16.1 Radian Measure16.1 Radian MeasureFinally, what that other mode on our Finally, what that other mode on our calculator is all about!calculator is all about!

14.1 Angles geometry review and more!

OBJECTIVESOBJECTIVES

Students will understand the meaning of Students will understand the meaning of radian measure radian measure

Students will convert between degrees and Students will convert between degrees and radiansradians

Students will find function values for angles in Students will find function values for angles in radiansradians

Geometry ReviewGeometry Review

-angles can be named with three or one -angles can be named with three or one lettersletters

Complementary angles add up to 90.Complementary angles add up to 90.

Supplementary angles add up to 180.Supplementary angles add up to 180.

Angle 1 and 2 are supplementary. Angle 1 Angle 1 and 2 are supplementary. Angle 1 measures 2x + 6, Angle 2 measures 4x. Find measures 2x + 6, Angle 2 measures 4x. Find x and the measure of the angles.x and the measure of the angles.

The measure of an angle is determined by rotating a The measure of an angle is determined by rotating a ray starting at one side of the angle called the ray starting at one side of the angle called the initial initial side side to the other side called the to the other side called the terminal sideterminal side. .

A counterclockwise rotation generates a positive A counterclockwise rotation generates a positive measure, a clockwise rotation generates a negative measure, a clockwise rotation generates a negative measure.measure.

An angle is said to lie in the quadrant in which its An angle is said to lie in the quadrant in which its terminal side lies. (ex. An acute angle is in quadrant terminal side lies. (ex. An acute angle is in quadrant 1.)1.)

-We will be using the Greek letter theta -We will be using the Greek letter theta θθ to name to name angles.angles.

Calculating with Calculating with degrees, minutes and degrees, minutes and

secondsseconds One minute (1’) is equal to 1/60 degrees, or One minute (1’) is equal to 1/60 degrees, or

there are 60 minutes in one degreethere are 60 minutes in one degree

One second (1”) is equal to 1/60 minutes and One second (1”) is equal to 1/60 minutes and 1/3600 degrees. There are 60 seconds in a 1/3600 degrees. There are 60 seconds in a minute.minute.

12º42’38” represents 12 degrees 42 minutes 12º42’38” represents 12 degrees 42 minutes 38 seconds38 seconds

Ex. Add : 51º29’ + 32º46’Ex. Add : 51º29’ + 32º46’

Converting between decimal Converting between decimal degrees and Degree, Minutes and degrees and Degree, Minutes and

SecondsSeconds Convert 74º8’14” to decimal degrees to the Convert 74º8’14” to decimal degrees to the

nearest thousandth.nearest thousandth.

Or use your calculator!Or use your calculator!

137.74

0039.01333.0743600

14

60

874

Quadrantal AnglesQuadrantal Angles

Angles are in standard position if the vertex is Angles are in standard position if the vertex is at the origin and the initial side lies on the at the origin and the initial side lies on the positive x axis.positive x axis.

Angles in standard position whose terminal Angles in standard position whose terminal sides lie on the x or y axis are quadrantal sides lie on the x or y axis are quadrantal angles. Ex. 90, 180 or 270 degrees or any angles. Ex. 90, 180 or 270 degrees or any multiple of these.multiple of these.

Coterminal Angles.Coterminal Angles.

The measures of coterminal angles differ by The measures of coterminal angles differ by 360º360º

Coterminal angles have the same initial side Coterminal angles have the same initial side and terminal side but have different amounts and terminal side but have different amounts of rotation.of rotation.

Ex. 60º and 420º are coterminal angles.Ex. 60º and 420º are coterminal angles.

Can you name another pair of coterminal angles?Can you name another pair of coterminal angles?

Finding measures of Finding measures of coterminal angles.coterminal angles.

Find angles of least possible positive measure Find angles of least possible positive measure coterminal with each angle.coterminal with each angle.

A) 908ºA) 908º B.) -75ºB.) -75º C.) -800ºC.) -800º

188, 285, 280

What is a radian?What is a radian?

Converting between Converting between degrees and radiansdegrees and radians

To convert from DEGREES to RADIANS:To convert from DEGREES to RADIANS: Multiply a degree measure by π/180 radians and Multiply a degree measure by π/180 radians and

then simplify.then simplify.To convert from RADIANS to DEGREES:To convert from RADIANS to DEGREES:

Multiply a radian measure by 180˚/π and then Multiply a radian measure by 180˚/π and then simplify.simplify.

Example: 45˚ to radians…45(π/180) = 45π/180=π/4 Example: 45˚ to radians…45(π/180) = 45π/180=π/4 radians (Notice I left it in fraction form)radians (Notice I left it in fraction form)

Example: 9π/4 to Example: 9π/4 to degrees…9π/4(180/π)=9(180)/4=405˚degrees…9π/4(180/π)=9(180)/4=405˚

You tryYou try

Convert 108˚ to radiansConvert 108˚ to radians

3π/5 radians (leave in fraction!)3π/5 radians (leave in fraction!)

Convert 11π/12 to degreesConvert 11π/12 to degrees

165˚165˚

You can use the angle key on your calculator, you You can use the angle key on your calculator, you must hit the 2must hit the 2ndnd key then the angle key. key then the angle key.

We can have negative radian measures, just like We can have negative radian measures, just like we can have negative degree measures! A we can have negative degree measures! A counterclockwise rotation generates a positive counterclockwise rotation generates a positive measure, a clockwise rotation generates a measure, a clockwise rotation generates a negative measure.negative measure.

NOTE:NOTE:

If there is no unit that is specified in If there is no unit that is specified in a problem or a diagram, then it is a problem or a diagram, then it is understood that the angle is understood that the angle is measured in radians.measured in radians.

So, only work in degrees from now So, only work in degrees from now on on if the problem if the problem specifically says specifically says to.to.

Don’t forget to use your mode key!Don’t forget to use your mode key!