6. (1+cosx)2tanx2\begin{align*}(1+ \cos x)^2 \tan \frac{x}{2}\end{align*}

Verify the following identities.

1. cotx2=sinx1cosx\begin{align*}\cot \frac{x}{2}=\frac{\sin x}{1- \cos x}\end{align*}
2. sinx1+cosx=1cosxsinx\begin{align*}\frac{\sin x}{1+ \cos x}=\frac{1- \cos x}{\sin x}\end{align*}
3. sin2x1+cos2x=tanx\begin{align*}\frac{\sin 2x}{1+ \cos 2x}= \tan x\end{align*}
4. (sinx+cosx)2=1+sin2x\begin{align*}(\sin x+ \cos x)^2=1+ \sin 2x\end{align*}
5. sinxtanx2+2cosx=2cos2x2\begin{align*}\sin x \tan \frac{x}{2}+2 \cos x=2 \cos^2 \frac{x}{2}\end{align*}
6. cotx+tanx=2csc2x\begin{align*}\cot x+ \tan x=2 \csc 2x\end{align*}
7. cos3x=4cos3x3cosx\begin{align*}\cos 3x=4 \cos^3x-3 \cos x\end{align*}
8. cos3x=cos3x3sin2xcosx\begin{align*}\cos 3x=\cos^3x-3 \sin^2x \cos x\end{align*}
9. sin2xtanx=tanxcos2x\begin{align*}\sin 2x-\tan x=\tan x \cos 2x\end{align*}
10. cos4xsin4x=cos2x\begin{align*}\cos^4x-\sin^4x=\cos 2x\end{align*}

## Solving Trig Equations using Double and Half Angle Formulas

Objective

Here you'll solve trig equations using the half and double angle formulas.

Guidance

Lastly, we can use the half and double angle formulas to solve trigonometric equations.

Example A

Solve tan2x+tanx=0\begin{align*}\tan 2x+\tan x=0\end{align*} when 0x<2π\begin{align*}0\le x <2\pi\end{align*}.

Solution: Change tan2x\begin{align*}\tan 2x\end{align*} and simplify.

tan2x+tanx2tanx1tan2x+tanx2tanx+tanx(1tan2x)2tanx+tanxtan3x3tanxtan3xtanx(3tan2x)=0=0=0Multiply everything by 1tan2x to eliminate denominator.=0=0=0

Set each factor equal to zero and solve.

3tan2x=0   tan2x=3tanx=0and tan2x=3x=0 and π tanx=±3 x=π3,2π3,4π3,5π3

Example B

Solve 2cosx2+1=0\begin{align*}2\cos \frac{x}{2}+1=0\end{align*} when 0x<2π\begin{align*}0\le x<2\pi\end{align*}.

Solution: In this case, you do not have to use the half-angle formula. Solve for x2\begin{align*}\frac{x}{2}\end{align*}.

2cosx2+12cosx2cosx2=0=1=12

Now, let’s find cosa=12\begin{align*}\cos a = -\frac{1}{2}\end{align*} and then solve for x\begin{align*}x\end{align*} by dividing by 2.

x2=2π3,4π3=4π3,8π3

Now, the second solution is not in our range, so the only solution is x=4π3\begin{align*}x=\frac{4\pi}{3}\end{align*}.

Example C

Solve 4sinxcosx=3\begin{align*}4\sin x \cos x = \sqrt{3}\end{align*} for 0x<2π\begin{align*}0\le x < 2\pi\end{align*}.

Solution: Pull a 2 out of the left-hand side and use the sin2x\begin{align*}\sin 2x\end{align*} formula.

4sinxcosx22sinxcosx2sin2xsin2x2xx=3=3=3=32=π3,5π3,7π3,11π3=π6,5π6,7π6,11π6

Guided Practice

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

1. sinx2=1\begin{align*}\sin \frac{x}{2}=-1\end{align*}

2. cos2xcosx=0\begin{align*}\cos 2x-\cos x=0\end{align*}

1.

sinx2x2x=1=3π2=3π

From this we can see that there are no solutions within our interval.

2.

cos2xcosx2cos2xcosx1(2cosx1)(cosx+1)=0=0=0

Set each factor equal to zero and solve.

2cosx12cosxcosxx=0=1cosx+1=0=12andcosx=1=π3,5π3 x=π

Problem Set

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

1. cosxcos12x=0\begin{align*}\cos x -\cos \frac{1}{2}x=0\end{align*}
2. sin2xcosx=sinx\begin{align*}\sin 2x \cos x=\sin x\end{align*}
3. cos3x=cos3x=3sin2xcosx\begin{align*}\cos 3x = \cos ^3x=3\sin ^2x\cos x\end{align*}
4. tan2xtanx=0\begin{align*}\tan 2x - \tan x =0\end{align*}
5. cos2xcosx=0\begin{align*}\cos 2x -\cos x =0\end{align*}
6. 2cos2x2=1\begin{align*}2\cos ^2\frac{x}{2}=1\end{align*}
7. tanx2=4\begin{align*}\tan \frac{x}{2}=4\end{align*}
8. cosx2=1+cosx\begin{align*}\cos \frac{x}{2}=1+\cos x\end{align*}
9. sin2x+sinx=0\begin{align*}\sin 2x +\sin x=0\end{align*}
10. cos2x=cos2x=0\begin{align*}\cos ^2x=\cos 2x =0\end{align*}
11. cos2xcos2x=1\begin{align*}\frac{\cos 2x}{\cos ^2x}=1\end{align*}
12. cos2x1=sin2x\begin{align*}\cos 2x-1=\sin^2x\end{align*}
13. cos2x=cosx\begin{align*}\cos 2x =\cos x\end{align*}
14. sin2xcos2x=1\begin{align*}\sin 2x-\cos 2x =1\end{align*}
15. sin2x2=cos2x\begin{align*}\sin^2x-2=\cos 2x\end{align*}
16. cotx+tanx=2csc2x\begin{align*}\cot x+\tan x=2\csc 2x\end{align*}

## Date Created:

Apr 23, 2013

Jun 11, 2015
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