Math 36 "Fall 08" 5.2 "Sum and Di erence Identities" * Find exact values of functions of rational multiples of by using sum and di erence identities.

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1 Math 36 "Fall 08" 5.2 "Sum and Di erence Identities" Skills Objectives: * Find exact values of functions of rational multiples of by using sum and di erence identities. * Develop new identities from the sum and di erence identities Conceptual Objectives: * Derive sum and di erence identities for the cosine function by using the distance formula * Use the sum and di erence identities for the cosine function to obtain the cofunction identities * Understand that a trigonometric function of a sum is not the sum of the trigonometric functions. Preliminaries: In the previous sections we discussed the more basic trigonometric identities. In this section we continue discussing trigonometric identities by deriving formulas for when the argument of the trigonometric function is a sum or di erence. First, it is important to note that function notation is not distributive: cos (A + B) 6= cos A + cos B In this section, we will derive some new and important identities which are: - Sum and di erence identities for cosine, sine, and tangent - Cofunction identities Page: 1

2 Sum and Di erence Identities for Cosine Develop the di erence identity for cosine function (Using the unit circle and the distance formula): Develop the sum identity for cosine function (Using the di erence identity for cosine and even/odd functions): Page: 2

3 Sum and Di erence Identities for Cosine Example 1: (Finding exact values for cosine) Use the sum or di erence identities for cosine to evaluate the cosine expressions exactly: a) cos 7 b) cos (15 ) c) cos Example 2: (Writing a sum or di erence as single cosine) Use the sum and di erence identities for cosine to write the expressions as a single cosine. a) sin 5x sin 2x + cos 5x cos 2x Page: 3

4 b) cos x cos 3x sin x sin 3x c) (sin A + sin B) 2 + (cos A + cos B) 2 2 Example 3: (Finding the exact value of the indicated expression using the given information and identities) Find the exact value of cos ( ) if cos = 1 3 and cos = 1 4 ; given the terminal side of lies in QIV and the terminal side of lies in QII Cofunction Identities In section 1.3, we discussed cofunction relationships for acute angles. Recall that the trigonometric function of an angle is equal to its cofunction value of the complementary angle. Cofunction Identities Page: 4

5 Sum and Di erence Identities for Sine Develop the sum identity for sine function (Using the cofunction identities and the di erence identities for cosine): Develop the di erence identity for sine function (Using the sum identity for sine and even/odd functions): Sum and di erence Identities for Sine Example 4: (Finding exact values for sine) Use the sum or di erence identities for sine to evaluate the sine expressions exactly: a) sin 5 Page: 5

6 b) sin c) sin ( 15) Example 5: (Writing a sum or di erence as a single sine) a) Graph y = 3 sin x cos 3x + 3 cos x sin 3x b) Graph y = cos 3 sin x cos x sin 3 Page: 6

7 Example 6: (Finding the exact value of the indicated expression using the given information and identities) Find the exact value of sin ( + ) if sin = 3 5 and sin = 1 5 ; given the terminal side of lies in QIII and the terminal side of lies in QII: Sum and Di erence Identities for Tangent Develop the sum and di erence identities for tangent function: Sum and Di erence Identities for Tangent Page: 7

8 Example 7: (Finding exact values for tangent) Use the sum or di erence identities for tangent to evaluate tan 5 Example 8: (Finding exact values for tangent) Find the exact value of tan ( ) if sin = 3 5 and cos = 1 4 ; given the terminal side of lies in QIII and the terminal side of lies in QII Page: 8

9 Example 9: (Determine if each equation is an identity) a) sin x + 2 = cos x + 2 b) cos 2x = cos 2 x sin 2 x c) tan (A ) = tan A Page: 9

Algebra and Trig. I. In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position.

Algebra and Trig. I. In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position. Algebra and Trig. I 4.4 Trigonometric Functions of Any Angle In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position. IN this section we