Pre-Calculus 11 Student Edition Chapter 2 Trigonometry
Page 75 Problem 1 Answer
Given that: Group A angles are in Standard Position.
Whereas, Group B angles are not in Standard Position.
Difference between Group A and Group B angles is as follows :
- All angles of Group A have their vertex located at the origin and one ray is on the positive x-axis.
- Whereas, no angle in Group B have their one ray on positive x-axis.
Characteristics of angles in Standard Position are :
1. An angle is in standard position if its vertex is located at the origin. And
- one ray is on the positive x-axis.
An angle which satisfies both the conditions is known as angle in Standard Position.
Difference between Group A and Group B angles is as follows :
- All angles of Group A have their vertex located at the origin and one ray is on the positive x-axis.
- Whereas, no angle in Group B have their one ray on positive x-axis.
Characteristics of angles in Standard Position are :
1. An angle is in standard position if its vertex is located at the origin. And
- one ray is on the positive x-axis.
An angle which satisfies both the conditions is known as angle in Standard Position.
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Page 75 Problem 2 Answer
The correct option is option B
The reason for the same is :
- The vertex is located at origin.
- One ray is on positive x-axis.
For other options, we have
Option A: The vertex is not located at origin.
Option C: no ray lies on positive x-axis.
The correct option is Option B as it satisfies all the conditions of angle in standard position.
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Page 75 Problem 3 Answer
Given Angle to be drawn 750,
Steps to draw:
- Create a line segment on x-axis.
- Put a proctor with center on (0,0)and aligned to the x-axis and see where the angle of 750lies.
- The terminal arm of angle lies in first quadrant as the angle is an acute angle.
The graph for the same looks as follows:
Given Angle to be drawn 1050
Steps to draw:
- Create a line segment on x-axis.
- Put a proctor with center on (0,0),and aligned to the x-axis and see where the angle of lies.
- The terminal arm of angle lies in second quadrant.
The graph for the same looks as follows:
Given Angle to be drawn 2250
Steps to draw:
- Create a line segment on x-axis.
- Put a proctor with center on (0,0),and aligned to the x-axis and see where the angle of lies.
- We will put the proctor upside down and mark at 450starting from third quadrant
- The terminal arm of angle lies in third quadrant .
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The graph for the same looks as follows:
Given Angle to be drawn 3200
Steps to draw:
- Create a line segment on x-axis.
- Put a proctor with center on (0,0),and aligned to the x-axis and see where the angle of lies.
- We will put the proctor upside down and mark at 400starting from fourth quadrant.
- The terminal arm of angle lies in fourth quadrant.
The graph for the same looks as follows:
a)750: The terminal arm lies in first quadrant
The graph looks like
- b) 1050: The terminal arm lies in second quadrant
The graph looks like
- c) 2250: The terminal arm lies in third quadrant
The graph looks like
d)3200: The terminal arm lies in third quadrant
The graph looks like
Page 75 Problem 4 Answer
Given: The measurement of Angle is 2900
We need to draw angles in standard position on XY Plane.
Given Angle to be drawn2900
Steps to draw:
- Create a line segment on x-axis.
- Put a proctor with center on (0,0),and aligned to the x-axis and see where the angle of lies.
3.We will put the proctor upside down and will mark at 1100 starting from third quadrant.
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OR
3.We will put the proctor upside down and will mark at 700 starting from third quadrant.
The graph for the same looks as follows:
The graph of angle of 2900 looks like
Page 75 Problem 5 Answer
Given- The angle 200 degrees.
To find- The drawing of the angle.
Explanation- The angle between the rays is on the common point that is called the vertex of the angle.
Make use of the protector to draw the required angle.
Draw the horizontal line first, which is called the arm of the angle.
Mark the dot at the angle of 200 degrees on the protector.
Now join the initial point of the arm with the dot to draw the required angle.
Mark the drawn angle.
The required drawn angle is
Page 75 Problem 6 Answer
Given- The angle is 130 degrees.
To find- The drawing of the angle.
Explanation- The angle between the rays is on the common point that is called the vertex of the angle.
Make use of the protector to draw the required angle.
Draw the horizontal line first, which is called the arm of the angle.
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Mark the dot at the angle of 130 degrees on the protector.
Now join the initial point of the arm with the dot to draw the required angle.
Mark the drawn angle.
The required drawn angle is
Page 75 Problem 7 Answer
Given- The angle is 325 degrees.
To find- The drawing of the angle.
Explanation- The angle between the rays is on the common point that is called the vertex of the angle.
Make use of the protector to draw the required angle.
Since the angle, 325 degrees is greater than 180 degrees and 325 degrees less than 360 degrees.
Draw the horizontal line first, which is called the arm of the angle.
