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Sum and Difference Identities

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Sum and Difference Identities

With your knowledge of special angles like the sine and cosine of 30^\circ  and 45^\circ , you can find the sine and cosine of 15^\circ , the difference of 45^\circ  and 30^\circ , and 75^\circ , the sum of 45^\circ  and 30^\circ .  Using what you know about the unit circle and the sum and difference identities, how do you determine \sin 15^\circ  and \sin 75^\circ

Watch This

http://www.youtube.com/watch?v=H-0jQTzfkWQ James Sousa: Sum and Difference Identities for Cosine

http://www.youtube.com/watch?v=hiNDDQyee2E James Sousa: Sum and Difference Identities for Sine

http://www.youtube.com/watch?v=OQP78bwYcWw James Sousa: Sum and Difference Identities for Tangent

Guidance

There are some intuitive but incorrect formulas for sums and differences with respect to trigonometric functions.  The form below does not work for any trigonometric function and is one of the most common incorrect guesses as to the sum and difference identity.

\sin (\theta + \beta) \neq \sin \theta + \sin \beta

First look at the derivation of the cosine difference identity:

\cos (\alpha - \beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta

Start by drawing two arbitrary angles \alpha  and \beta .  In the image above \alpha  is the angle in red and \beta  is the angle in blue.  The difference \alpha-\beta  is noted in black as \theta .  The reason why there are two pictures is because the image on the right has the same angle \theta  in a rotated position.  This will be useful to work with because the length of \overline{AB}  will be the same as the length of \overline{CD} .

\overline{AB} &= \overline{CD}\\\sqrt{(\cos \alpha-\cos \beta)^2+(\sin \alpha-\sin \beta)^2} &= \sqrt{(\cos \theta -1)^2+(\sin \theta-0)^2}\\(\cos \alpha-\cos \beta)^2+(\sin \alpha-\sin \beta)^2 &= (\cos \theta-1)^2+(\sin \theta-0)^2

(\cos \alpha)^2-2 \cos \alpha \cos \beta+(\cos \beta)^2+(\sin \alpha)^2-2 \sin \alpha \sin \beta+(\sin \beta)^2 &= (\cos \theta-1)^2+(\sin \theta)^2

2-2 \cos \alpha \cos \beta-2 \sin \alpha \sin \beta &= (\cos \theta)^2-2 \cos \theta+1+(\sin \theta)^2\\2-2 \cos \alpha \cos \beta-2 \sin \alpha \sin \beta &= 1 - 2 \cos \theta+1\\-2 \cos \alpha \cos \beta-2 \sin \alpha \sin \beta &= -2 \cos \theta\\\cos \alpha \cos \beta+\sin \alpha \sin \beta &= \cos \theta\\&= \cos (\alpha-\beta)

The proofs for sine and tangent are left to examples and exercises.  They are listed here for your reference.  Cotangent, secant and cosecant are excluded because you can use reciprocal identities to get those once you have sine, cosine and tangent. 

Summary:

  • \cos (\alpha \pm \beta)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta
  • \sin (\alpha \pm \beta)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta
  • \tan (\alpha \pm \beta)=\frac{\sin (\alpha \pm \beta)}{\cos (\alpha \pm \beta)}=\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}

Example A

Prove the cosine of a sum identity. 

Solution:  Start with the cosine of a difference and make a substitution.  Then use the odd-even identity.

\cos \alpha \cos \beta+\sin \alpha \sin \beta=\cos (\alpha -\beta)

Let \gamma=-\beta

\cos \alpha \cos (-\gamma)+\sin \alpha \sin (-\gamma) &= \cos (\alpha + \gamma)\\\cos \alpha \cos \gamma-\sin \alpha \sin \gamma &= \cos (\alpha+\gamma)

Example B

Find the exact value of \tan 15^\circ  without using a calculator.

Solution: \tan 15^\circ =\tan (45^\circ-30^\circ)=\frac{\tan 45^\circ-\tan 30^\circ}{1+\tan 45^\circ \tan 30^\circ}=\frac{1-\frac{\sqrt{3}}{3}}{1+1 \cdot \frac{\sqrt{3}}{3}}=\frac{3-\sqrt{3}}{3+\sqrt{3}}

A final solution will not have a radical in the denominator.  In this case multiplying through by the conjugate of the denominator will eliminate the radical.  This technique is very common in PreCalculus and Calculus.

&= \frac{(3 - \sqrt{3}) \cdot (3-\sqrt{3})}{(3+\sqrt{3}) \cdot (3-\sqrt{3})}\\&= \frac{(3-\sqrt{3})^2}{9-3}\\&= \frac{(3-\sqrt{3})^2}{6}

Example C

Evaluate the expression exactly without using a calculator.

\cos 50^\circ \cos 5^\circ+\sin 50^\circ \sin 5^\circ

Solution:  Once you know the general form of the sum and difference identities then you will recognize this as cosine of a difference.

