menu

arrow_back Graph each of the (oriented) angles below in standard position and classify them according to where their terminal side lies. Find three coterminal angles, at least one of which is positive and one of which is negative.

by Platinum
(106,848 points)
in Mathematics
57 views
Graph each of the (oriented) angles below in standard position and classify them according to where their terminal side lies. Find three coterminal angles, at least one of which is positive and one of which is negative.

1. \(\alpha=60^{\circ}\)
2. \(\beta=-225^{\circ}\)
3. \(\gamma=540^{\circ}\)
4. \(\varphi=-750\)

1 Answer

Best answer
0 like 0 dislike
 
Best answer

1. To graph \(\alpha=60^{\circ}\), we draw an angle with its initial side on the positive \(x\)-axis and rotate counter-clockwise \(\frac{60^{\circ}}{360^{\circ}}=\frac{1}{6}\) of a revolution. We see that \(\alpha\) is a Quadrant I angle.
To find angles which are coterminal, we look for angles \(\theta\) of the form \(\theta=\alpha+360^{\circ} \cdot k\) for some integer \(k\).
- When \(k=1\), we get \(\theta=60^{\circ}+360^{\circ}=420^{\circ}\).
- Substituting \(k=-1\) gives \(\theta=60^{\circ}-360^{\circ}=-300^{\circ}\).
- If we let \(k=2\), we get \(\theta=60^{\circ}+720^{\circ}=780^{\circ}\).

 

2. Since \(\beta=-225\) is negative, we start at the positive \(x\)-axis and rotate clockwise \(\frac{225}{360^{\circ}}=\frac{5}{8}\) of a revolution. We see that \(\beta\) is a Quadrant II angle.

To find coterminal angles, we proceed as before and compute \(\theta=-225^{\circ}+360^{\circ} \cdot k\) for integer values of \(k\). Letting \(k=1\)
\(k=-1\) and \(k=2\), we find \(135^{\circ},-585^{\circ}\) and \(495^{\circ}\) are all coterminal with \(-225^{\circ}\).

3. Since \(\gamma=540\) is positive, we rotate counter-clockwise from the positive \(x\)-axis. One full revolution accounts for \(360^{\circ}\), with \(180^{\circ}\), or half of a revolution, remaining. Since the terminal side of \(\gamma\) lies on the negative \(x\)-axis, \(\gamma\) is a quadrantal angle.

All angles coterminal with \(\gamma\) are of the form \(\theta=540^{\circ}+360^{\circ} \cdot k\), where \(k\) is an integer. Working through the arithmetic, we find three such angles: \(900^{\circ}, 180^{\circ}\) and \(-180^{\circ}\).

4. The Greek letter \(\varphi\) is pronounced 'fee' or 'fie'9 and since \(\varphi=-750 \quad\) is negative, we begin our rotation clockwise from the positive \(x\)-axis. Two full rotations account for \(720^{\circ}\), with just \(30^{\circ}\) or \(\frac{1}{12}\) of a revolution to go.
We find that \(\varphi\) is a Quadrant IV angle. To find coterminal angles, we compute \(\theta=-750^{\circ}+360^{\circ} \cdot k\) for a few integers \(k\) and obtain \(-390^{\circ},-30^{\circ}\) and \(330^{\circ} .\)

by Platinum
(106,848 points)

Related questions


Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
What is the difference between an initial side and a terminal side of an oriented angle?
1 answer 61 views
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
In relation to the points $P_{1}$ and $P_{2}$, what can you say about the terminal point of the following vector if its initial point is at the origin?

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Find the terminal point of the vector that is equivalent to $\mathbf{u}=(1,2)$ and whose initial point is $A(1,1)$.
0 answers 5 views
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Find the initial point of the vector that is equivalent to $\mathbf{u}=(1,1,3)$ and whose terminal point is $B(-1,-1,2)$.
0 answers 5 views
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Find the initial point of the vector that is equivalent to $\mathbf{u}=(1,2)$ and whose terminal point is $B(2,0)$.
0 answers 5 views
close

Join the MathsGee Homework Help Q&A club where you get study support for success from our verified experts. Subscribe for only R100 per month or R960 per year.


On the MathsGee Homework Help Q&A learning community, you can:


  1. Ask questions
  2. Answer questions
  3. Comment on Answers
  4. Vote on Questions and Answers
  5. Donate to your favourite users
  6. Create/Take Live Video Lessons

Posting on the MathsGee Homework Help Q&A learning community


  1. Remember the human
  2. Behave like you would in real life
  3. Look for the original source of content
  4. Search for duplicates before posting
  5. Read the community's rules

MathsGee is Zero-Rated (You do not need data to access) on: Telkom |Dimension Data | Rain | MWEB

MathsGee Tools

Math Worksheet Generator

Math Algebra Solver

Trigonometry Simulations

Vectors Simulations

Matrix Arithmetic Simulations

Matrix Transformations Simulations

Quadratic Equations Simulations

Probability & Statistics Simulations

PHET Simulations

Visual Statistics

Management Leadership | MathsGee ZOOM | eBook | H5P