1. Warm Up Write answers in reduced pi form.
Convert 8𝜋 9 to degrees
Convert -2560̊ to radians
Find the arc length
The square has side lengths 14. The two curves are each of a circle with radius 14. Find the area of the shaded region.
𝟗𝟖𝝅−𝟏𝟗𝟔

2. Section 7-3 The Sine and Cosine Functions
Objective: To use the definitions of sine and cosine to find values of these functions and to solve simple trigonometric equations.

3. 𝑠𝑖𝑛𝜃=
𝑜𝑝𝑝 ℎ𝑦𝑝 =
𝑦 𝑟 r 
𝑎𝑑𝑗 ℎ𝑦𝑝 =
𝑥 𝑟
cos 𝜃=

4. Example 1
If the terminal ray of an angle θ in standard position passes through (-3, 2), find sin θ and cos θ.
Solution: On a grid, locate (-3,2).
Use this point to draw a right triangle, where one side is on the x-axis, and the hypotenuse is line segment between (-3,2) and (0,0).
Start of Day 2

5. 𝑟= −3 2 + 2 2 𝒙=−𝟑 𝒓= 𝟏𝟑 𝐲=𝟐 = 2 13 13 = 2 13 𝑠𝑖𝑛𝜃= 𝑦 𝑟 = −3 13 13= =
𝑠𝑖𝑛𝜃= 𝑦 𝑟
= −
= −3 13
𝑐𝑜𝑠𝜃= 𝑥 𝑟
𝑟= −
𝒙=−𝟑
𝒓= 𝟏𝟑
𝐲=𝟐

6. Example 2
If the 𝑠𝑖𝑛𝜃=− 5 13 , what quadrant is the angle in?

7. = 12 13
𝑐𝑜𝑠𝜃= 𝑥 𝑟
𝐲=−𝟓
𝒓=𝟏𝟑
𝑥= − −5 2
𝒙=𝟏𝟐
𝒙=±𝟏𝟐
4th Quadrant, so

8. 𝑠𝑖𝑛𝜃= cos 𝜃= 𝑦 𝑟 𝑥 𝑟 𝑜𝑝𝑝 ℎ𝑦𝑝 = 𝑎𝑑𝑗 ℎ𝑦𝑝 =r 
𝑎𝑑𝑗 ℎ𝑦𝑝 =
𝑥 𝑟
cos 𝜃=
When the radius =1 on the unit circle,
𝑠𝑖𝑛𝜃= 𝑦 1 =𝑦
𝑐𝑜𝑠𝜃= 𝑥 1 =𝑥

9. Unit Circle
The circle x2 + y2 = 1 has radius 1 and is therefore called the unit circle. This circle is the easiest one with which to work because sin θ and cos θ are simply the y- and x-coordinates of the point where the terminal ray of θ intersects the circle.
When the radius =1 on the unit circle,
𝑠𝑖𝑛𝜃= 𝑦 1 =𝑦
𝑐𝑜𝑠𝜃= 𝑥 1 =𝑥

10. 1
1 2
1 2

11. II I III IV (−,+) (+,+) (−,−) (+,−) On Your Unit Circle:
Label the quadrants.
Note the positive or negative x and y values in each quadrant.
(cos, sin)
(cos, sin)
(−,+)
(+,+) II I
III IV
(−,−)
(+,−)
(cos, sin)
(cos, sin)

12. You can determine the exact value of sine and cosine for many angles on the unit circle.1 -1 −
2 2
Find:
sin 90°
sin 450°
cos (-π)
sin (− 2𝜋 3 )
cos -315°
Refer to graph file “UC Quadrantal Angles”

13. Example 3 Solve sin θ = 1 for θ in degrees and radians.

14. Multiple solutions to trig equations

15. Expressing multiple solutions to trig equations
Degrees: 𝜃=90˚±360𝑛
𝜃= 𝜋 2 ±2𝑛𝜋
Radians:
Where n can be any integer value

16. Repeating Sin and Cos Values
For any integer n,
𝑠𝑖𝑛 (𝜃 ± 360°𝑛) = 𝑠𝑖𝑛⁡𝜃
𝑐𝑜𝑠 (𝜃 ± 360°𝑛) =𝑐𝑜𝑠⁡𝜃
𝑠𝑖𝑛 (𝜃 ±2𝜋𝑛) = 𝑠𝑖𝑛⁡𝜃
𝑐𝑜𝑠 (𝜃 ±2𝜋𝑛) =𝑐𝑜𝑠⁡𝜃
The sine and cosine functions are periodic. They have a fundamental period of 360˚ or 2 radians.

17. In-class practice part 1
P271 class exercises 1-9

18. Section 7-4 Evaluating & Graphing Sine and Cosine
Objective: To use reference angles and the unit circle to find values of the sine and cosine functions.

19. The reference angle is always less than 90˚
The smallest angle that the terminal side of a given angle makes with the x-axis.
The reference angle is always less than 90˚

20. Reference Angles 60 ̊ Locate 5𝜋 4 on your unit circle. 𝝅 𝟒
How far is 5𝜋 4 from the x-axis?
𝝅 𝟒 is the reference angle.
Locate -240 ̊on your unit circle.
How far is -240 ̊from the x-axis?
60 ̊
60 ̊ is the reference angle.

21. Finding the reference angle, , for angle 𝜽.
0<𝜽< 360˚(2𝜋) 𝜽   𝜽
Quadrant I:
=𝜽 𝜽 𝜽
Quadrant II:
=𝟏𝟖𝟎−𝜽
=𝝅−𝜽  
Quadrant III:
=𝜽−𝟏𝟖𝟎
=𝜽−𝝅
Quadrant IV:
=𝟑𝟔𝟎−𝜽
=𝟐𝝅−𝜽

22. Reference Angle,  The reference angle for 60˚ is 60˚

23. Reference Angle,  The reference angle for 240˚ is 60˚

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

25. 30˚ 150˚ 210˚ 330˚ Name 4 angles between 0˚ and 360˚ that have a
reference angle of 30˚.
150˚
30˚
210˚
330˚

26. Draw each angle, then Find the reference angle
Reference Angles
𝟎<𝜶< 𝝅 𝟐
Draw each angle, then Find the reference angle
° 2. −37° °
4. −155° 5. −350° 6. −543°
𝜋 − 7𝜋 𝜋 6
𝜋 𝜋 − 𝜋 4
24°
37°
48.4°
25°
10° 3°
𝝅 𝟑
𝝅 𝟔
𝝅 𝟔
𝝅 𝟒
𝝅 𝟑
𝝅 𝟒

27. Classwork In class practice part II
Section 7.3 & 7.4 Practice Worksheet

28. Homework Suggested practice problems
7.3: p272: 1-19all, odd this is a lot of problems, but they are almost all, answer with the unit circle.
7.4: p278: 1-17odd