1. 9.2: Sum and Differences of Trig Functions
Homework
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions

2. Sum and Differences Section 9.2
Pre-Calculus AB PreAP/Dual, Revised ©2014
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9.2: Sum and Differences of Trig Functions

3. 9.2: Sum and Differences of Trig Functions
Identity
Question 1: What is Cosine 45? Question 2: What is Cosine 30? Question 3: What is Cosine 15?
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9.2: Sum and Differences of Trig Functions

4. Addition and Subtraction Identities
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9.2: Sum and Differences of Trig Functions

5. 9.2: Sum and Differences of Trig Functions
Identity
Consider these angles a and b and the distance in the following unit circle is equal. How can one measure distance?
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9.2: Sum and Differences of Trig Functions

6. 9.2: Sum and Differences of Trig Functions
Identity
Consider these two 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?
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9.2: Sum and Differences of Trig Functions

7. 9.2: Sum and Differences of Trig Functions
Identity
Consider these two angles a and b and the distance in the following two unit circles are equal.
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions

8. 9.2: Sum and Differences of Trig Functions
Identity
Consider these two angles a and b and the distance in the following two unit circles are equal.
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions

9. 9.2: Sum and Differences of Trig Functions
Identity
Consider these two angles a and b and the distance in the following two unit circles are equal.
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions

10. 9.2: Sum and Differences of Trig Functions
Identity
Consider these two angles a and b and the distance in the following two unit circles are equal.
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9.2: Sum and Differences of Trig Functions

11. 9.2: Sum and Differences of Trig Functions
Identity
Consider these two angles a and b and the distance in the following two unit circles are equal.
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9.2: Sum and Differences of Trig Functions

12. 9.2: Sum and Differences of Trig Functions
Identity
Continue to prove the identity.
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9.2: Sum and Differences of Trig Functions

13. 9.2: Sum and Differences of Trig Functions
Identity
Continue to prove the identity.
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9.2: Sum and Differences of Trig Functions

14. 9.2: Sum and Differences of Trig Functions
Steps
A. Using the Addition and Subtraction Identities
Identify the parts of the given
Utilize any parts of the addition/subtraction identities; BREAK IT DOWN
Convert into the unit circle, if possible. If not, draw a picture.
Simplify
B. Hints
Sine: SIGNS STAY THE SAME = Sine and functions alternate
Cosine: Cosine = CHANGE and functions use the opposite sign and do not alternate functions
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9.2: Sum and Differences of Trig Functions

15. 9.2: Sum and Differences of Trig Functions
Example 1
Find the exact value of cos 15 without a calculator.
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9.2: Sum and Differences of Trig Functions

16. 9.2: Sum and Differences of Trig Functions
Earlier…
Question 1: What is Cosine 45? Question 2: What is Cosine 30? Question 3: What is Cosine 15?
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions

17. 9.2: Sum and Differences of Trig Functions
Earlier…
Find the exact value of cos 15
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9.2: Sum and Differences of Trig Functions

18. 9.2: Sum and Differences of Trig Functions
Example 2
Find the exact value of 𝐬𝐢𝐧 𝟓𝝅 𝟏𝟐 without a calculator.
75° ??
30°
45°
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9.2: Sum and Differences of Trig Functions

19. 9.2: Sum and Differences of Trig Functions
Example 2
Find the exact value of 𝐬𝐢𝐧 𝟓𝝅 𝟏𝟐 without a calculator.
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9.2: Sum and Differences of Trig Functions

20. 9.2: Sum and Differences of Trig Functions
Example 2 – Another Way
Find the exact value of 𝐬𝐢𝐧 𝟓𝝅 𝟏𝟐 without a calculator.
120°
75° ??
45°
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9.2: Sum and Differences of Trig Functions

21. 9.2: Sum and Differences of Trig Functions
Example 2 – Another Way
Find the exact value of 𝐬𝐢𝐧 𝟓𝝅 𝟏𝟐 without a calculator.
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9.2: Sum and Differences of Trig Functions

22. 9.2: Sum and Differences of Trig Functions
Example 3
Find the exact value of 𝐜𝐨𝐬 𝟏𝟏𝝅 𝟏𝟐 without a calculator.
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9.2: Sum and Differences of Trig Functions

