1. Warm Up Solve for x: 𝟓 𝟐 𝟔𝒙− 𝟑 𝟓 +𝟐𝒙= 𝟏 𝟒 𝒙+𝟖 𝟓 𝟒 𝒙+𝟐 =−𝟏𝟓
I can complete the unit circle
Warm Up
Solve for x:
𝟓 𝟐 𝟔𝒙− 𝟑 𝟓 +𝟐𝒙= 𝟏 𝟒 𝒙+𝟖
𝟓 𝟒 𝒙+𝟐 =−𝟏𝟓

2. Warm Up
Solve for x:
𝟓 𝟐 𝟔𝒙− 𝟑 𝟓 +𝟐𝒙= 𝟏 𝟒 𝒙+𝟖

3. Warm Up
Solve for x:
𝟓 𝟒 𝒙+𝟐 =−𝟏𝟓

4. Homework Questions

5. What is the Unit Circle?

6. Is there another way to solve?
Unit Circle
Solve for 𝑦
sin 30 =𝑦
𝑦= 1 2
Solve for 𝑥
60°
cos 30 =𝑥
𝑥=.86660
Is there another way to solve? 1 𝑦
30° 𝑥

7. Unit Circle 1 2 2 + 𝑥 2 = 1 2 Solve for 𝑥 1 4 + 𝑥 2 =1 𝑥 2 = 3 4
𝑥 2 = 1 2
1 4 + 𝑥 2 =1
𝑥 2 = 3 4
𝑥= 3 4 𝑥=
Solve for 𝑥
60° 1 𝑦
30° 𝑥

8. What we know so far…

9. What we know so far…
𝟑 𝟐
𝟏 𝟐 𝟑𝟎

10. Unit Circle
Solve for 𝑥
cos 45 =𝑥 𝑥=
45° 1 𝑦
45° 𝑥

11. Unit Circle Solve for 𝑥 Is there another way to solve for 𝑥?
𝑥 2 + 𝑦 2 = 1 2
𝑥 2 + 𝑥 2 =1
2 𝑥 2 =1
𝑥 2 = 1 2
𝑥= 1 2 𝑥=
45° 1 𝑥 𝑦
45° 𝑥

12. Unit Circle
Solve for 𝑥
cos 45 =𝑥 𝑥=
45° 1 𝑦
45° 𝑥

13. Unit Circle Solve for 𝑥 Is there another way to solve for 𝑥?
𝑥 2 + 𝑦 2 = 1 2
𝑥 2 + 𝑥 2 =1
2 𝑥 2 =1
𝑥 2 = 1 2
𝑥= 1 2 𝑥=
45° 1 𝑥 𝑦
45° 𝑥

14. Unit Circle
Solve for 𝑥 and 𝑦
𝑥= 1 2 𝑦=
30° 1 𝑦
60° 𝑥

15. Unit Circle
Using what we have discussed about patterns, try to fill in the first quadrant

16. What we know so far…
𝟐 𝟐
𝟐 𝟐
𝟑 𝟐
𝟏 𝟐 𝟒𝟓 𝟑𝟎

17. Unit Circle
Solve for 𝑥 and 𝑦
𝑥= 1 2 𝑦=
30° 1 𝑦
60° 𝑥

18. What we know so far… 𝟎 𝟏
𝟑 𝟐 𝟗𝟎
𝟏 𝟐
𝟐 𝟐
𝟐 𝟐 𝟔𝟎
𝟑 𝟐
𝟏 𝟐 𝟒𝟓 𝟑𝟎 𝟎 𝟏 𝟎

19. In this case, 𝜃, 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑎𝑛𝑔𝑙𝑒, and the calculation gives the angle of rotation:
165°
180−𝜃°
195°
360−𝜃°
245°
180+𝜃° 𝜃°
15°

20. Reference Angles
A devious person created a new ride similar to The Screamer. On this ride, the circle rotates every 12°. At 12° a rider is 15 feet off the ground. What other angles is the rider the same distance from the ground? 180−12°=168° °=192° 360−12°=348°

21. Reference Angles
A rider is 45 feet from the ground when the angle of rotation is 134°. What other angles is the rider the same distance from the ground? 180−𝜃°=134° 𝜃= °=226° 360−46°=314°

22. Reference Angles
A reference angle helps find repeating patterns in a circle.𝑖𝑓 𝜃° is the given angle on the unit circle: the reference angle, 𝛼, will be: Quadrant II: 180−𝜃° Quadrant III: 180+𝜃° Quadrant IV: 360−𝜃° }
And the 𝑠𝑖𝑛𝜃 𝑜𝑟 𝑐𝑜𝑠𝜃 is equal to sin 𝛼 𝑜𝑟 𝑐𝑜𝑠𝛼

23. More on reference angles
A reference angle is always a 1st quadrant angle
A reference angle is always positive
Reference angles are always determined by measuring to the X-axis, NEVER to the Y-axis

24. Reference Angles
Identify the reference angle for each angle (Hint what quadrant would the angle be in?):
133°
245°
18°
359°
47°
65°
18° 1°

25. Unit Circle
Using what we have discussed about patterns and reference angles, try to fill in the rest of the unit circle keeping in mind when values are positive or negative based on the coordinate grid.

26. 𝟎 𝟏
𝟑 𝟐
𝟏 𝟐 𝟗𝟎
𝟐 𝟐
𝟐 𝟐
𝟏𝟐𝟎 𝟔𝟎
𝟑 𝟐
𝟏 𝟐
𝟏𝟑𝟓 𝟒𝟓
𝟏𝟓𝟎 𝟑𝟎
𝟏𝟖𝟎 𝟎

27. 𝟎 𝟏
𝟑 𝟐
−𝟏 𝟐
𝟑 𝟐
𝟏 𝟐 𝟗𝟎
𝟐 𝟐
𝟐 𝟐
− 𝟐 𝟐
𝟐 𝟐
𝟏𝟐𝟎 𝟔𝟎
𝟑 𝟐
− 𝟑 𝟐
𝟏 𝟐
𝟏 𝟐
𝟏𝟑𝟓 𝟒𝟓
𝟏𝟓𝟎 𝟑𝟎
𝟏𝟖𝟎 𝟎 −𝟏 𝟎 𝟏 𝟎

28. 𝟑𝟔𝟎
𝟐𝟏𝟎
𝟑𝟑𝟎
𝟐𝟐𝟓
𝟑𝟏𝟓
𝟐𝟒𝟎
𝟑𝟎𝟎
𝟐𝟕𝟎

29. 𝟑𝟔𝟎
𝟐𝟏𝟎
𝟑𝟑𝟎
𝟐𝟐𝟓
𝟑𝟏𝟓
− 𝟑 𝟐
−𝟏 𝟐
𝟑 𝟐
−𝟏 𝟐
𝟐𝟒𝟎
𝟑𝟎𝟎
− 𝟐 𝟐
− 𝟐 𝟐
− 𝟐 𝟐
𝟐 𝟐
− 𝟑 𝟐
−𝟏 𝟐
− 𝟑 𝟐
𝟐𝟕𝟎
𝟏 𝟐 𝟎 −𝟏

30. Notes: Unit Circle
Points on the unit circle are (𝑐𝑜𝑠𝜃, 𝑠𝑖𝑛𝜃) Find the exact value of sin⁡(150°) sin 150° = 1 2 Find the exact value of cos⁡(210°) cos 210° =− 3 2

31. Homework
Textbook:
, , 8-67