Mark the dot at the angle of 35 degrees on the protector from the opposite side.
Now join the initial point of the arm with the dot to draw the required angle.
Mark the drawn angle.
The required drawn angle is
Page 81 Problem 8 Answer
Given: an angle of 60 degrees.
To determine: the angle when the given angle is reflected in the y-axis.
Summary: We will reflect the angle of 60 degrees in y-axis that is we will subtract 60 degrees from 180 degrees and get the required angle.
On subtracting 60 degrees from 180 degrees, we have, 180−60=120.
Thus, the required angle when an angle of 60 degrees is reflected in y-axis is 120 degrees.
Hence, the required angle in standard position is 120 degrees.
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Page 81 Problem 9 Answer
Given: an angle of 60 degrees.
To determine: the angle in standard position when the given angle is reflected in the x-axis.
Summary: We will reflect the angle of 60 degrees in x-axis that is we will subtract 60 degrees from 360 degrees and get the required angle.
On subtracting 60 degrees from 360 degrees, we have, 360−60=300.
Thus, the required angle when an angle of 60 degrees is reflected in x-axis is 300 degrees.
Hence, the required angle in standard position is 300 degrees.
Page 81 Problem 10 Answer
We are given the angle θ=60∘
We have to find standard angle in the y-axis and then in the x-axis after reflection.
Reflecting an angle of 60∘in the y-axis and then in the x-axis will result in a reference angle of 60∘in quadrant III.
The measure of an angle in standard position for quadrant III is 180∘+60∘=240∘
The measure of an angle in standard position for quadrant III is 240∘.
Page 82 Problem 11 Answer
Given that the tempo is adjusted so that the arm of the metronome swings from 45° to 135°.
We have to find what exact horizontal distance does the tip of the arm travel in one beat.
Find the horizontal distance a.
cos45∘=adjacent/hypotenuse
1/√2=a/10
a=10/√2
a=5√2
Because the reference angle for 135° is 45°, the tip moves the same horizontal distance past the vertical position to reach B.
The exact horizontal distance travelled by the tip of the arm in one beat is 2(√5)×2=4√5 cm.
The exact horizontal distance travelled by the tip of the arm in one beat is 4√5 cm.
Page 83 Problem 12 Answer
Given : an angle = 150∘
With the help of the figure given below we can easily match the given angle with its diagram.
Thus the correct diagram of given angle is
Hence the given angle is matched with the correct diagram.
Page 83 Problem 13 Answer
Given : an angle = 180∘
With the help of the figure given below we can easily match the given angle with its diagram.
Hence the given angle is matched with the correct diagram.
Page 83 Problem 14 Answer
Given : an angle = 45∘
With the help of the figure given below we can easily match the given angle with its diagram.
Hence the given angle is matched with the correct diagram.
Page 83 Problem 15 Answer
Given : an angle = 320∘
With the help of the figure given below we can easily match the given angle with its diagram.
Hence the given angle is matched with the correct diagram.
Page 83 Problem 16 Answer
Given: An angle is given as 215∘.
With the help of the figure given below, the given angle can be matched easily with its diagram.
Observe the position of the angle 215∘
from the diagram of the cartesian plane.
It is observed that 215∘is in the range of 180∘<θ<270∘, thus the angle215∘
lies in the third quadrant. Therefore the correct option is B.
Hence the angle 215∘matches with diagram B of the angle in standard position.
Page 83 Problem 17 Answer
Given: An angle is given as 270∘.
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With the help of the figure given below, the given angle can be matched easily with its diagram.
Observe the position of the angle270∘
from the diagram of the cartesian plane.
Thus the correct option is E.
Hence the angle270∘matches with diagram E of the angle in standard position.
Page 83 Problem 18 Answer
Given: Measure of angles are given.
To find: The quadrant in which the terminal arm of each angle lies in standard position.
Observe the diagram of the cartesian plane to find the quadrant in which the terminal arm of each angle lies in standard position.
The diagram of the cartesian plane is shown below:
(a)From the diagram is observed that the angle 48∘is in the range of 0∘<θ<90∘, hence the angle lies in the first quadrant.
(b)From the diagram is observed that the angle300∘is in the range of 270∘<θ<360∘
hence the angle lies in the fourth quadrant.
From the diagram is observed that the angle185∘ is in the range of 180∘<θ<270∘hence the angle lies in the third quadrant.
From the diagram is observed that the angle 75∘is in the range of 0∘<θ<90∘hence the angle lies in the first quadrant.
From the diagram is observed that angle 220∘is in the range of180∘<θ<270∘
hence the angle lies in the third quadrant.
From the diagram is observed that the angle 160∘is in the range of 90∘<θ<180∘
hence the angle lies in the second quadrant.
