SUM AND DIFFERENCES. Section 5.3 Precalculus PreAP/Dual, Revised 2017
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1 SUM AND DIFFERENCES Section 5. Precalculus PreAP/Dual, Revised /1/ :41 AM 5.4: Sum and Differences of Trig Functions 1
2 IDENTITY Question 1: What is Cosine 45? Question 2: What is Cosine 0? Question : What is Cosine 15? ?? 2 2 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 2
3 ADDITION AND SUBTRACTION IDENTITIES (FILL IN CHART) sin x + = sin x cos + cos x sin sin x = sin x cos cos x sin cos x + = cos x cos sin x sin cos x = cos x cos + sin x sin tan x + = tan x = tan x + tan 1 tan x tan tan x tan 1 + tan x tan 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions
4 IDENTITY Consider these angles a and b and the distance in the following unit circle is equal. How can one measure distance? 2 2 d x x cos cos sin sin 2 2 d x x 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 4
5 IDENTITY Consider these angles a and b and the distance in the following unit circle is equal. What happens when the point is moved towards the x-axis? 2 2 d x x d cos x 1 sin x 0 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 5
6 IDENTITY Consider these two angles a and b and the distance in the following two unit circles are equal. 2 2 cos cos sin sin d x x d d cos x 1 sin x 0 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 6 d 2 2 cos x cos sin x sin cos x 1 sin x 0 2
7 IDENTITY Consider these two angles a and b and the distance in the following two unit circles are equal cos x cos sin x sin cos x 1 sin x cos x cos sin x sin cos x 1 sin x cos x 2 cos x cos cos sin x 2sin x sin sin 2 2 cos x 2 cos x 1 sin x cos sin 2 cos cos sin sin cos sin cos sin 2 cos 1 2 x 2 x x x 2 x x 2 2 cos x cos sin x sin 2 2 cos x cos x cos x cos sin x sin 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 7
8 IDENTITY Consider these two angles a and b and the distance in the following two unit circles are equal. cos x cos x cos sin x sin x x x cos cos cos sin sin cos cos and sin sin Negative Angle Identit cos x cos x cos sin x sin 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 8
9 IDENTITY Consider these two angles a and b and the distance in the following two unit circles are equal. cos x cos x cos sin x sin sin x cos x,cos x sin x, and cos x sin x Cofunction Identities cos x cos x cos sin x sin sin x sin x cos cos xsin 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 9 x x 2
10 IDENTITY Consider these two angles a and b and the distance in the following two unit circles are equal. sin x sin x cos cos xsin x x x sin sin cos cos sin cos cos and sin sin Negative Angle Identit sin x sin x cos cos xsin 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 10
11 IDENTITY Continue to prove the identit. tan tan tan x sin x x cos x sin x cos sin cos x cos x cos sin sin x sin x cos sin cos x cos xcos x cos x cos sin sin x cos xcos tan sin x cos sin cos x cos x cos cos x cos x cos x cos sin xsin cos x cos cos x cos tan tan sin x sin cos x cos x sin xsin 1 cos x cos x tan x tan 1 tan xtan 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 11
12 IDENTITY Continue to prove the identit. tan tan tan x x sin cos x x sin x cos sin cos x cos x cos sin sin x sin x cos sin cos x cos xcos x cos x cos sin sin x cos xcos tan sin x cos sin cos x cos x cos cos x cos x cos x cos sin xsin cos x cos cos x cos tan tan sin x sin cos x cos x sin xsin 1 cos x cos x tan x tan 1 tan xtan 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 12
13 STEPS A. Using the Addition and Subtraction Identities 1. Identif the parts of the given 2. Utilize an parts of the addition/subtraction identities; BREAK IT DOWN. Convert into the unit circle, if possible. If not, draw a picture. 4. Simplif B. Hints 1. Sine: SIGNS STAY THE SAME = Sine and functions alternate 2. Cosine: Cosine = CHANGE and functions use the opposite sign and do not alternate functions 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 1
14 EXAMPLE 1 Find the exact value of cos 15 without a calculator. cos15 cos 45 cos 0 cos cos cos x cos x cos sin x sin cos cos sin sin /1/ :41 AM 5.4: Sum and Differences of Trig Functions 14
15 Question 1: What is Cosine 45? Question 2: What is Cosine 0? EARLIER Question : What is Cosine 15? 2?? /1/ :41 AM 5.4: Sum and Differences of Trig Functions 15
16 Find the exact value of cos EARLIER 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 16
