1. Sketching Angles And Coterminal Angles

2. Standard Position y vertex x initial side
An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis. y
vertex x
initial side

3. Link to

4. Quadrantal Angles Quadrant II 90° < θ < 180° Quadrant I
Angles in standard position having their terminal sides along the x-axis or y-axis are called Quadrantal angles. (shown in red)
90°
Quadrant II
90° < θ < 180°
Quadrant I
0° < θ < 90°
180°
360°
Quadrant III
180° < θ < 270°
Quadrant IV
270° < θ < 360°
270°

5. Let’s Practice
Take your white board and sketch each angle in standard position. Indicate its rotation by a curved arrow. Classify each angle by it’s quadrant.
1) 50°
2) °
3) °
4) °

6. Answers
Take your white board and sketch each angle in standard position. Indicate its rotation by a curved arrow. Classify each angle by it’s quadrant.
1) 50° I
2) ° II
3) ° III
4) ° IV

7. Review
A quadrantal angle is one that has its terminal side on one of the coordinate axes.
What are the four quadrantal angles?

8. 4 Quadrantal angles 90° 180° (1/2 revolution) 270°
360° (1 complete revolution)

9. Practice - Sketch each angle in standard position
Practice - Sketch each angle in standard position. Indicate it’s rotation by a curved arrow. Classify each angle by its quadrant. If the angle is a quadrantal angle, say so. This is on your worksheet.
1) 135° 2) -240
3) ) -300
5) ) 315
7) ) -90
10) 495
11) ) 750

10. ANSWERS 1) II 2) II 3) I 4) I 5) Quadrantal Angle 6) IV
7) IV 8) Quadrantal Angle
10) II
11) Quadrantal Angle 12) I

11. Coterminal Angles
Coterminal angles are angles that have the same terminal side (unlimited).
60° and 420°
( )
One revolution + 60
HAVE THE SAME TERMINAL SIDE

12. The UNIT CIRCLE
This unit circle will be essential to our study in trig.
You will want to study it daily because we will be adding new pieces of information to it constantly. Yes, you will need to memorize it!
Today we will label the positive angles.

13. Try this example
Find the angle of smallest positive measure coterminal with this angle.
908°

14. Try this example
Find the angle of smallest positive measure coterminal with this angle.
908°
Hint: subtract off 360 as many times as need to obtain an angle with measure greater than 0 but less than 360

15. Try this example
Find the angle of smallest positive measure coterminal with this angle.
908°
908 – 360 = 548
548 – 360 = 188
So 188° is coterminal with an angle of 908°

16. How about a negative angle?
Find the angle of smallest positive measure coterminal with this angle.
-75°
-75°

17. How about a negative angle?
Find the angle of smallest positive measure coterminal with this angle. HINT: use of rotation of 360°
-75°
360 + (-75) = 285°
285°
-75°

18. Coterminal
Let n represent any integer, then all angles coterminal with an angle of 150° can be expressed as 150° + (n x 360°)
N = 0 then 150° + (0 x 360°) = 150 °
N = 1 then 150° + (1 x 360°) = 510 °
N = 2 then 150° + (2 x 360°) = 870 °
N = -1 then 150° + (-1 x 360°) = -210 °

19. Find the angle of smallest positive measure coterminal with each angle
If positive angle, keep subtracting off 360° until it is in the range 0 < θ < 360°
If negative angle, then add 360°
-40 ° 5) 539 °
-98 ° 6) 699 °
-125 ° 7) 850 °
-203 ° 8) °

20. Answers
1) 320 ° 5) 179 °
2) 262 ° 6) 339 °
3) 235 ° 7) 130 °
4) 157 ° 8) 280 °

21. Now try in radians
If positive angle, keep subtracting off 2π until it is in the range 0 < θ < 2π
If negative angle, then add 2π
12π/5
17π/4
-7π/3

22. Answers 1) 12π/5 - 10π/5 = 2π /5 2) 17π/4 - 8π/4 = 9π/4 now too large
1) 12π/5 - 10π/5 = 2π /5
2) 17π/4 - 8π/4 = 9π/4 now too large
So subtract off 2 π and get 9π/4 -8π/4 = π/4
3) -7π/3 + 6π/3 = - π /3 Now less than 0 so add 2 π π/3 + 6 π /3 = 5π/3
Fractions are fun!

23. Homework
Page 453 #2-36 even
Packet p. 6
Practice unit circle