Central Angles and Arcs

Transcription

1 Advance Algebra & Trigonometry Angles & Circular Functions Central Angles and Arcs Arc Length 1. A central angle an angle whose vertex lies at the center of the circle: A B ABC is a central angle C 2. An arc is part of a circle, we say that an arc subtends its central angle. Hence in the diagram above, AC subtends ABC. 3. The length of an arc can be found by using the formula, s= rθ, where: s is the length of the arc. r is the radius of the circle. θ is the radian measure of the central angle. 4. Examples 1) Find the length of an arc that subtends a central angle of 5π 6 in a circle with radius of 12 cm. s = rθ = 12 5π 6 = 10π cm

2 2) Find the length of an arc that subtends a central angle of 120 in a circle with radius of 6 m. First, convert 120 to radian measure. π θ = = 2π 3 Then calculate the arclength. s = rθ = 6 2π = 4π m 3 5. Here are some examples for you to try: Find the length of the arc given the following central angles and radii. 1. θ = 3π 4 with a radius of 5 cm 2. θ = 11π 6 with a radius 14 inches 3. θ = 114 with a radius of 10 inches 4. θ = 50 with a radius of 7 cm

3 Linear and Angular Velocities 1. Most of us are accustomed to thinking in terms of linear measurement. It is possible however to relate measures of linear motion or velocity and measures of circular or angular motion or velocity. If an object is moving in a circular motion, we can use the formula, V = r θ t, where V is the linear velocity, for example 15 cm sec. r is the radius of the circular motion. θ t is the angular velocity in radians per unit of time, for example 24π rad min. 2. Examples 1) A pulley connected to a drive shaft with a radius of 10 cm turns at 12 revolutions per second. What is the linear velocity of the belt driving the pulley? Note: One revolution is 2π radians, hence 12 revolutions per second is 12 2π or 24π radians per second. V = r θ t = 10 cm 24π rad 1 sec = 240π cm sec cm sec 2. A trigonometry student notices that the waterwheel at an old iron works spins at a rate of 160 per second. The radius of the wheel is 10 feet. What is the equivalent linear velocity for this wheel in feet per second? 160 /sec is equivalent to: 160 π 180Þ = 8π 9 radians/sec V = r θ t = 10 ft 8π 9 ra d sec = 80π 9 ft sec ft sec

4 Area of a Circular Sector 1. A sector of a circle is a region bounded by a central angle and the intercepted arc. B A ABC is a sector of circle B C 2. We can find the area of a circular sector using the formula, A = 1 2 r 2 θ, where A is the area of the sector r is the radius of the circle θ is the measure of the central angle in radians 3. Examples 1) A sector has an arc length of 44 cm and a central angle measuring 2.25 radians. Find the radius and the area of the sector. First find the radius using s = rθ : 44 = r r 2.25 = r r = cm To find the area use: A = 1 2 r 2 θ A = 1 2 (19.56) A = cm 2

5 2) Find the area of a sector if the central angle θ = 7π 12 is 10 feet. First, note that the radius is 10 or 5 ft. 2 and the diameter of the circle A = 1 2 r 2 θ A = 1 2 (5)2 7π 12 A = 175π 24 A = ft 2 3. Here are some examples for you to try. 1. Find the area of a sector with a central angle = 3π 8 with a radius of 24 feet. 2. Find the area of a sector with a central angle = 58 and a radius of 6 inches. Summary 1. Find the length of an arc given the radian measure of the central angle. s= rθ 2. Find linear and angular velocities: v = r θ t 3. Find the area of a sector: A = 1 2 r 2 θ