2. Section 7-4 Evaluating and Graphing Sine and Cosine
Objective:
Day 1: Reference angles.
Day 2: Parent Graphs of sine and cosine function
Day 3: UC and parent graphs; application problems.

3. The curve bank!

4. The site below demonstrates reference angles.

5. Evaluating Sine and Cosine
CAH
SOH
First we must Understand the
Reference Angles

6. What is a Reference Angle?
A reference angle is the positive acute angle formed by the terminal side of the angle and the x-axis.
Terminal Side
125o
55o
Reference Angle

7. Finding the Reference Angle Quadrant I
In the first quadrant the angle is its own reference angle!

8. Quadrant II
In the 2nd quadrant, the reference angle is the angle formed by the terminal side of the 120o angle and the x-axis.
To find the reference angle measure:
180o – 120o.
180o – 1200 = 60o

9. Quadrant III
In the 3rd quadrant, the reference angle is the angle formed by the terminal side of the 240o angle and the x-axis.
To find the reference angle measure:
240o – 180o.
240o – 1800 = 60o

10. Quadrant IV To find the reference angle measure: 360o – 315o.
In the 4th quadrant, the reference angle is the angle formed by the terminal side of the 315o angle and the x-axis.
To find the reference angle measure:
360o – 315o.
360o – 3150 = 60o

11. Reference angles The acute angle 𝛼 is the reference angle for:
𝜃= 180 ∘ −𝛼 𝜃=180°+ α 𝜃=360−𝛼
And all their coterminal angles.
The acute angle α is the reference angle for:
θ=– α ; 𝜃= + α 𝜃=2 – α
And all their coterminal angles.

12. Reference Angles 𝑄 𝐼𝐼: 180° − 𝛼 𝑄𝐼𝐼𝐼 180° +𝛼 𝑄𝐼𝑉: 360° −𝛼
In general, the acute angle  is the reference angle for the angles:
𝑄 𝐼𝐼: 180° − 𝛼
𝑄𝐼𝐼𝐼 180° +𝛼
𝑄𝐼𝑉: 360° −𝛼
as well as all coterminal angles. In other words, the reference angle for any angle θ is the acute positive angle  formed by the terminal ray of θ and the x-axis.

13. Remember:  The reference angle is measured from the terminal side of the original angle "to" the x-axis (not the y-axis).

16. More on Reference angles
Consider angle 30°.
Draw the line segments that represent sin 30° and cos 30°.

17. Can you find angle  whose terminal ray is in the 2nd quadrant such that :
The reference angle is 30°
How does sin 30° = compare to sin 150° ?
How does cos 150° compares to cos 30° ?

19. Can you find angle  whose terminal ray is in 3rd quadrant such that :
30° is its reference angle.
How does sin 210° compare to sin 30° ?
How does cos 210° compare to cos 30° ?

21. 4th quadrant: Find a 4th quadrant angle for which:
30° is its reference angle.
How does sin 330° compare to sin 30° ?
How does cos 330° compare to cos 30° ?

23. The important part of reference angles
How does the value of: 𝑠𝑖𝑛𝜃 (𝑜𝑟 𝑐𝑜𝑠𝜃 ) compare to the 𝑠𝑖𝑛𝛼 (or cos𝛼) if
𝛼 is the reference angle for 𝜃 ?
Answer: they are equal!
So knowing the value for one angle, gives you the value for many other angles.

24. A graphic visual of reference angles

25. What if the angle is bigger than 360o or 2π?
Find an angle between 0o and 360o that is co-terminal.
695o – 360o = 335o
To find the reference angle for 695o -
Find the reference angle for 335o :
360o – 335o = 25o

26. Reference Angles Express each in terms of a reference angle: sin 695°
cos 124°
sin -190°

27. Reference Angles sin 25°= 0.4226 sin 155° sin 205° sin 335 ° sin -25 °
Without using a calculator find the following:
sin 155°
sin 205°
sin 335 °
sin -25 °
sin 515 °

28. Reference Angles Sin 25°= 0.4226 Cos 155° Cos 205° Cos 335 °
Use your calc to find: Cos 25°.
Without using a calculator find the following:
Cos 155°
Cos 205°
Cos 335 °
After this slide use the slide for section 7.4

29. Sec 7.4 Day 2
Review: express each in terms of the reference angle:
sin 473°
cos -123 °

30. The unit circle
Now that we understand reference angles we can build the unit circle.
We will determine the key values of 1st quadrant angles, and then use reference angles to determine the key values of the 2nd, 3rd, and 4th quadrants.

