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Double Angle Identities

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Double, Half, and Power Reducing Identities

These identities are significantly more involved and less intuitive than previous identities.  By practicing and working with these advanced identities, your toolbox and fluency substituting and proving on your own will increase.  Each identity in this concept is named aptly.  Double angles work on finding \sin 80^\circ  if you already know \sin 40^\circ .  Half angles allow you to find  \sin 15^\circ if you already know \sin 30^\circ .  Power reducing identities allow you to find \sin ^2 15^\circ  if you know the sine and cosine of 30^\circ

What is \sin ^2 15^\circ

Watch This

http://www.youtube.com/watch?v=-zhCYiHcVIE James Sousa: Double Angle Identities

http://www.youtube.com/watch?v=Rp61qiglwfg James Sousa: Half Angle Identities

Guidance

The double angle identities are proved by applying the sum and difference identities.  They are left as practice problems.  These are the double angle identities.

  • \sin 2x=2 \sin x \cos x
  • \cos 2x=\cos ^2x-\sin ^2 x
  • \tan 2 x=\frac{2 \tan x}{1-\tan ^2 x}

The power reducing identities allow you to write a trigonometric function that is squared in terms of smaller powers.  The proofs are left as guided practice and practice problems.

  • \sin^2x=\frac{1-\cos 2x}{2}
  • \cos^2 x=\frac{1+\cos 2x}{2}
  • \tan^2 x=\frac{1-\cos 2x}{1+\cos 2x}

The half angle identities are a rewritten version of the power reducing identities.  The proofs are left as practice problems.

  • \sin \frac{x}{2} = \pm \sqrt{\frac{1-\cos x}{2}}
  • \cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}
  • \tan \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{1+\cos x}}

Example A

Rewrite \sin^4 x  as an expression without powers greater than one. 

Solution:  While \sin x \cdot \sin x \cdot \sin x \cdot \sin x  does technically solve this question, try to get the terms to sum together not multiply together.

\sin^4 x &= (\sin^2 x)^2\\&= \left(\frac{1-\cos 2x}{2}\right)^2\\&= \frac{1-2 \cos 2x+\cos ^2 2x}{4}\\&= \frac{1}{4} \left(1-2 \cos 2x+\frac{1+\cos 4x}{2}\right)

Example B

Write the following expression with only \sin x  and \cos x : \sin 2x+\cos 3x .

Solution:

\sin 2x+\cos 3x &= 2 \sin x \cos x+\cos (2x+x)\\&= 2 \sin x \cos x+\cos 2x \cos x-\sin 2x \sin x\\&= 2 \sin x \cos x + (\cos^2 x-\sin^2 x) \cos x-(2 \sin x \cos x) \sin x\\&= 2 \sin x \cos x+\cos^3 x-\sin^2 x \cos x-2 \sin^2 x \cos x\\&= 2 \sin x \cos x +\cos^3 x-3 \sin^2 x \cos x

Example C

Use half angles to find an exact value of \tan 22.5^\circ  without using a calculator. 

Solution: \tan \frac{x}{2} =\pm \sqrt{\frac{1-\cos x}{1+\cos x}}

\tan 22.5^\circ=\tan \frac{45^\circ}{2}=\pm \sqrt{\frac{1-\cos 45^\circ}{1+\cos 45^\circ}}=\pm \sqrt{\frac{1-\frac{\sqrt{2}}{2}}{1+\frac{\sqrt{2}}{2}}}=\pm \sqrt{\frac{\frac{2}{2}-\frac{\sqrt{2}}{2}}{\frac{2}{2}+\frac{\sqrt{2}}{2}}}=\pm \sqrt{\frac{2-\sqrt{2}}{2+\sqrt{2}}}

Sometimes you may be requested to get all the radicals out of the denominator.

Concept Problem Revisited

In order to fully identify \sin^2 15^\circ  you need to use the power reducing formula.

\sin^2 x &= \frac{1-\cos 2x}{2}\\\sin^2 15^\circ &= \frac{1-\cos 30^\circ}{2}=\frac{1}{2}-\frac{\sqrt{3}}{4}

Vocabulary

An identity is a statement proved to be true once so that it can be used as a substitution in future simplifications and proofs. 

Guided Practice

1.  Prove the power reducing identity for sine.

\sin^2 x=\frac{1-\cos 2x}{2}

2.  Simplify the following identity: \sin^4 x-\cos ^4 x .

3. What is the period of the following function?

f(x)=\sin 2x \cdot \cos x+\cos 2x \cdot \sin x

Answers:

1.  Start with the double angle identity for cosine.

\cos 2x &= \cos^2 x-\sin^2x\\\cos 2x &= (1-\sin^2 x)-\sin^2 x\\\cos 2x &= 1-2 \sin^2 x

This expression is an equivalent expression to the double angle identity and is often considered an alternate form.

2 \sin^2 x &= 1-\cos 2x\\\sin^2 x &= \frac{1-\cos 2x}{2}

2.  Here are the steps:

\sin^4 x-\cos^4 x &= (\sin^2 x-\cos^2 x)(\sin^2 x+\cos^2 x)\\&= -(\cos^2 x-\sin^2 x)\\&= -\cos 2x

3. f(x)=\sin 2x \cdot \cos x+\cos 2x \cdot \sin x  so f(x)=\sin (2x+x)=\sin 3x .  Since b=3  this implies the period is \frac{2 \pi}{3}

Practice

Prove the following identities.

1.  \sin 2x=2 \sin x \cos x

2. \cos 2x=\cos^2 x-\sin^2 x

3. \tan 2x=\frac{2 \tan x}{1-\tan^2 x}

4.  \cos^2 x=\frac{1+\cos 2x}{2}

5.  \tan^2 x=\frac{1-\cos 2x}{1+\cos 2x}

6.  \sin \frac{x}{2} =\pm \sqrt{\frac{1-\cos x}{2}}

7.  \cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}

8.  \tan \frac{x}{2}=\pm \sqrt{\frac{1-\cos x}{1+\cos x}}

9.  \csc 2x=\frac{1}{2} \csc x \sec x

10.  \cot 2x=\frac{\cot^2 x-1}{2 \cot x}

Find the value of each expression using half angle identities.

11.  \tan 15^\circ

12.  \tan 22.5^\circ

13.  \sec 22.5^\circ

14.  Show that \tan \frac{x}{2}=\frac{1-\cos x}{\sin x} .

15.  Show that \tan \frac{x}{2}=\frac{\sin x}{1+\cos x} .

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