Lesson Objectives
- Learn about the standard position of an angle
- Learn how to find the measures of coterminal angles
Standard Position
An angle is in standard position if its vertex is at the origin and its initial side lies on the positive x-axis. When we have an angle that is in standard position, it is said to lie in the quadrant where its terminal side lies. 
We can see in our image above that an acute angle is in quadrant I. 
We can see in our image above that an obtuse angle is in quadrant II. 
If we continue rotating the ray, we can produce an angle with a measure that is larger than 360°. For example, if we performed an extra quarter of a full rotation, we would produce an angle that is 450° (360° + 90° = 450°). 
Coterminal angles are angles with the same initial side and the same terminal side but differ by amounts of rotation. Their measures will differ by a multiple of 360°
As an example, 55° and 415° are coterminal angles. They have the same initial side and the same terminal side, and their measures differ by an amount of 360.
415° - 55° = 360°
We can see in the above image that the 415° angle can be formed by making one full rotation (360°) along with a 55° rotation. Let's look at an example.
Example #1: State if the given angles are coterminal. $$90°, -630°$$ We want to determine if the difference between the two angles is a multiple of 360. $$-630° - 90°=-720°$$ Once we have our difference, we can check to see if dividing by 360 gives us a remainder. $$-720 \div 360=-2$$ Since there is no remainder from the division, we can say these angles are coterminal. The angle measures differ by a multiple of 360°. Note, reversing the order doesn't change the result. Only the sign will change, but there will still be no remainder from our division. $$90° - (-630°)=720°$$ $$720 \div 360=2$$ Example #2: State if the given angles are coterminal. $$55°, -485°$$ We want to determine if the difference between the two angles is a multiple of 360. $$55° - (-485°)=540°$$ Once we have our difference, we can check to see if dividing by 360 gives us a remainder. $$540 \div 360=1 R \hspace{.15em}180$$ Since there is a remainder from the division, we can say these angles are not coterminal.
In some cases, we may be asked to find a coterminal angle between 0° and 360°. Let's look at an example.
Example #3: Find a coterminal angle between 0° and 360°. $$1015°$$ Here, we would subtract 360° as many times as needed to obtain an angle measure that is greater than 0° but less than 360° $$1015° - 360°=655°$$ Although 655° and 1015° are coterminal angles, our angle measure doesn't fall in the desired range, we perform the operation again. $$655° - 360°=295°$$ 295° is our desired angle.
We can see in our image above that an acute angle is in quadrant I. We can see in our image above that an obtuse angle is in quadrant II. Quadrantal Angles
Quadrantal Angles are angles that are in standard position and the terminal side lies on the x-axis or the y-axis. The angle measures of such angles would be 90°, 180°, 270°, and so on and so forth...Finding the Measures of Coterminal Angles
At this point, we know that a complete rotation of a ray represents an angle, whose measure is 360°. If we continue rotating the ray, we can produce an angle with a measure that is larger than 360°. For example, if we performed an extra quarter of a full rotation, we would produce an angle that is 450° (360° + 90° = 450°). Coterminal angles are angles with the same initial side and the same terminal side but differ by amounts of rotation. Their measures will differ by a multiple of 360° As an example, 55° and 415° are coterminal angles. They have the same initial side and the same terminal side, and their measures differ by an amount of 360.
415° - 55° = 360°
We can see in the above image that the 415° angle can be formed by making one full rotation (360°) along with a 55° rotation. Let's look at an example. Example #1: State if the given angles are coterminal. $$90°, -630°$$ We want to determine if the difference between the two angles is a multiple of 360. $$-630° - 90°=-720°$$ Once we have our difference, we can check to see if dividing by 360 gives us a remainder. $$-720 \div 360=-2$$ Since there is no remainder from the division, we can say these angles are coterminal. The angle measures differ by a multiple of 360°. Note, reversing the order doesn't change the result. Only the sign will change, but there will still be no remainder from our division. $$90° - (-630°)=720°$$ $$720 \div 360=2$$ Example #2: State if the given angles are coterminal. $$55°, -485°$$ We want to determine if the difference between the two angles is a multiple of 360. $$55° - (-485°)=540°$$ Once we have our difference, we can check to see if dividing by 360 gives us a remainder. $$540 \div 360=1 R \hspace{.15em}180$$ Since there is a remainder from the division, we can say these angles are not coterminal.
In some cases, we may be asked to find a coterminal angle between 0° and 360°. Let's look at an example.
Example #3: Find a coterminal angle between 0° and 360°. $$1015°$$ Here, we would subtract 360° as many times as needed to obtain an angle measure that is greater than 0° but less than 360° $$1015° - 360°=655°$$ Although 655° and 1015° are coterminal angles, our angle measure doesn't fall in the desired range, we perform the operation again. $$655° - 360°=295°$$ 295° is our desired angle.
Skills Check:
Example #1
Determine if the given angles are coterminal. $$180°, 720°$$
Please choose the best answer.
A
Yes
B
No
Example #2
Find the coterminal angle between 0° and 360°. $$695°$$
Please choose the best answer.
A
95°
B
115°
C
335°
D
245°
E
195°
Example #3
Find the coterminal angle between 0° and 360°.
995° 37' 15"
Please choose the best answer.
A
115° 18' 22"
B
65° 13' 19"
C
275° 37' 15"
D
205° 23' 45"
E
155° 37' 15"
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