31. Sine and Cosine of Special Angles3 1 2 2 1 3 2
30o-60o-90o
30o-60o-90o
45o-45o-90o 45o
Sin 60o = 𝑦 𝑟 =
Cos 60o = 𝑥 𝑟 = 1 2
Sin 45o = 𝑦 𝑟 =
Cos 45o = 𝑥 𝑟 =
Sin 30o = 𝑦 𝑟 = 1 2
Cos 30o = 𝑥 𝑟 =

32. Notice a pattern?
Ѳ degrees
Ѳ radians
Sin Ѳ
Cos V 1 30
𝜋 6
1 2
3 2 45
𝜋 4
2 2 60
𝜋 3 90
𝜋 2
0 2
1 2
2 2
3 2
4 2
4 2
3 2
2 2
1 2
0 2

33. Find the exact value of each:
sin 𝜋 6 = so,
sin 7𝜋 6 =− 1 2
sin 7𝜋 6
7𝜋 6 − 6𝜋 6 = 𝜋 6
6𝜋 6 is 𝜋
𝜋 6 is the reference angle
Sin is negative in
the 3rd Quadrant
is in the 3rd Quadrant
cos 7𝜋 6 =−cos 𝜋 6 =
Let log =𝑥
4 𝑥 = 1 2
2 2𝑥 = 2 −1
2𝑥=−1
𝑥=− 1 2 = log 4 𝑠𝑖𝑛150° =− 1 2
𝑠𝑖𝑛150°= sin 30°= 1 2
log 4 𝑠𝑖𝑛150°

34. Graphing sine and cosine functions
Graphing using your calculator.
When angle measure is in degrees or in radians.
Graphing without your calculator.
When angle measure is in degrees or in radians.

35. Graphing with the 5 key points
1 complete period of Sine or Cosine can be graphed using the 5 key points.
For each specific equation, the horizontal spacing between each key point is constant. i.e. if it is 3 units between point 2 to point 3, then it is also 3 units between point 4 and point 5.
For both sine and cosine, the 5 key points will always be; maximum values, minimum values, and points on the axis of the wave. (the middle)
the axis of the wave for the parent graphs is the X-axis.

36. Critical values of the parent graph of the sine function:
Radians
Degrees
Notes
The Period
The amplitude
The coordinates of the starting point
aka Y-intercept
aka The maximum
First x intercept
The minimum point
Second x intercept
End point

37. Critical values of the parent graph of the sine function:
Radians
Degrees
Notes
The Period 2π
360
The amplitude 1
The coordinates of the starting point
aka Y-intercept
(0,0)
Sine “starts” in the middle and increases)
key point #1
The maximum
( 𝜋 2 , 0)
(90, 1)
Key point #2
Second x intercept
(𝜋, 0)
(180,0)
Key point #3
The minimum point
( 3𝜋 2 , −1)
(270, -1)
Key point # 4
End point (3rd x-intercept)
(2𝜋, 0)
(360, 0)
Key point #5

39. Critical values of the parent graph of the cosine function:
Radians
Degrees
Notes
The Period
The amplitude
The coordinates of the starting point
aka Y-intercept
aka The maximum
First x intercept
The minimum point
Second x intercept
End point

40. Critical values of the parent graph of the cosine function:
Radians
Degrees
Notes
The Period 2π
360
The amplitude 1
The coordinates of the starting point
aka Y-intercept
Aka the maximum
(0,1)
cosine “starts” at the maximum and decreases
key point #1
The first x-intercept
( 𝜋 2 , 0)
(90, 0)
Key point #2
The minimum point
(𝜋, −1)
(180,-1)
Key point #3
The second x intercept
( 3𝜋 2 , 0)
(270, 0)
Key point # 4
End point (back to max)
(2𝜋, 1)
(360, 1)
Key point #5

44. All at once! What do you think:
The coordinates of the intersection point are?
Where would you find the intersection points on the UC?

45. All at once but more than once!

46. Simulation of sine and cosine graphs
See the site below for cool demonstartion
Simulation of sine and cosine graphs

47. How to use your calculator to find sin  and cos 
Before doing any calculations involving trig functions always check the calculator mode.

48. Make sure to check the mode then evaluate the expressions below:
Find the value of each expression to three decimal places.
A.) sin 122°
B.) cos 237°
C.) cos 5
D.) sin (-2)

49. Latitude
The latitude of a point on Earth is the degree measure of the shortest arc from that point to the equator. For example, the latitude of point P in the diagram equals the degree measure of arc PE.

52. How far is Rome (aka Roma) from the equator?
The Latitude of Rome is approximately 42 N.
The radius of earth is approximately 3963 miles
The longitude of Barcelona is about 2 degrees East.
Remember s=r where  is measured in radians.

53. How far is Santiago, Chile from the equator?
The latitude of Santiago is 33º 28´ S.
The longitude of Santiago is about 71 degrees W.

54. Homework written exercises sec 7.4
Part 1:
#1-17 odds
#21-24 ALL
Use your 4-day weekend wisely.
Part 2: #26-31 All