150 Lecture Notes - Section 6.1 Angle Measure

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1 c Marcia Drost, February, 008 Definition of Terms 50 Lecture Notes - Section 6. Angle Measure ray a line angle vertex two rays with a common endpoint the common endpoint initial side terminal side Standard Position initial side: positive x-axis positive angles: measured counter-clockwise negative angles: measured clockwise Measure of an Angle: amount of rotation from initial side to terminal side r = radius πr = circumference s = arc length radian measure of a central angle = arc length s radius r θ = s r one revolution = πr = π radians r revolution = π radians revolution = π 4 radians revolution = π 6 3 radians Acute angles measure between 0 and π. Obtuse angles measure between π and π.

2 c Marcia Drost, February, 008 Finding Coterminal Angles: two angles in standard position are coterminal if their sides coincide. i) 3π 6 π = 3π 6 π 6 = π 6 ii) 3π 4 π = 3π 4 8π 4 = 5π 4 iii) π 3 + π = π 3 + 6π 3 = 4π 3 Coterminal Angles To find angles that are coterminal with angle θ, add or subtract any integer multiple of π or 360 o. Example: Find a positive angle less than 360 O that is coterminal with the angle of measure 0 o in standard position. Complementary Angles two positive angles whose sum is π Supplementary Angles two positive angles whose sum is π Example: Find the complement of 5 π 5 π + α = π α = π π 5 α = 5π 0 4π 0 = π 0 Example: Find the supplement of π 5 π 5 + β = π β = π π 5 β = 5π 5 π 5 = 3π 5

3 c Marcia Drost, February, Relationship between Degrees and Radians 80 o = π radians ( ) o 80 = rad π o = π 80 rad A. To convert from degrees to radians: multiply degrees by π 80 o rad. B. To convert from radians to degrees: multiply radians by 80o π rad. 35 o = 35 o ( π rad 80 o ) = 3π 4 rad 540 o = 540 o ( π rad ) = 3π rad 80o 70 o = 70 o ( π rad 80 ) = 3π o rad π 4 rad = π 80o rad( 4 π rad ) = 45o 9π rad = 9π rad( 80o π rad ) = 80o Relationship between Degrees, Minutes and Seconds degree = 60 minutes minute = 60 seconds Convert from degrees/minutes/seconds to decimal degrees: θ = 3 o degrees, 0 minutes, 5 seconds θ = θ = θ = o

4 c Marcia Drost, February, Length of a Circular Arc S = θ * circumference of the circle, where θ is measured in radians π S = θ (πr) = rθ π θ r S Another useful formula is found by solving for θ. S = rθ S r = θ Arc Length and Angle Measure Find the length of an arc of a circle with radius m that subtends a central angle of 45 o. Area of a Circular Sector A = θ area of a circle π A = θ π πr = r θ Example: Find the area of a sector of a circle with a central angle of 80 o if the radius of the circle is 4 m. Example: A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 40 o. s = r θ where θ is measured in radians

5 c Marcia Drost, February, distance = rate time rate = speed = distance time speed = v = s t ω = θ t Linear speed Angular speed Relationship between linear and angular speed: If a point moves along a circle of radius r with angular speed ω, then its linear speed v is given by v = r ω Example: If the second hand on a clock is 8.5 centimeters long, find the speed of the tip of the second hand. The time for the second hand to make one full revolution is t = 60 seconds = minute The distance traveled by the tip of the second hand in one revolution is: s = π(radius) = π(8.5) = 7π centimeters. 7 π centimeters Therefore: the speed = 60 seconds s = centimeters per second Finding Linear Speed from Angular Speed A woman is riding a bicycle whose wheels are 4 inches in diameter. If the wheels rotate at 0 rpm, find the speed at which she is traveling in mph.