mcgrawhill ryerson precalculus 11 chapter 2 trigonometry · chapter 2 trigonometry. section 2.1...
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McGrawHill Ryerson
PreCalculus 11
Chapter 2Trigonometry
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Focus On ...
Chapter 2
• sketching an angle from 0° to 360° in standard position and determining its reference angle• determining the quadrant in which an angle in standard position terminates• determining the exact values of the sine, cosine, and tangent ratios of a given angle with reference angle 30°, 45°, or 60°• solving problems involving trigonometric ratios
2.1 Angles in Standard Position
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Definitions:
Initial Arm: the arm of an angle in standard position that lies on the xaxis
Terminal Arm: the arm of an angle in standard position that meets the initial arm at the origin to form an angle
Angle in Standard Position: The position of an angle when its initial arm is on the positive xaxis and its vertex is at the origin.
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R
Click here for the solution.
2.1 Angles in Standard Position
Initial Arm
Terminal Arm
Reference Angle
Angle in Standard Position
QuadrantsI II III IV
Drag to label the parts of the above diagram.
Hint
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Reference Angles• Reference
angle is always positive
• Measures between 0 and 90 degrees
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Click here for the solution.
Example 1 Sketch an Angle in Standard Position, 0° ≤ θ < 360° Sketch each angle in standard position. State the quadrant in which the terminal arm lies.
0
00
c) 315°
a) 36° b) 210°
0 360
10 350
20340
30
33040
32050
31060
300
70
290
80
280
90
270
0°
100
260
110
250
120
240
130
230
140
220
150
210
160200
170190
180180
190170
200160
210
150220
140 230
130240
120
250
110
260
100
270
90
280
80
290
70
300
60
310
50
320
40
330
30
34020
350 10
Use the protractor to measure the indicated angle. • Drag the green dot to the desired angle.• Press the green arrow to draw the angle.• Drag the angle to the coordinate grid.
Note that if the green arrow and dot are not visible on the protractor, click in the centre of the protractor.
Hint
2.1
36°
210°
315°
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2.1
Example 1: Your Turn
Sketch each angle in standard position. State the quadrant inwhich the terminal arm lies.
0 0
0
a) 150° b) 60°
c) 240°
Click here for the solution.
0 360
10 350
20340
30
33040
32050
31060
300
70
290
80
280
90
270
0°
100
260
110
250
120
240
130
230
140
220
150
210
160200
170190
180180
190170
200160
210
150220
140 230
130240
120
250
110
260
100
270
90
280
80
290
70
300
60
310
50
320
40
330
30
34020
350 10
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Example 2 Determine a Reference Angle Determine the reference angle θR for each angle θ. Sketch θ in standard position and label the reference angle θR.
a) 130° b) 300°
2
130°
2
300°
00
Hint
2.1
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2.1
Click here for the solution.
Example 2: Your TurnDetermine the reference angle θR for each angle θ. Sketch θ and θR in standard position.
0 0
a) θ = 75° b) θ = 240°
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Example 3 Determine the Angle in Standard Position Determine the angle in standard position when an angle of 40° is reflected.
00
a) in the yaxis b) in the xaxis
Continue Next Page
2.1
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0
c) in the yaxis and then in the xaxis
2.1 Example 3 Continued
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2.1
Click here for the solution.
Example 3: Your TurnDetermine the angle in standard position when an angle of 60° is reflected.
0 0 0
c) in the yaxis and then in the xaxis
b) in the xaxisa) in the yaxis
60° 60° 60°
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Section 2.1(Continued)
Angles in Standard Position
Review
Angle in Standard Position: The position of an angle when its initial arm is on the positive xaxis and its vertex is at the origin.
Standard Position Not Standard Position
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Lesson Focus: Numbering Quadrants and Range of Angles
2.1 Angles in Standard Position
2.1 Angles in Standard Position
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Reference Angles• Reference
angle is always positive
• Measures between 0 and 90 degrees
Note: The angle between the terminal ray and the xaxis!!
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Lesson Focus: Reference Angle Calculations (see page 78)
2.1 Angles in Standard Position
2.1 Angles in Standard Position
θR= 180o - θ θR= θ
θR= θ - 180oθR= 360o - θ
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Find the reference angle for the following.
11003300
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Find all four angles within 3600 that have a reference angle of 500.
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Find the angle of standard position for a terminal arm which has a angle of 200 is reflected about the • xaxis• yaxis• x and y axes Reflected about the xaxis
Reflected about the yaxis Reflected about both
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Find x, then find the three trig ratios of angle A.
3
4x
A
B
C
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Exact Values: Answers involving radicals or fractions are exact unlike approximated decimal values.
Examples:
Exact Not Exact
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Find x, then find the three trig ratios of angle A. Give exact answers
2
5x
A
B
C
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Very Special Triangles
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Lesson Focus: Special Triangles
2.1 Angles in Standard Position
2.1 Angles in Standard Position
45o 45o 90o Triangle
45o
45o
1
1
Calculate c
c
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Lesson Focus: Investigate
2.1 Angles in Standard Position
2.1 Angles in Standard Position
Equilateral triangle:
60o 60o
60o
2
22
a
Calculate the altitude of the equilateral triangle
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Lesson Focus: Special Triangles and Exact Values
2.1 Angles in Standard Position
2.1 Angles in Standard Position
30o
60o
2
1
45o
45o
1
1
30o 45o 60o
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Trigonometric Ratios:
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01
2
34 5
67
89
2345°30°
60°
Special Right Triangles
Use the symbols and values to label the following triangles. Symbols may be used more than once, or not at all.
Click here for the solution.
2.1
45°30°60°
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Special Right Triangles
For angles of 30°, 45°, and 60°, you can determine the exact values of trigonometric ratios. Determine what should go in each cell, and then click on the button in each cell to check.
θ sin θ cos θ tan θ 30°
45°
60°
2.1
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A 40 cm wiper rotates from 300 to 1500. Find the exact horizontal distance it covers.
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A 12 m boom lowers from 600 to 450. Find the change in vertical height. Give an exact answer.
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2.1
Example 4: Find an exact Distance
Allie is learning to play the piano. Her teacher uses a metronome to help her keep time. The pendulum arm of the metronome is 10 cm long. For one particular tempo, the setting results in the arm moving back and forth from a start position of 60 to 120. What horizontal distance does the tip of the arm move in one beat? Give an exact answer.
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Your Turn
The tempo is adjusted so that the arm of the metronome swings from 45 to 135. What exact horizontal distance does the tip of the arm travel in one beat?
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2.1
The following pages are solution pages
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Initial Arm
Terminal Arm
Reference AngleAngle in Standard Position
III
III IV
R
Click here to return
2.1
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60°
30°
1
2 345°
45°1
1 2
Click here to return
2.1
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c) 315°
315°
Quadrant IV
b) 210°
Quadrant III
210°
a) 36°
36°
Quadrant I
2.1
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2.1
0
III
III IV
a) 150° b) 60° c) 240°
150°
0
III
III IV
0
III
III IV
60°240°
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Since 90° < θ < 180°, the terminal arm of θ lies in quadrant II.
Since 0° < θ < 90°, the terminal arm of θ lies in quadrant I.
Since 180° < θ < 270°, the terminal arm of θ lies in quadrant III.
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2.1
a) θ = 75°
75°
b) θ = 60°
240°
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R R
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2.1
0
60°
0
60°300°
0
60°240°
c) in the yaxis and then in the xaxis
b) in the xaxisa) in the yaxis
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