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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
Page 1
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
Page 1
3.−50 −50 ∙𝜋
180¿−50𝜋
180¿−5𝜋
18
Page 1
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.
Page 1
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
Page 1
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
Page 1
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