Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚...
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Transcript of Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚...
![Page 1: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/1.jpg)
Section 4.3 The Unit Circle
![Page 2: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/2.jpg)
(0, 1)
(0, -1)
(1, 0)(-1, 0)
0, 0˚ 2π, 360˚
π2
, 90˚
π, 180˚
3π 2
, 270˚
Quadrantal Angles
“All Students Take Calculus” tells you where each trig function is positive.
AS
T C
![Page 3: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/3.jpg)
, 180 0, 02, 360
3
2
2
Let’s begin with an easy family… 4
2
2
2
2
1
45
2
2,
2
2
What are the coordinates?
4
Now, reflect the triangle to the second quadrant…
![Page 4: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/4.jpg)
What are the coordinates?
, 180 0, 02, 360
3
2
2
2
2
2
2
1
45
2
2,
2
2
4
Now, reflect the triangle to the third quadrant…
1
-2
2
2
2
-2
2,
2
2 3
4
![Page 5: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/5.jpg)
What are the coordinates?
Now, reflect the triangle to the fourth quadrant…
, 180 0, 02, 360
3
2
2
2
2
2
2
1
45
2
2,
2
2
4
1
-2
2
2
2
-2
2,
2
2 3
4
-2
2, -
2
2 5
4
![Page 6: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/6.jpg)
What are the coordinates?
, 180 0, 02, 360
3
2
2
2
2
2
2
1
45
2
2,
2
2
4
1
-2
2
2
2
-2
2,
2
2 3
4
-2
2, -
2
2 5
4
2
2, -
2
2 7
4
![Page 7: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/7.jpg)
EX. Evaluate the six trig functions at θ = .
**REMEMBER!!! cos θ is represented by the x-coordinates and sin θ is represented by the y-coordinates.**
4
![Page 8: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/8.jpg)
EX. Evaluate the six trig functions at θ = .3
4
Homework: worksheet
![Page 9: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/9.jpg)
30
1 1
2
3
2
6
3
2,1
2
Now, reflect the triangle to the second quadrant.
Complete the family… 6
![Page 10: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/10.jpg)
30
1 1
2
3
2
6
3
2,1
2
Now, reflect the triangle to the third quadrant.
5
6
1
2
-3
2
-3
2,1
2
![Page 11: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/11.jpg)
30
1 1
2
3
2
6
3
2,1
2
Now, reflect the triangle to the fourth quadrant.
5
61
2
-3
2
-3
2,1
2
-3
2, -
1
2
What are the coordinates?
7
6
![Page 12: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/12.jpg)
What are the coordinates?
30
11
2
3
2
6
3
2,1
2 5
6
1
2
-3
2
-3
2,1
2
-3
2, -
1
2 7
6
3
2, -
1
2 11
6
![Page 13: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/13.jpg)
EX. Evaluate the six trig functions at θ = .
6
![Page 14: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/14.jpg)
EX. Evaluate the six trig functions at θ = .
Homework: worksheet
7
6
![Page 15: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/15.jpg)
3
3
2
1
2
1
2,
3
2
3
1
, 180 0, 02, 360
3
2
2
Complete the family…
Now, reflect the triangle to the second quadrant.
60˚
![Page 16: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/16.jpg)
1
, 180 0, 02, 360
3
2
2
60
3
2
1
2
1
2,
3
2
3
1
-1
2
What are the coordinates?
-1
2,
3
2 2
3
3
2
Now, reflect the triangle to the third quadrant.
![Page 17: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/17.jpg)
1
, 180 0, 02, 360
3
2
2
60
3
2
1
2
1
2,
3
2
3
13
2
-1
2
-1
2,
3
2 2
3
What are the coordinates?
-1
2, -
3
2 4
3
Now, reflect the triangle to the fourth quadrant.
![Page 18: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/18.jpg)
1
, 180 0, 02, 360
3
2
2
60
3
2
1
2
1
2,
3
2
3
13
2
-1
2
-1
2,
3
2 2
3
-1
2, -
3
2 4
3
1
2, -
3
2
What are the coordinates?
5
3
![Page 19: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/19.jpg)
Shortcut to remember ordered pairs in quadrant 1
1 32 23 1
1) Parenthesis2) Fractions with denominator of 23)
4) Square roots
![Page 20: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/20.jpg)
EX. Evaluate the six trig functions at θ = .
3
![Page 21: Section 4.3 The Unit Circle. (0, 1) (0, -1) (1, 0)(-1, 0) 0, 0˚ 2π, 360˚ π2π2, 90˚ π, 180˚ 3π 2, 270˚ Quadrantal Angles “All Students Take Calculus” tells.](https://reader033.fdocuments.in/reader033/viewer/2022061416/56649d1b5503460f949f0c2a/html5/thumbnails/21.jpg)
EX. Evaluate the six trig functions at θ = .
Homework: worksheet
2
3