INTRODUCTION TO ANGLES
Example 1:
If possible, express 164^{o} in EXACT radians ( is included in the answer!) reduced to lowest terms.
Given , we find the following:
Example 2:
Express 46.52^{o} in radians rounded to two decimal places. Use the button on your calculator instead of 3.14.
Given , we find the following:
Calculator Input: 46.52 / 180 Use the button on the calculator!
We find that 0.81 is its decimal approximation.
Example 3:
If possible, express the radian measure in EXACT degree measure. Reduce to lowest terms if necessary.
Given , we find the following:
Example 4:
Express the radian measure in degree measure rounded to two decimal places. Use the button on your calculator instead of 3.14.
Given , we find the following:
NOTE: Whenever possible, use the button on your calculator instead of 3.14 because this will result in more exact calculations.
Example 5:
Express the radian measure 4.8 in degree measure rounded to two decimal places. Use the button on your calculator instead of 3.14 .
Given , we find the following:
Example 6:
Express the radian measure 5 in degree measure rounded to two decimal places. Use the button on your calculator instead of 3.14 .
Given , we find the following:
Example 7:
If possible, express 630^{o} in EXACT radians ( is included in the answer!) reduced to lowest terms.
Given , we find the following:
Here we had to reduce fractions!
Alternative method since we notice that 630^{o} is a multiple of the quadrantal angle. It does not require reducing fractions.
We notice that 630^{o} = 7(90^{o}).
We know that . Therefore, we can immediately state that 7(90^{o}) is equivalent to or .
Example 8:
If possible, express 4 in EXACT degree measure.
Given , we find the following:
Here we had to reduce fractions!
Alternative method since we notice that 4 is a multiple of the quadrantal angle. It does not require reducing fractions.
1. We notice that 4is actually 4
We know that . Therefore, we can immediately state that 4is equivalent to 4(180^{o}) = 720^{o}.
2. We could also notice that 4= 2(2)
We know that . Therefore, we can immediately state that 4is equivalent to 2(360^{o}) = 720^{o}.
Example 9:
If possible, express 315^{o} in EXACT radians ( is included in the answer!) reduced to lowest terms.
Given , we find the following:
Here we had to reduce fractions!
Since we were asked to find EXACT radians we are going to leave the answer in terms of a fraction containing the number .
Alternative method since we notice that 315^{o} is a multiple of a special angle. It does not require reducing fractions.
We notice that 315^{o} = 7(45^{o}).
We know that . Therefore, we can immediately state that 7(45^{o}) is equivalent to or .
Example 10:
If possible, express 330^{o }in EXACT radians ( is included in the answer!) reduced to lowest terms.
Given , we find the following:
Here we had to reduce fractions!
Alternative method since we notice that 330^{o} is a multiple of a special angle. It does not require reducing fractions.
We notice that 330^{o} = 11(30^{o})
We know that . Therefore, we can immediately state that 11(30^{o}) is equivalent to or .
Example 11:
If possible, express 120^{o} in EXACT radians ( is included in the answer!) reduced to lowest terms.
Given , we find the following:
Here we had to reduce fractions!
Alternative method since we notice that 120^{o} is a multiple of a special angle. It does not require reducing fractions.
We notice that 120^{o} = 2(60^{o})
We know that . Therefore, we can immediately state that 2(60^{o}) is equivalent to or .
Example 12:
If possible, express the radian measure in EXACT degree measure reduced to lowest terms.
Given , we find the following:
Here we had to reduce fractions!
Alternative method since we notice that is a multiple of a special angle. It does not require reducing fractions.
We notice that .
We know that . Therefore, we can immediately state that is equivalent to
4(60^{o}) = 240^{o}.
Example 13:
If possible, express the radian measure in EXACT degree measure reduced to lowest terms.
Given , we find the following:
Alternative method since we notice that t is a multiple of a special angle. It does not require reducing fractions.
We notice that .
We know that . Therefore, we can immediately state that is equivalent to
3(45^{o}) = 135^{o}.
Example 14:
Which angles are quadrantal angles?
