Radian and Degree Measure

Section 4.1Radian and Degree MeasureRadian and Degree MeasureWe will begin our study of precalculus by focusing on the topic of trigonometry

Literal meaning of trigonometryThe measurement of triangles

Thus, we will be spending a lot of time working with and studying different trianglesRadian and Degree MeasureTo begin our study on trigonometry, we first start with angles and their measures

An angle is determined by rotating a ray (half-line) about its endpoint.Initial SideTerminal SideVertexRadian and Degree MeasureStandard PositionAn angle in standard position has 2 characteristics:Initial side lies on the x-axisVertex is at the originxyRadian and Degree MeasureStandard PositionAn angle in standard position has 2 characteristics:Initial side lies on the x-axisVertex is at the originxyRadian and Degree MeasureStandard PositionAn angle in standard position has 2 characteristics:Initial side lies on the x-axisVertex is at the originxyRadian and Degree MeasurePositive AnglesRotate clockwiseIn standard position, start by going up

Negative AnglesRotate counterclockwiseIn standard position, start by going downRadian and Degree MeasureNegative AngleRadian and Degree MeasurePositive AngleRadian and Degree MeasureAngles can be measured in one of two units:Degrees

Radians

One full revolution of a central angle would be equal to:360

2 radians (or 6.28 radians)Radian and Degree MeasureIn radians, there are common angles that will need to be memorized

= 180

= 90

= 270

Radian and Degree MeasureIn addition to our quadrant angles, there are 3 more angles that we will be using throughout the year.

= 30

= 45

= 60

Radian and Degree MeasureCoterminal AnglesTwo angles that have the same:VertexInitial SideTerminal SideAll angles have an infinite number of coterminal anglesRadian and Degree MeasureFinding Coterminal anglesTo find a positive coterminal angleAdd 2 (or 360) to the given angle

To find a negative coterminal angleSubtract 2 (or 360) from the given angleRadian and Degree MeasureGraph the following angle and determine two coterminal angles, one positive and one negative.

Radian and Degree MeasureGraph the following angles and find two coterminal angles, one positive and one negative.

Radian and Degree Measure

Radian and Degree Measure

Radian and Degree Measure

Radian and Degree Measure

Section 4.1Radian and Degree MeasureRadian and Degree MeasureGraph the following angles and find two coterminal angles, one positive and one negative.

Radian and Degree Measure

Radian and Degree MeasureYesterday we covered:Angles in degrees and radiansCoterminal angles

Today we are going to cover:Complementary and supplementary anglesConverting between degrees and radiansConverting minutes & seconds to degreesRadian and Degree MeasureComplementary AnglesTwo positive angles whose sum is (or 90)

Supplementary AnglesTwo positive angles whose sum is of (180)

Radian and Degree MeasureFind the complement and supplement to the following angle.

supplement:Radian and Degree MeasureFind the complement and supplement of the following angles:

Radian and Degree MeasureConversions between degrees and radians

To convert degrees to radians, multiply degrees by:

To convert radians to degrees, multiply radians by:

28Radian and Degree Measure

Degrees to RadiansRadians to DegreesRadian and Degree MeasureConvert the following from degrees to radians.

Radian and Degree MeasureConvert the following from radians to degrees.

Section 4.1Radian and Degree MeasureRadian and Degree MeasureFind the complement, supplement, and two coterminal angles of the following angle.

Convert the angle above to degrees.

Radian and Degree MeasureSo far in this section, we have:Graphed angles in both radians & degreesFound positive and negative coterminal anglesFound complementary and supplementary anglesConverted between radians and degrees

Today we are going to apply this to different word problems (arc length, linear speed, angular speed)Radian and Degree MeasureArc LengthThe distance along the circumference of a circle with a central angle of

Given by the formula: s = r

Where: s = the arc length r = the radius of the circle = the central angle in radians

Radian and Degree MeasureA circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240.

Convert the angle to radians.

Apply the formula.

S =

(4) =16.7 inchesRadian and Degree MeasureOn a circle with a radius of 9 inches, find the length of the arc intercepted by a central angle of 140.

S =

(9) = 22 inchesRadian and Degree MeasureLinear & Angular Speed

Linear speed measures how fast a particle is moving along the circular arc of a circle with radius r

Given by the formula:

Radian and Degree MeasureLinear & Angular Speed

Angular speed measures how fast the angle changes

Given by the formula:

Radian and Degree MeasureThe second hand of a clock is 11 inches long. Find the linear speed of the tip of this second hand as it passes around the clock face.

In one revolution, how far does the tip travel?s = 2 r = 22 inches

What is the time required to travel this distance?t = 60 seconds

r = 11 inchesRadian and Degree MeasureThe second hand of a clock is 11 inches long. Find the linear speed of the tip of this second hand as it passes around the clock face.

s = 22 inchest = 60 seconds

Linear Speed =

Radian and Degree MeasureA car is moving at a rate of 65 mph, and the diameters of its wheels is 2.5 feet.

Find the number of revolutions per minute the wheels are rotating.

Find the angular speed of the wheels in radians per minute.Radian and Degree MeasureA car is moving at a rate of 65 mph, and the diameters of its wheels is 2.5 feet.Find the number of revolutions per minute the wheels are rotating.Find the arc length for one revolution:S = r = (1.25) (2)= 2.5 feet per revolutionHow many feet per hour is the car traveling?(65 mph )(5,280 feet)= 343,200 feet/hour= 5,720 feet/min= 728.3 revolutionsRadian and Degree MeasureA car is moving at a rate of 65 mph, and the diameters of its wheels is 2.5 feet.b) Find the angular speed of the wheels in radians per minute.(728.3 revolutions)(2)= 4,576 radians

Angular Speed

Radian and Degree MeasureA car is moving at a rate of 35 mph, and the radius of its wheels is 2 feet.

Find the number of revolutions per minute the wheels are rotating.

Find the angular speed of the wheels in radians per minute.245.1 revolutions per minute

Radian and Degree Measure