Post on 12-Jan-2016
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Introduction to Trigonometry
Angles and Radians
(MA3A2): Define an understand angles measured in degrees and radians.
the rotating ray and the positive half of the x-axis
angle
a positive angle whose vertex is at the center of a circle
central angle
2 angles with a sum of 90 degrees
complementary angles
2 angles in standard position that have the same terminal side
coterminal angles
The beginning ray of an angular rotation; the positive half of the x-axis
for an angle in standard position
initial side
a clockwise measurement of an angle
negative degree measure
Moving around a circle toward the right; a negative rotation
Clockwise
a counterclockwise measurement of an angle
positive degree measure
Opposite to clockwise; moving around the circle toward the left; a
positive rotation
counterclockwise
An angle in standard position with a terminal side that coincides with one
of the four axes
quadrantal angle
One of the 4 regions into which the x- and y-axes divide the coordinate
plane
quadrant
A unit of angle measure
radian
The position of an angle with vertex at the origin, initial side on the positive x-axis, and terminal side in the plane
standard position
2 angles with a sum of 180 degrees
supplementary angles
The ending ray of an angular rotation; rotating ray
terminal side
1 full revolution
Obj: Find the quadrant in which the terminal side of an angle lies.
half a revolution
¼ of a revolution
Obj: Find the quadrant in which the terminal side of an angle lies.
triple revolution
Quad. IQuad. II
Quad. III Quad. IV
0◦/360°
90◦
180◦
270◦
III
III IV
0◦/360◦
90◦
180◦
270◦
In which quadrant does the terminal side of the angle lie?
EX: 53°
III
III IV
0◦/360◦
90◦
180◦
270◦
In which quadrant does the terminal side of each angle lie?
EX: 253°
III
III IV
0◦/360◦
90◦
180◦
270◦
EX: In which quadrant does the terminal side of each angle lie?
EX: -126°
III
III IV
0◦/360◦
90◦
180◦
270◦
EX: In which quadrant does the terminal side of each angle lie?
EX: -373°
III
III IV
0◦/360◦
90◦
180◦
270◦
EX: In which quadrant does the terminal side of each angle lie?
EX: 460°
III
III IV
0◦/360◦
90◦
180◦
270◦
ON YOUR OWN: In which quadrant does the terminal side of each angle lie?
1. 47° 2. 212° 3. -43°
4. -135° 5. 365°
III
III IV
0◦/360◦
90◦
180◦
270◦
ON YOUR OWN: In which quadrant does the terminal side of each angle lie?
1. 47° 2. 212° 3. -43°
4. -135° 5. 365°
I III IV
III I
ON YOUR OWN: Draw each angle.
7. -90°6. 45°
8. 225° 9. 405°
ON YOUR OWN: Draw each angle.
7. -90°6. 45°
8. 225° 9. 405°
Θ (theta)
Lowercase Greek letters are used to denote angles
α (alpha)
β (beta)
γ (gamma)
EX: Θ = 60°.
Finding Coterminal Angles (angles that have the same terminal side):
EX: Θ = 790°.
Finding Coterminal Angles (angles that have the same terminal side):
1) Add and subtract from 360°, OR2) Add and subtract from 2∏.
EX: Θ = 440°.
EX: Θ = -855°.
1. Θ = 790°.
ON YOUR OWN: Find one positive and one negative coterminal angle.
Reference Angle: the acute angle formed by the terminal side and the closest x-axis.
EX: Find the reference angle for each angle. Θ = 115°.
**Reference angles are ALWAYS positive!!
**Subtract from 180° or 360°.
Reference Angle: the acute angle formed by the terminal side and the closest x-axis.
EX: Θ = 225°.
**Reference angles are ALWAYS positive!!
**Subtract from 180° or 360°.
Reference Angle: the acute angle formed by the terminal side and the closest x-axis.
EX: Θ = 330°
**Reference angles are ALWAYS positive!!
**Subtract from 180° or 360°.
EX: Θ = -150°.
Finding Reference Angles fpr Negative Angles:
1) Add 360° to find the coterminal angle.2) Subtract from closest x-axis (180° or 360°) .
EX: Θ = 60°.
**If Θ lies in the first quadrant, then the angle is it’s own reference angle.
11. Θ = 210 °10. Θ = 405°
12. Θ = -300 ° 13. Θ = -225 °
ON YOUR OWN: Find the reference angle for the following angles of rotation.
11. Θ = 210 °10. Θ = 405°
12. Θ = -300 ° 13. Θ = -225 °
ON YOUR OWN: Find the reference angle for the following angles of rotation.
Converting Degrees to Radians
*Multiply by ∏ radians 180 degrees
EX: 60°
EX: 225°
EX: 300°
EX: -315°
ON YOUR OWN: Convert to radian measure. Give answer in terms of ∏.
14. 15°
15. 30°
16. 100°
17. -220°
18. -85°
ON YOUR OWN: Convert to radian measure. Give answer in terms of ∏.
14. 15°
15. 30°
16. 100°
17. -220°
18. -85°
12
6
9
5
9
11
36
17
Converting Radians to Degrees
*Multiply by 180 degrees ∏ radians
EX:
EX:
EX:
EX:
4
33
4
2
55
4
ON YOUR OWN: Convert to Degrees
6
719.
20.
21.
22.
23.
12
4
3
4
5
2
3
-210°
-2160°
135°
225°
270°
ON YOUR OWN: Convert to Degrees
6
719.
20.
21.
22.
23.
12
4
3
4
5
2
3
Finding Coterminal Angles
*add or subtract from 360°*_____degrees ± 360
EX: 390° EX: 140°
EX: -100°
Finding Complements and Supplements
*To find the complement: subtract from 90°*To find the supplement: subtract from 180°
EX: 35° EX: 120°