Unit: More Trig Functions

Converting between Degrees and Radians:

Radian: the measure of an angle that, when drawn as a central angle of a circle, would intercept an arc whose length is equal to the length of a radius of the circle.

measure.radian in anglean of examplean is 3

Lets determine how many degrees are in 1 radian…

How many degrees are there once around a circle?

360

In radians, once around the circle is

360 °=2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠360 °

2=

2𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠2

180 °=𝜋𝑟𝑎𝑑𝑖𝑎𝑛𝑠180 °𝜋

=𝜋 𝑟𝑎𝑑𝑖𝑎𝑛𝑠

π

180 °𝜋

=1𝑟𝑎𝑑𝑖𝑎𝑛 OR

It may be necessary to convert from radian measure to degree measure.

radians toDegrees 2. degrees toRadians 1.

know. toneed that wesconversion twoare There

Radians to degrees:

.180

by anglegiven the

multiply wedegrees, toradians convertingWhen

Change each angle from radian measure to degree measure.

2

3 .11

180

2

3 .expression hesimplify t

and thecancelcan weso thisdo We

2

1803

2

540 270 270 radians

2

3

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13.5𝜋4

5𝜋4∙180𝜋¿

5 ∙1804

¿900

4¿225

15.5𝜋6

5𝜋6∙180𝜋¿

5 ∙1806

¿900

6¿150

17.𝜋5

𝜋5∙

180𝜋¿

1805 ¿36

19.−𝜋6−

𝜋6∙180𝜋¿−

1806 ¿−30

Degrees to radians:

.180

by angle given the

multiply weradians, todegrees converting When

Change each angle from degree measure to radian measure.

1. 120 120 ∙𝜋

180¿

120𝜋180

We need to simplify this fraction by using the calculator.

¿2𝜋3

Notice that we keep in the final answer

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3.−50 −50 ∙𝜋

180¿−50𝜋

180¿−5𝜋

18

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5.−135−135 ∙𝜋

180¿−135𝜋

180¿−3𝜋

4

7. 330 330 ∙𝜋

180¿

330𝜋180

¿11𝜋

6

9.−45−45 ∙𝜋

180¿−45𝜋

180¿−𝜋

4

Finding the measure of an arc on a circle.

𝑠=𝑟 𝜃This formula is used when trying to find one of three things:

To use this formula, the measure of the central angle MUST BE IN RADIANS.

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21. Find the length of the radius of a circle in which a central angle of 4.5 radians intercepts an arc of 9 meters.

𝑠=𝑟 𝜃 Is the central angle in radians?

9=𝑟 ( 4.5 )9

4.5=𝑟 ( 4.5 )

4.5

2=𝑟

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21. What is the measure of an angle formed by the hands of a clock at 5:00?

a. degrees?b. radians?

2560

=𝑥

360

6 0𝑥=9000

6 0 𝑥60

=900060

𝑥=150

2560

=𝑥

2𝜋

6 0𝑥=50𝜋

6 0 𝑥60

=50𝜋60

𝑥=5𝜋6

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27. Circle O has a radius of 10 inches. What is the length, in inches, of the arc subtended by a central angle measuring 2.5 radians?

1 0𝜃=2.5

𝑠𝑢𝑏𝑡𝑒𝑛𝑑

𝑒𝑑𝑎𝑟𝑐

𝑠=𝑟 𝜃

𝑠=(10 ) (2.5 )𝑠=25

21. Find the radius of a circle on which a central angle measuring 135 intercepts an arc on the circle with a length of 24 yds. [Answer may be expressed in terms of ]

Page 2

𝑠=𝑟 𝜃

24=𝑟 ∙135

24=𝑟 ∙135 ∙𝜋

180

24=𝑟3𝜋4

43𝜋

∙24=𝑟3𝜋4∙

43𝜋

32𝜋

=𝑟

Homework

Page 1#2-20 even, 26,28,29,30