Sum and Difference Formulas Trigonometry MATH 103 S. Rook

Overview • Section 5.2 in the textbook: – Sum and difference formulas for cosine – Sum and difference formulas for sine – Sum and difference formulas for tangent 2

Sum and Difference Formulas for Cosine

• • Sum and Difference Formulas in General Consider cos(A + B) or cos(A – B) – Sum or difference as an argument to the cosine Does cos(A + B) = cos A + cos B ?

– Let A = B = 45°: cos  45   45    cos 45   cos 45  cos 90   0  2 2  2 2 2 • However, the sum and difference of trigonometric functions occurs often enough that we do want an equivalent formulas called

sum and difference formulas

4

cos(A + B) • • Any of the sum or difference formulas can be derived by using angles in standard position along with geometry – We will not cover this, but those interested can ask me or consult section 5.2 of the book We will start with cos(A + B) and derive the other sum and difference formulas: cos 

A



B

  cos

A

cos

B

 sin

A

sin

B

5

cos(A – B) • • To derive cos(A – B): – Consider cos(A + B) and let B = -B: cos 

A

     cos

A

cos    sin

A

sin  cos

A

cos

B

 sin

A

sin

B

 Therefore, cos 

A



B

  cos

A

cos

B

 sin

A

sin

B

6

Sum and Difference Formulas for the Cosine (Example)

Ex 1:

Use a sum or difference formula to find the exact value: cos 15° 7

Sum and Difference Formulas for Sine

sin(A + B) • • • Recall the

Cofunction Theorem

: sin x = cos(90° – x) To derive sin(A + B): sin 

A



B

     cos cos cos cos   90  90    90   90      

A A A A

   

B



B

 cos

B

 

B

  sin  90    sin

A

cos

B

 cos

A

sin

B A

 sin Therefore, sin 

A



B

  sin

A

cos

B

 cos

A

sin

B B

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sin(A – B) • • To derive sin(A – B), we will use the same process when we derived cos(A – B): – Use sin(A + B) and let B = -B: sin 

A

     sin

A

cos  cos  sin

A

cos

B

 cos

A

sin

A

sin

B

Therefore, sin 

A



B

  sin

A

cos

B

 cos

A

sin

B

10

Sum and Difference Formulas for the Sine (Example)

Ex 2:

Find the exact value: sin 11  12 11

Sum and Difference Formulas for Tangent

tan(A + B) • • • Recall that tan   sin  cos  Possible to evaluate tan 

A



B

  sin cos  

A A

 

B B

  To derive tan(A + B)

in terms of tangents

: tan 

A



B

  sin cos  

A A

 

B B

   sin  cos

A

sin

B

cos

A

cos

B A

cos

B

 sin

A

sin

B

  sin cos cos cos tan 1 

A

tan cos

A A

cos cos

B B



A A

cos 

B

tan

B A B

tan 

B

cos cos sin cos

A

sin

B A

cos

A

sin

B B A

cos

B

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tan(A – B) • To derive tan(A – B), we will use the same process when we derived cos(A – B): – Use tan(A + B) and let B = -B: tan 

A

     1 tan 

A

tan 

A

tan tan      tan 1 

A

tan  tan

A

tan

B B

• Therefore, tan 

A



B

  tan 1 

A

tan 

A

tan tan

B B

14

Sum and Difference Formulas for the Tangent (Example)

Ex 3:

Find the exact value: tan 105° 15

Summary of Sum and Difference Formulas • • Summary of the sum and difference formulas for sine and cosine: cos cos  

A A

 

B B

    cos cos

A A

cos cos

B B

  sin sin

A

sin

A

sin

B B

sin sin  

A A

 

B B

    sin sin

A

cos

B

 cos

A

sin

B A

cos

B

 cos

A

sin

B

Summary of the sum and difference formulas for tangent: tan 

A

 tan 

A



B

 

B

  tan

A

 tan

B

1  tan tan

A



A

tan tan

B B

1  tan

A

tan

B

16

Additional Examples

Ex 4:

Write each expression as a single trigonometric function and then simplify if possible: a) sin 8

x

cos

x

 cos 8

x

sin

x

b) cos 15  cos 75   sin 15  sin 75  17

Ex 5:

Find: Additional Examples

A

13

A

 sin

B

 3 , 5

B

 QI a) sin(A – B) b) sec(A – B) c) tan(A – B) d) The quadrant where A – B terminates 18

Summary • • • After studying these slides, you should be able to: – Apply the sum and difference formulas for sine and cosine – Apply the sum and difference formulas for tangent Additional Practice – See the list of suggested problems for 5.2

Next lesson – Double-Angle Formulas (Section 5.3) 19