We have heard sometimes people say they’ve done ‘a complete 180?’ What does that really mean? For example, if you are facing North and you complete a 180-degree rotation, you’d now be facing South. It doesn’t matter which direction, clockwise or counterclockwise, you turn because 180 degrees is exactly halfway from completing a full revolution. You can think about the coordinate plane in terms of direction. We can see North, South, East and West.
We can turn in two different directions, clockwise or counterclockwise.
All angles have an initial side and a terminal side. The initial side of the angle is where the angle starts. Angles on the coordinate plane have an initial side on the positive x-axis. If we think of it in terms of directions, all angles start on East.
Angle measurements are done corresponding to the axes on the coordinate plane. One full revolution, or circle, measures 360 degrees. If we only turn halfway, we are making an angle of 180 degrees. If we go only one quarter, we make an angle of 90 degrees. The angle measured is positive when we measure in an anti-clockwise direction with respect to the positive x-axis as shown below.
Similarly, if we complete these turns in a clockwise direction, the angles become negative.
In trigonometry, angles are placed on coordinate axes. The vertex is always placed at the origin and one ray is always placed on the positive x-axis. This ray is called the initial side of the angle. The other ray is called the terminal side of the angle. This positioning of an angle is called the standard position. The Greek letter theta theta is often used to represent an angle measure. Two angles in standard position are shown below.
When an angle is drawn in a standard position, it has a direction. Notice that there are little curved arrows in the above drawing. The one on the left goes counterclockwise and is defined to be a positive angle. The one on the right goes clockwise and is defined to be a negative angle. Why is counterclockwise positive? This is just a convention that mathematicians have agreed on because one way has to be positive and the other way negative.
Since the x and y axes are perpendicular, each axis then represents an increment of ninety degrees of rotation. The diagrams below show a variety of angles formed by rotating a ray through the quadrants of the coordinate plane.
An angle of rotation can be described infinitely in many ways. It can be described by a positive or negative angle of rotation or by making multiple full-circle rotations through 360circ. The example below illustrates this concept.
For the angle 525circ, an entire 360circ rotation is made and then we keep going another 165circ to 525circ. Therefore, the resulting angle is equivalent to 525circ−360circ, or 165circ. In other words, the terminal side is in the same location as the terminal side for a 165circ angle. If we subtract 360circ again, we get a negative angle, – 195circ. Since they all share the same terminal side, they are called coterminal angles.
Two angles in a standard position that share a common terminal side are said to be coterminal. The angles in the figure below are all coterminal with an angle that measures 30circ.
All angles that are coterminal with thetacirc can be written as
thetacirc+ncdot360circ
where n is an integer (positive, negative, or zero).
Example 1: textIsananglemeasuring200circtextcoterminalwithananglemeasuring940circtext?
Solution: If an angle measuring 940circ and an angle measuring 200circ were coterminal, then
begingathered940circ=200circ+ncdot360circ 740circ=ncdot360circendgathered
Because 740circ is not a multiple of 360circ , these angles are not coterminal.
Example 2: textName5textanglesthatarecoterminalwith−70circtext.
Solution:
begingathered−70circ+(1)360circ=290circ −70circ+(2)360circ=650circ −70circ+(3)360circ=1010circ −70circ+(−1)360circ=−430circ −70circ+(−2)360circ=−790circendgatheredA reference angle is an acute angle between the terminal side of an angle and the x− axis. The diagram below shows the reference angles for the terminal sides of angles in each of the four quadrants.
Note: A reference angle is never determined by the angle between the terminal side and the y – axis. This is a common error for students, especially when the terminal side appears to be closer to the y – axis than the x – axis.
For example, let’s determine the quadrant in which −745circ lies and hence determine the reference angle.
Since our angle is more than one rotation, we need to add 360circ until we get an angle whose absolute value is less than 360circ:−745circ+360circ=−385circ, again −385circ+360circ=−25circ
Now we can plot the angle and determine the reference angle:
Note that the reference angle is positive 25circ. All reference angles will be positive as they are acute angles (between 0circ and 90circ ).
Finally, let’s give two coterminal angles to 595circ, one positive and one negative, and find the reference angle. To find the coterminal angles we can add/subtract 360circ. In this case, our angle is greater than 360circ so it makes sense to subtract 360circ to get a positive coterminal angle: 595circ−360circ=235circ. Now subtract again to get a negative angle: 235circ−360circ=−125circ.
