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Chapter 7

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

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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.

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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
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

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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) =
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:

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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.
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:

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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
Ex: It is roughly true that the earth revolves once every 24 hours on an axis through the north and south poles.

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