1. Sum and Differences Section 5.3 Precalculus PreAP/Dual, Revised ©2017
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§5.4: Sum and Differences of Trig Functions

2. §5.4: 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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§5.4: Sum and Differences of Trig Functions

3. Addition and Subtraction Identities (fill in chart)
𝐬𝐢𝐧 𝒙+𝒚 =
𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒚 +𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒚
𝐬𝐢𝐧 𝒙−𝒚 =
𝐬𝐢𝐧 𝒙 𝐜𝐨𝐬 𝒚 −𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒚
𝐜𝐨𝐬 𝒙+𝒚 =
𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝒚 −𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒚
𝐜𝐨𝐬 𝒙−𝒚 =
𝐜𝐨𝐬 𝒙 𝐜𝐨𝐬 𝒚 +𝐬𝐢𝐧 𝒙 𝐬𝐢𝐧 𝒚
𝐭𝐚𝐧 𝒙+𝒚 =
𝐭𝐚𝐧 𝒙 +𝐭𝐚𝐧 𝒚 𝟏−𝐭𝐚𝐧 𝒙 𝐭𝐚𝐧 𝒚
𝐭𝐚𝐧 𝒙−𝒚 =
𝐭𝐚𝐧 𝒙 −𝐭𝐚𝐧 𝒚 𝟏+𝐭𝐚𝐧 𝒙 𝐭𝐚𝐧 𝒚
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§5.4: Sum and Differences of Trig Functions

4. §5.4: 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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§5.4: Sum and Differences of Trig Functions

5. §5.4: Sum and Differences of Trig Functions
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?
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§5.4: Sum and Differences of Trig Functions

6. §5.4: 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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§5.4: Sum and Differences of Trig Functions

7. §5.4: 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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§5.4: Sum and Differences of Trig Functions

8. §5.4: 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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§5.4: Sum and Differences of Trig Functions

9. §5.4: 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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§5.4: Sum and Differences of Trig Functions

10. §5.4: 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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§5.4: Sum and Differences of Trig Functions

11. §5.4: Sum and Differences of Trig Functions
Identity
Continue to prove the identity.
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§5.4: Sum and Differences of Trig Functions

12. §5.4: Sum and Differences of Trig Functions
Identity
Continue to prove the identity.
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§5.4: Sum and Differences of Trig Functions

13. §5.4: 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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§5.4: Sum and Differences of Trig Functions

14. §5.4: Sum and Differences of Trig Functions
Example 1
Find the exact value of 𝐜𝐨𝐬⁡𝟏𝟓 without a calculator.
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§5.4: Sum and Differences of Trig Functions

15. §5.4: 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?
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§5.4: Sum and Differences of Trig Functions

16. §5.4: Sum and Differences of Trig Functions
Earlier…
Find the exact value of 𝐜𝐨𝐬⁡𝟏𝟓
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§5.4: Sum and Differences of Trig Functions

17. §5.4: 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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§5.4: Sum and Differences of Trig Functions

18. §5.4: Sum and Differences of Trig Functions
Example 2
Find the exact value of 𝐬𝐢𝐧 𝟓𝝅 𝟏𝟐 without a calculator.
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§5.4: Sum and Differences of Trig Functions

19. §5.4: 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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§5.4: Sum and Differences of Trig Functions

20. §5.4: 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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§5.4: Sum and Differences of Trig Functions

21. §5.4: Sum and Differences of Trig Functions
Example 3
Find the exact value of 𝐜𝐨𝐬 𝟏𝟏𝝅 𝟏𝟐 without a calculator.
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§5.4: Sum and Differences of Trig Functions

22. §5.4: Sum and Differences of Trig Functions
Your Turn
Find the exact value of 𝐬𝐢𝐧 𝟕𝝅 𝟏𝟐 without a calculator.
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§5.4: Sum and Differences of Trig Functions

23. §5.4: 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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§5.4: Sum and Differences of Trig Functions

24. §5.4: Sum and Differences of Trig Functions
Review Example
Rationalize the denominator & Simplify 𝟏 𝟒+ 𝟑
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§5.4: Sum and Differences of Trig Functions

25. §5.4: Sum and Differences of Trig Functions
Review Example
Rationalize the denominator & Simplify 𝟏 𝟒+ 𝟑
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§5.4: Sum and Differences of Trig Functions

26. §5.4: Sum and Differences of Trig Functions
Review Example
Rationalize the denominator & Simplify 𝟏 𝟒+ 𝟑
We try to get rid of the radicals on the bottom so the bottom will be even
With the CONJUGATE
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§5.4: Sum and Differences of Trig Functions

27. §5.4: 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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§5.4: Sum and Differences of Trig Functions

28. §5.4: Sum and Differences of Trig Functions
Example 4
Find the exact value of 𝐭𝐚𝐧 𝟕𝝅 𝟏𝟐 without a calculator.
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§5.4: Sum and Differences of Trig Functions

29. §5.4: Sum and Differences of Trig Functions
Example 4
Find the exact value of 𝐭𝐚𝐧 𝟕𝝅 𝟏𝟐 without a calculator.
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§5.4: Sum and Differences of Trig Functions

30. §5.4: Sum and Differences of Trig Functions
Your Turn
Find the exact value of 𝐭𝐚𝐧 𝟏𝟏𝝅 𝟏𝟐 without a calculator.
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§5.4: Sum and Differences of Trig Functions

31. §5.4: Sum and Differences of Trig Functions
Example 5
Find the exact value of 𝐬𝐢𝐧 𝝅 𝟐 +𝒙 without a calculator.
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§5.4: Sum and Differences of Trig Functions

32. §5.4: Sum and Differences of Trig Functions
Example 6
Find the exact value of 𝐜𝐨𝐬 𝒙+𝒚 −𝐜𝐨𝐬 𝒙−𝒚 without a calculator.
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§5.4: Sum and Differences of Trig Functions

33. §5.4: Sum and Differences of Trig Functions
Example 7
Write the expression as the trig function of the angle of without a calculator: 𝐜𝐨𝐬 𝟐𝟓° 𝐜𝐨𝐬 𝟏𝟓°−𝐬𝐢𝐧 𝟐𝟓° 𝐬𝐢𝐧 𝟏𝟓°
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§5.4: Sum and Differences of Trig Functions

34. §5.4: Sum and Differences of Trig Functions
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
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§5.4: Sum and Differences of Trig Functions

35. §5.4: Sum and Differences of Trig Functions
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°
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§5.4: Sum and Differences of Trig Functions

36. 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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§5.4: Sum and Differences of Trig Functions

37. §5.4: Sum and Differences of Trig Functions
Example 9
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 .
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§5.4: Sum and Differences of Trig Functions

38. §5.4: Sum and Differences of Trig Functions
Example 9
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 .
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§5.4: Sum and Differences of Trig Functions

39. §5.4: Sum and Differences of Trig Functions
Example 10
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 .
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§5.4: Sum and Differences of Trig Functions

40. §5.4: Sum and Differences of Trig Functions
Your Turn
Given 𝐬𝐢𝐧 𝒙=− 𝟕 𝟐𝟓 from 𝝅≤𝒙≤ 𝟑𝝅 𝟐 and 𝐜𝐨𝐬 𝒚=− 𝟒 𝟓 from 𝝅≤𝒙≤ 𝟑𝝅 𝟐 , identify the value of 𝐭𝐚𝐧⁡(𝒙−𝒚).
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§5.4: Sum and Differences of Trig Functions

41. §5.4: Sum and Differences of Trig Functions
Assignment
Page 379 7, 9, 13, 19, 21, odd
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§5.4: Sum and Differences of Trig Functions