All quadrantal angles are integer multiples of 90^{o}!
a. 388^{o}
4(90^{o}) + 28^{o }, QI angle
b. 450^{o}
5(90^{o}), quadrantal angle
c. 720^{o}
8(90^{o}), quadrantal angle
d. 390^{o }
4(90^{o}) + 30^{o}, QIII angle
e. 540^{o}
6(90^{o}), quadrantal angle
f. 225^{o}
2(90^{o}) 45^{o}, QII angle
Example 15:
Which angles are quadrantal angles? Sometimes it is easier to change the angles to degrees before you decide!
All quadrantal angles are integer multiples of /2 90^{o}!
a. /8
22.5^{o}, QI angle
b. 5/2
5/2 590^{o}, quadrantal angle
c. 3/2
3/2 390^{o}, quadrantal angle
d. 6/5
216^{o}, QIII angle
e.
2/2 290^{o}, quadrantal angle
Example 16:
Which angles are integer multiples of special angles?
Special angles are 30^{o}, 45^{o}, and 60^{o}.
a. 388^{o}
not a multiple of special angle since neither 30, nor 45, nor 60 divides into angle evenly
b. 330^{o}
11(30^{o}), multiple of special angle
c. 225^{o}
5(45^{o}), multiple of special angle
d. 119^{o }
not a multiple of special angle since neither 30, nor 45, nor 60 divides into angle evenly
e. 120^{o}
2(60^{o}), multiple of special angle
f. 150^{o}
5(30^{o}), multiple of special angle
Example 17:
Which angles are integer multiples of special angles? Sometimes it is easier to change the angles to degrees before you decide!
Special angles are 30^{o }/6 and 45^{o} /4 and 60^{o} /3.
a. /8
22.5^{o}, not a special angle
b. 5/3
5/3 560^{o}, multiple of special angle
c. 3/4
3/4 345^{o}, multiple of special angle
d. /5
36^{o}, not a special angle
e. 7/6
7/6 730^{o}, multiple of special angle
f. 4/9
80^{o}, not a special angle
Example 18:
Find the angle measures from each graph.
A.
= 360^{o} 60^{o} = 300^{o}
B.
Here we must first find the angle between the terminal side of angle and the negative xaxis. Given is 115^{o}. We can change it to the sum of 90^{o} and 25^{o}. Therefore, tha angle between the terminal side of and the negative xaxis is 25^{o}.
Then = 180^{o} +25^{o} = 155^{o}
C.
We see that angle has two full rotations, which equals 360^{o}(2). Remaining is a rotation of 180^{o}.
Then = 360^{o}(2) +180^{o} = 900^{o}
Example 19:
Find the measure of the angle below in radians. Express the angle as an improper fraction in terms of .
The direction of the angle is counterclockwise. Therefore, we are dealing with a positive angle. We are counting two full rotations which is 2 + 2 = 4. To it we will add the measure of the arc .
We are supposed to express the answer as an improper fraction in terms of . When adding fractions, we need a common denominator. In our case, it is 4.
The measure of the angle in radians is .
Example 20:
Find the location of the terminal side of the following angles.
Hint: Graph them!
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a. 77^{o}  Quadrant I angle b. 77^{o}  Quadrant IV angle c. 125^{o}  Quadrant II angle d. 125^{o}  Quadrant III angle e. 216^{o}  Quadrant III angle f. 216^{o}  Quadrant II angle g. 330^{o}  Quadrant IV angle
i. 960^{o} = 360^{o}(2) + 240^{o}
Quadrant III angle using the angle 240^{o}
which is coterminal to the angle 960^{o}h. 330^{o}  Quadrant I angle
j. 376^{o} = 360^{o}(1) + (16^{o})
Quadrant IV angle using the angle 16^{o}
which is coterminal to the angle 376^{o}
Example 21:
Find the location of the terminal side of the following angles.
Hint: If necessary, convert the angles to decimal form, then graph them.
Remember that and and and .
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b. 0.81  Quadrant IV angle


d. 2.45  Quadrant III angle


f.  Quadrant II angle