\cos 50^\circ \cos 5^\circ + \sin 50^\circ \sin 5^\circ=\cos (50^\circ-5^\circ)=\cos 45^\circ=\frac{\sqrt{2}}{2}

Concept Problem Revisited

In order to evaluate \sin 15^\circ  and \sin 75^\circ  exactly without a calculator, you need to use the sine of a difference and sine of a sum.

\sin (45^\circ-30^\circ) &= \sin 45^\circ \cos 30^\circ-\cos 45^\circ \sin 30^\circ=\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}=\frac{\sqrt{6}-\sqrt{2}}{4}\\\sin (45^\circ+30^\circ) &= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ=\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2} \cdot \frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}

Vocabulary

The Greek letters used in this concept refer to unknown angles.  They are \alpha - alpha , \beta - beta , \theta - theta , \gamma - gamma

The symbol \pm  is short hand for “ plus or minus .”  The symbol \mp  is shorthand for “ minus or plus .”  The order is important because for cosine of a sum, the negative sign is used on the other side of the identity.  This is the opposite of sine of a sum, where a positive sign is used on the other side of the identity. 

Guided Practice

1. Prove the sine of a difference identity. 

2. Use a sum or difference identity to find an exact value of \cot \left(\frac{5 \pi}{12}\right) .

3. Prove the following identity:

\frac{\sin (x-y)}{\sin (x+y)}=\frac{\tan x-\tan y}{\tan x+\tan y}

Answers:

1. Start with the cofunction identity and then distribute and work out the cosine of a sum and cofunction identities.

\sin (\alpha-\beta) &= \cos \left(\frac{\pi}{2}-(\alpha - \beta)\right)\\&= \cos \left(\left(\frac{\pi}{2}-\alpha \right)+\beta \right)\\&= \cos \left(\frac{\pi}{2}-\alpha \right) \cos \beta-\sin \left(\frac{\pi}{2}-\alpha \right) \sin \beta\\&= \sin \alpha \cos \beta - \cos \alpha \sin \beta

2.Start with the definition of cotangent as the inverse of tangent. 

\cot \left(\frac{5 \pi}{12}\right) &= \frac{1}{\tan \left(\frac{5 \pi}{12}\right)}\\&= \frac{1}{\tan \left(\frac{9 \pi}{12}-\frac{4 \pi}{12}\right)}\\&= \frac{1}{\tan (135^\circ-60^\circ)}\\&= \frac{1+\tan 135^\circ \tan 60^\circ}{\tan 135^\circ - \tan 60^\circ}\\&= \frac{1+(-1) \cdot \sqrt{3}}{(-1) - \sqrt{3}}\\&= \frac{(1-\sqrt{3})}{(-1-\sqrt{3})}\\&= \frac{(1-\sqrt{3})^2}{(-1+\sqrt{3}) \cdot (1-\sqrt{3})}\\&= \frac{(1-\sqrt{3})^2}{-(1-3)}\\&= \frac{(1-\sqrt{3})^2}{2}

3. Here are the steps:

\frac{\sin (x-y)}{\sin (x+y)} &= \frac{\tan x-\tan y}{\tan x+\tan y}\\\frac{\sin x \cos y-\cos x \sin y}{\sin x \cos y+\cos x \sin y} &=\\\frac{\sin x \cos y-\cos x \sin y}{\sin x \cos y+\cos x \sin y} \cdot \frac{\left(\frac{1}{\cos x \cdot \cos y}\right)}{\left(\frac{1}{\cos x \cdot \cos y}\right)} &=\\\frac{\left(\frac{\sin x \ \cancel{\cos y}}{\cos x \cdot \cancel{\cos y}}\right)-\left(\frac{\cancel{\cos x} \ \sin y}{\cancel{\cos x} \cdot \cos y}\right)}{\left(\frac{\sin x \ \cancel{\cos y}}{\cos x \cdot \cancel{\cos y}}\right)+\left(\frac{\cancel{\cos x} \ \sin y}{\cancel{\cos x} \cdot \cos y}\right)} &=\\\frac{\tan x -\tan y}{\tan x +\tan y}&=

Practice

Find the exact value for each expression by using a sum or difference identity.

1. \cos 75^\circ

2.  \cos 105^\circ

3.  \cos 165^\circ

4.  \sin 105^\circ

5.  \sec 105^\circ

6.  \tan 75^\circ

7. Prove the sine of a sum identity.

8. Prove the tangent of a sum identity.

9. Prove the tangent of a difference identity.

10. Evaluate without a calculator: \cos 50^\circ \cos 10^\circ-\sin 50^\circ \sin 10^\circ .

11. Evaluate without a calculator: \sin 35^\circ \cos 5^\circ-\cos 35^\circ \sin 5^\circ .

12. Evaluate without a calculator: \sin 55^\circ \cos 5^\circ+\cos 55^\circ \sin 5^\circ .

13. If \cos \alpha \cos \beta=\sin \alpha \sin \beta , then what does \cos (\alpha + \beta)  equal?

14. Prove that \tan \left(x+\frac{\pi}{4}\right)=\frac{1+\tan x}{1-\tan x} .

15. Prove that \sin (x+\pi)=-\sin x .

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