17 EXAMPLE 2 Find the exact value of sin 5π without a calculator. 75?? 0 sin x sin sin 5 sin /1/ :41 AM 5.4: Sum and Differences of Trig Functions 17
18 Find the exact value of sin 5π 12 EXAMPLE 2 without a calculator. 5 sin sin sin x sin x cos cos xsin sin sin cos cos sin /1/ :41 AM 5.4: Sum and Differences of Trig Functions 18
19 EXAMPLE 2 ANOTHER WAY Find the exact value of sin 5π sin 2 x sin 4 without a calculator ?? 45 sin 5 2 sin /1/ :41 AM 5.4: Sum and Differences of Trig Functions 19
20 EXAMPLE 2 ANOTHER WAY Find the exact value of sin 5π 12 without a calculator. 5 2 sin sin 12 4 sin x sin x cos cos xsin sin sin cos cos sin /1/ :41 AM 5.4: Sum and Differences of Trig Functions 20
21 EXAMPLE Find the exact value of cos 11π 12 without a calculator /1/ :41 AM 5.4: Sum and Differences of Trig Functions 21
22 YOUR TURN Find the exact value of sin 7π 12 without a calculator /1/ :41 AM 5.4: Sum and Differences of Trig Functions 22
23 THE CONJUGATE A. Rule: No RADICAL in the denominator B. Conjugate is the complex number s opposite sign C. When we multipl a compound equation with a radical, we must multipl the OPPOSITE to get rid of the radical to both the NUMERATOR (TOP) and DENOMINATOR (BOTTOM) 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 2
24 REVIEW EXAMPLE Rationalize the denominator & Simplif /1/ :41 AM 5.4: Sum and Differences of Trig Functions 24
25 REVIEW EXAMPLE Rationalize the denominator & Simplif /1/ :41 AM 5.4: Sum and Differences of Trig Functions 25
26 REVIEW EXAMPLE Rationalize the denominator & Simplif We tr to get rid of the radicals on the bottom so the bottom will be even With the CONJUGATE 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 26
27 EXAMPLE 4 Find the exact value of tan 7π 12 without a calculator tan x tan 4 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 27
28 Find the exact value of tan 7π 12 tan tan tan 4 1 tan tan 4 x EXAMPLE 4 without a calculator. 7 tan tan 12 4 tan x tan 1 tan xtan /1/ :41 AM 5.4: Sum and Differences of Trig Functions 28
29 EXAMPLE 4 Find the exact value of tan 7π 12 without a calculator /1/ :41 AM 5.4: Sum and Differences of Trig Functions 29
30 YOUR TURN Find the exact value of tan 11π 12 without a calculator. 2 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 0
31 EXAMPLE 5 Find the exact value of sin π 2 + x without a calculator. sin x sin x cos cos xsin sin cos x cos sin cos x 0 sin x x cos x 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 1
32 EXAMPLE 6 Find the exact value of cos x + cos x without a calculator. 2sin x sin 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 2
33 EXAMPLE 7 Write the expression as the trig function of the angle of without a calculator: cos 25 cos 15 sin 25 sin 15 cos x cos sin xsin cos x cos25 cos15 sin 25 sin15 cos40 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions
34 EXAMPLE 8 Write the expression as the trig function of the angle of without a calculator: sin π 4 cos 7π 6 + cos π 4 sin 7π 6 17 sin 12 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 4
35 YOUR TURN Write the expression as the trig function of the angle of without a calculator: tan 140 tan 60 1+tan 140 tan 60 tan80 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 5
36 STEPS IN TRIANGLES USING SUM/DIFFERENCES A. Draw the triangle(s) B. Identif the missing side C. Determine all three main trig functions D. Identif and plug into formula 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 6
37 EXAMPLE 9 Given sin x = 4 from π 5 x π and sin = from π π, identif the value of cos x cos x 5 4 tan x cos tan /1/ :41 AM 5.4: Sum and Differences of Trig Functions 7
38 EXAMPLE 9 Given sin x = 4 from π 5 x π and sin = from π π, identif the value of cos x +. 4 sin x 5 cos x 5 4 tan x 5 sin 1 12 cos 1 5 tan 12 cos x cos x cos sin x sin /1/ :41 AM 5.4: Sum and Differences of Trig Functions 8
39 EXAMPLE 10 Given sin x = 5 from π x π and cos = from π identif the value of tan x +. x π, /1/ :41 AM 5.4: Sum and Differences of Trig Functions 9
40 YOUR TURN Given sin x = 7 25 from π x π 2 and cos = 4 5 from π x π 2, identif the value of tan(x ) /1/ :41 AM 5.4: Sum and Differences of Trig Functions 40
41 ASSIGNMENT Page 79 7, 9, 1, 19, 21, 5-51 odd 8/1/ :41 AM 5.4: Sum and Differences of Trig Functions 41
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