By plotting any of these angles we can see that the terminal side lies in the third quadrant as shown. Since the terminal side lies in the third quadrant, we need to find the angle between 180circ and 235circ, so 235circ−180circ=55circ.
Example 3: Draw a 160circ angle in standard position.
Solution: The angle is positive, so you start at the x-axis and go 160circ counterclockwise.
Example 4: Draw a −200circ angle in standard position.
Solution: The angle is negative, so you start at the x-axis and go 200circ clockwise. Remember that 180circ is a straight line. That will bring you to the negative x axis, and then you have to go 20circ farther.
Notice that the terminal sides in the two examples above are the same, but they represent different angles. Such pairs of angles are said to be coterminal angles.
For each angle drawn in the standard position, there is a related angle known as a reference angle. This is the angle formed by the terminal side and the x-axis. The reference angle is always considered to be positive and has a value anywhere from 0circ to 90circ. Two angles are shown below in the standard position.
You can see that the terminal side of the 135° angle and the x-axis form a 45° angle (this is because the two angles must add up to 180°). This 45° angle, shown in red, is the reference angle for 135°. The terminal side of the 205° angle and the x-axis form a 25° angle. It is 25° because. This 25° angle, shown in red, is the reference angle for 205°.
Note: Think “reference” means to refer to the nearest part of the x-axis
To determine the other angles in other quadrants using reference angles follow the table below.
Example 5: Determine the reference angle that corresponds to each of the following angles.
a) 165circ
b) 249circ
c) 328circ
Solution:
a) 165circ is in quadrant II left(90circ<165circ<180circright)
The reference angle is 180circ−165circ=15circ
b) 249circ is in quadrant III left(180circ<249circ<270circright)
The reference angle is 249circ−180circ=69circ
c) 328circ is in quadrant III left(270circ<328circ<360circright)
The reference angle is 360circ−328circ=32circ
Example 6: textInwhichquadrant,dothefollowingangleslie?(a)420circtext(b)−570circtext(c)600circ
Solution:
(a) Look at the figure below, the angle 420circ starts from OA and counterclockwise rotates a full circle to the OA position and continues to rotate to the mathrmOB position. Since 420circ=360circ+60circ, so, the terminal side of 420circ lies in quadrant mathrmI. That is, 420circ is the angle of quadrant mathrmI.
Note: Both angles 420circ and 60circ have the same terminal side that lies in quadrant I and they are the angles of the quadrant mathrmI.
(b) Look at the figure below, the angle −570circ starts from OA and clockwise rotates two full circles then change direction to counterclockwise rotate to the OB position. Since −570circ=−2times360circ+150circ, so, the terminal side of −570circ lies in quadrant II. That is, −570circ is the angle of quadrant II.
Note: Both angles −570circ and 150circ have the same terminal side that lies in quadrant II and they are the angles of the quadrant II.
(c) Look at the figure below, the angle 600circ starts from OA and counterclockwise rotates two full circles then change direction to clockwise rotate to the mathrmOB position. Since 600circ=2times360circ−120circ, so, the terminal side of 600circ lies in quadrant III. That is, 600circ is the angle of quadrant III.
Note: Both angles 600circ and −120circ have the same terminal side that lies in quadrant III and they are the angles of the quadrant III.
Example 7: textfindthereferenceangleof−500circtextandfindthequadrantinwhichtheterminalsidelies.
Solution:
Since our angle is more than one rotation, we need to add 360circ until we get an angle whose absolute value is less than 360circ:−500circ+360circ=−200circ.
If we plot this angle we see that it is −200circ clockwise from the origin or 160circ counterclockwise. 160circ lies in the second quadrant.
Now determine the reference angle: 180circ−160circ=20circ.
Example 8: textFindtwocoterminalanglesto138circtext,onepositiveandonenegative.
Solution:
138circ+360circ=498circtextand138circ−360circ=−222circExample 9: textFindthereferenceanglefor895circtext.
Solution:
895circ−360circ=535circ,535circ−360circ=175circ. The terminal side lies in the second quadrant, so we need to determine the angle between 175circ and 180circ, which is 5circ.
Example 10: textFindthereferenceanglefor343circtext.
Solution:
343circtextisinthefourthquadrantsoweneedtofindtheanglebetween343circtextand360circtextwhichis17circtext.
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