23. 9.2: Sum and Differences of Trig Functions
Your Turn
Find the exact value of 𝐬𝐢𝐧 𝟕𝝅 𝟏𝟐 without a calculator.
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9.2: Sum and Differences of Trig Functions

24. 9.2: Sum and Differences of Trig Functions
The Conjugate
Rule: No RADICAL in the denominator
Conjugate is the complex number’s opposite sign
When we multiply a compound equation with a radical, we must multiply the OPPOSITE to get rid of the radical to both the NUMERATOR (TOP) and DENOMINATOR (BOTTOM)
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9.2: Sum and Differences of Trig Functions

25. 9.2: Sum and Differences of Trig Functions
Review Example
Rationalize Denominator & Simplify
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9.2: Sum and Differences of Trig Functions

26. 9.2: Sum and Differences of Trig Functions
Review Example
Rationalize Denominator & Simplify
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9.2: Sum and Differences of Trig Functions

27. 9.2: Sum and Differences of Trig Functions
Review Example
Rationalize Denominator & Simplify
We try to get rid of the radicals on the bottom so the bottom will be even
With the CONJUGATE
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9.2: Sum and Differences of Trig Functions

28. 9.2: Sum and Differences of Trig Functions
Example 4
Find the exact value of 𝐭𝐚𝐧 𝟕𝝅 𝟏𝟐 without a calculator.
105°
60°
45°
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9.2: Sum and Differences of Trig Functions

29. 9.2: Sum and Differences of Trig Functions
Example 4
Find the exact value of 𝐭𝐚𝐧 𝟕𝝅 𝟏𝟐 without a calculator.
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions

30. 9.2: Sum and Differences of Trig Functions
Example 4
Find the exact value of 𝐭𝐚𝐧 𝟕𝝅 𝟏𝟐 without a calculator.
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9.2: Sum and Differences of Trig Functions

31. 9.2: Sum and Differences of Trig Functions
Your Turn
Find the exact value of 𝐭𝐚𝐧 𝟏𝟏𝝅 𝟏𝟐 without a calculator.
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9.2: Sum and Differences of Trig Functions

32. Steps in Triangles using Sum/Differences
Draw the triangle(s)
Identify the missing side
Determine all three main trig functions
Identify and plug into formula
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9.2: Sum and Differences of Trig Functions

33. 9.2: Sum and Differences of Trig Functions
Example 5
Given 𝐬𝐢𝐧 𝒙= 𝟒 𝟓 from 𝝅 𝟐 ≤𝒙≤𝝅 and 𝐬𝐢𝐧 𝒚=− 𝟓 𝟏𝟑 from 𝝅≤𝒚≤ 𝟑𝝅 𝟐 , identify the value of cos(x + y).
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions

34. 9.2: Sum and Differences of Trig Functions
Example 5
Given 𝐬𝐢𝐧 𝒙= 𝟒 𝟓 from 𝝅 𝟐 ≤𝒙≤𝝅 and 𝐬𝐢𝐧 𝒚=− 𝟓 𝟏𝟑 from 𝝅≤𝒚≤ 𝟑𝝅 𝟐 , identify the value of cos(x + y).
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions

35. 9.2: Sum and Differences of Trig Functions
Example 6
Given 𝐬𝐢𝐧 𝒙= 𝟓 𝟏𝟑 from 𝝅 𝟐 ≤𝒙≤𝝅 and 𝐜𝐨𝐬 𝒚=− 𝟑 𝟓 from 𝝅 𝟐 ≤𝒙≤𝝅, identify the value of tan(x + y).
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions

36. 9.2: Sum and Differences of Trig Functions
Your Turn
Given 𝐬𝐢𝐧 𝒙=− 𝟕 𝟐𝟓 from 𝝅≤𝒙≤ 𝟑𝝅 𝟐 and 𝐜𝐨𝐬 𝒚=− 𝟒 𝟓 from𝝅≤𝒙≤ 𝟑𝝅 𝟐 , identify the value of tan(x – y).
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions

37. 9.2: Sum and Differences of Trig Functions
Assignment
Worksheet
12/26/2018 5:43 PM
9.2: Sum and Differences of Trig Functions