Hence the terminal arm of the angles in standard position lie:
(a)48∘ in the first quadrant.
(b)300∘in the fourth quadrant.
(c) 185∘in the third quadrant.
(d)75∘in the first quadrant.
(e)220∘in the third quadrant.
(f)160∘ in the second quadrant.
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Page 83 Problem 19 Answer
Given: An angle is given as 70∘.
To sketch: An angle in standard position of the angle 70∘.
Sketch the angle 70∘in standard position.
Since 70∘is in the range of 0∘<θ<90∘, so the angle 70∘lies in the first quadrant.
Hence the sketch of the angle 70∘in standard position is
Page 83 Problem 20 Answer
Given: An angle is given as 310∘.
To sketch An angle 310∘in standard position.
Sketch the angle 310∘in standard position.
Since310∘is in the range of270∘<θ<360∘, so 310∘
lies in the fourth quadrant.
Hence the sketch of 310∘in standard position is
Page 83 Problem 21 Answer
Given: An angle 225°
To sketch: An angle in standard position of the angle 225°.
Now, let us sketch the angle in standard position.
Since,180<θ<270,the terminal arm of θ the lies in the third quadrant.
Hence, The sketch an angle of 225° in standard position is
Page 83 Problem 22 Answer
Given: An angle 165°
To sketch: An angle in standard position of the angle 165°.
Now, let us sketch the angle in standard position.
Since,90<θ<180,the terminal arm of the θ lies in the second quadrant.
Hence, The sketch an angle of 165° in standard position is
Page 83 Problem 23 Answer
Given: An angle 170°
To find: The reference angle for the angle in standard position.
For every, angle in standard position there exist an acute angle called reference angle.
The reference angle is formed between the terminal angle and x axis.
Given: An angle 170°
To find: The reference angle for the angle in standard position.
First, let us sketch the angle in standard position.
The reference angleθR
=180−170
=10
Hence,the reference angle for each angle in standard position 170° is 10°.
Page 83 Problem 24 Answer
Given: An angle 345°
To find: The reference angle for the angle in standard position.
For every, angle in standard position there exist an acute angle called reference angle.
The reference angle is formed between the terminal angle and x axis.
Given: An angle 345°
To find: The reference angle for the angle in standard position.
First, let us sketch the angle in standard position.
The reference angleθR
=360−345
=15.
Hence,the reference angle for each angle in standard position 345° is 15°.
Page 83 Problem 25 Answer
Given: An angle 72°
To find: The reference angle for the angle in standard position.
For every, angle in standard position there exist an acute angle called reference angle.
The reference angle is formed between the terminal angle and xaxis.
Given: An angle 72°
To find: The reference angle for the angle in standard position.
First, let us sketch the angle in standard position.
The reference angle θR=72
The value of the reference angle and the value of the angle is same if the angle lies on the first quadrant.
Hence,the reference angle for each angle in standard position 72° is 72°.
Page 83 Problem 26 Answer
Given: An angle 215∘
To find: The reference angle for the angle in standard position.
For every, angle in standard position there exist an acute angle called reference angle.
The reference angle is formed between the terminal angle and x-axis.
First, let us sketch the angle in standard position.
The reference angle θR=215∘−180∘
θR=35∘
Hence,the reference angle for an angle 215∘ in standard position is 35∘
Page 83 Problem 27 Answer
Given: θR=45∘
On adding and subtracting the reference angle from 180∘or 360∘
we get the required standard angles.
First angle is 180∘−45∘=135∘
Second angle is 180∘+45∘=225∘
Third angle is 360∘−45∘=315∘
Hence the three other angles in standard position of the given reference angle are 135∘,255∘,315∘
Page 83 Problem 28 Answer
Given : θR=60∘
On adding and subtracting the reference angle from 180∘or 360∘
we get the required standard angles.
First angle is 180∘−60∘=120∘
Second angle is 180∘+60∘=240∘
Third angle is 360∘−60∘=300∘
Hence the three other angles in standard position of the given reference angle are 120∘.240∘,300∘
Page 83 Problem 29 Answer
Given : θR=30∘
On adding and subtracting the reference angle from 180∘or 360∘
we get the required standard angles.
First angle is 180∘−30∘=150∘
Second angle is 180∘+30∘=210∘
Third angle is 360∘−30∘=330∘
Hence the three other angles in standard position of the given reference angle are 150∘,210∘,330∘.
Page 83 Problem 30 Answer
Given : θR=75∘
On adding and subtracting the reference angle from 180∘or 360∘
we get the required standard angles.
First angle is 180∘−75∘=105∘
Second angle is 180∘+75∘=255∘
Third angle is 360∘−75∘=285∘
Hence the three other angles in standard position of the given reference angle are 105∘,255∘,285∘