INTRODUCTION TO ANGLES

Example 1:

If possible, express 164o in EXACT radians ( is included in the answer!) reduced to lowest terms.

Given , we find the following:

Example 2:

Express 46.52o 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 630o 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 630o is a multiple of the quadrantal angle. It does not require reducing fractions.

We notice that 630o = 7(90o).

We know that . Therefore, we can immediately state that 7(90o) 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(180o) = 720o.

2. We could also notice that 4= 2(2)

We know that . Therefore, we can immediately state that 4is equivalent to 2(360o) = 720o.

Example 9:

If possible, express 315o 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 315o is a multiple of a special angle. It does not require reducing fractions.

We notice that 315o = 7(45o).

We know that . Therefore, we can immediately state that 7(45o) is equivalent to or .

Example 10:

If possible, express 330o 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 330o is a multiple of a special angle. It does not require reducing fractions.

We notice that 330o = 11(30o)

We know that . Therefore, we can immediately state that 11(30o) is equivalent to or .

Example 11:

If possible, express 120o 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 120o is a multiple of a special angle. It does not require reducing fractions.

We notice that 120o = 2(60o)

We know that . Therefore, we can immediately state that 2(60o) 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(60o) = 240o.

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(45o) = 135o.

Example 14:

All quadrantal angles are integer multiples of 90o!

a. 388o

4(90o) + 28o , QI angle

b. 450o

c. 720o

d. 390o

4(90o) + 30o, QIII angle

e. 540o

f. 225o

2(90o) 45o, 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 90o!

a. /8

22.5o, QI angle

b. 5/2

c. 3/2

d. 6/5

216o, QIII angle

e.

Example 16:

Which angles are integer multiples of special angles?

Special angles are 30o, 45o, and 60o.

a. 388o

not a multiple of special angle since neither 30, nor 45, nor 60 divides into angle evenly

b. 330o

11(30o), multiple of special angle

c. 225o

5(45o), multiple of special angle

d. 119o

not a multiple of special angle since neither 30, nor 45, nor 60 divides into angle evenly

e. 120o

2(60o), multiple of special angle

f. 150o

5(30o), 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 30o /6 and 45o /4 and 60o /3.

a. /8

22.5o, not a special angle

b. 5/3

5/3 560o, multiple of special angle

c. 3/4

3/4 345o, multiple of special angle

d. /5

36o, not a special angle

e. 7/6

7/6 730o, multiple of special angle

f. 4/9

80o, not a special angle

Example 18:

Find the angle measures from each graph.

A.

= 360o 60o = 300o

B.

Here we must first find the angle between the terminal side of angle and the negative x-axis. Given is 115o. We can change it to the sum of 90o and 25o. Therefore, tha angle between the terminal side of and the negative x-axis is 25o.

Then = 180o +25o = 155o

C.

We see that angle has two full rotations, which equals 360o(2). Remaining is a rotation of 180o.

Then = 360o(2) +180o = 900o

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 counter-clockwise. 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.      77o - Quadrant I angle b.     77o - Quadrant IV angle c.   125o  - Quadrant II angle d.   125o - Quadrant III angle e.   216o  - Quadrant III angle f.    216o - Quadrant II angle g.   330o - Quadrant IV angle i.   960o = 360o(2) + 240o     Quadrant III angle using the angle 240o     which is coterminal to the angle 960o h.   330o - Quadrant I angle   j.   376o = 360o(1) + (16o)       Quadrant IV angle using the angle 16o       which is coterminal to the angle 376o

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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 a.   /8  - Quadrant I angle       (between 0 and 1.57) b.  0.81 - Quadrant IV angle (between 0 and 1.57) c.    - Quadrant II angle       (between 1.57 and 3.14) d. 2.45 - Quadrant III angle (between 1.57 and 3.14) e.   4.6 - Quadrant III angle      (between 3.14 and 4.71) f.  - Quadrant II angle   (between 3.14 and 4.71