{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter 7 - Chapter7: RadianMeasure(7.1 Definition:...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Chapter 7: Trigonometric Functions of Real Numbers: Radian Measure (7.1) Definition: Place the vertex of the angel at the center of a circle with radius r as shown below. Let s denote the length of the arc intercepted by the angel θ . The radian measure of the angel θ = s r = arc length of s radius r Ex: Find θ for s = 200cm, r = 1m Note: 1) Radian measure is a “dimensionless” quantity. It is the ratio of lengths with same units.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
2) Radian measure of angel θ is independent of the size of the circle in which it is placed. See below: 3) For a full circle ( θ =360 0 ) The conversion formula between radian and degree can be derived with the information above. 1 radian = 180 0 π ( to convert angel to degree measure multiply by this factor) Similarly 1 0 = π 180 0 ( to convert angel to radian measure multiply by this factor) θ = s r = 2 π r r = 2 π θ = 360 0 in degree = 2 π in radian π in radian = 360 2 = 180 0
Background image of page 2
Here is a table with the Unit Circle special angels expressed in both radian and degree measure. Ex: Express 5 0 in radian measure Ex: Express 11 π 6 radian to degree measure
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Ex: Compute the value of the following trigonometric functions: cos( π 3 ) = sin( π 3 ) = cos( 2 π 3 ) = sin( 2 π 3 ) tan( 7 π 4 ) cot( 7 π 4 ) cos4 = sin 4 = cos( 1) = sin( 1) =
Background image of page 4
Radian Measure and Geometry (7.2) Remember: In a circle of radius r, the arc length s determined by a central angel θ is θ = s r (s and r have the same units) (see picture below) s = r θ The formula to find the arc length of a circular sector with radius r and intercepted angel θ The area of a sector with central angel θ is A = 1 2 r 2 θ Proof:
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Ex: Given r=6cm, θ = 2 π 3 Find s and A Ex: Given r = 4 inches, θ =75 0 Find s and A. Ex: In a circle of radius 3 m, the area of a certain sector is 20 m 2 . Find the degree measure of the central angel.
Background image of page 6
Definition: Angular speed and Linear speed Suppose a wheel rotates about its axis at a constant rate . If a radial line turns through an angel θ in time t (see picture) the angular speed ω = θ t = angle travelled time taken Suppose a wheel turns about its axis at a constant rate. It the point P on the rotating wheel turns a distance s in time t, the linear speed of P, v = s t = arc length travelled time taken Note: In both definitions, θ will be in radian measure. Claim: v = r ω Linear speed=(radius)(angular speed) Proof:
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Ex: A circular gear in a motor rotates at the rate of 100 rpm (revolutions per minute). a) What is the angular speed of the gear, in radians per minute? b) Find the linear speed of a point on the gear 4 cm from the center. Ex: The rotation of a wheel is 1080 0 /sec, r = 25 cm. a) Find the angular speed in radian/sec b) Find the linear speed in cm/sec of a point on the circumference of the wheel
Background image of page 8
Ex: It is roughly true that the earth revolves once every 24 hours on an axis through the north and south poles.
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}