Chapter 6: Trigonometry 6.3: Angles and Radian Measure Essential Questions: How many degrees are in a circle? How many radians are in a circle?

6.3: Angles and Radian Measure In Geometry & Triangle Measurement, angles are formed by two rays that meet at an endpoint In trigonometry, angles are formed by taking an initial ray (called the initial side) and rotating it around itself, the point after rotation being called the terminal side. A trigonometry angle can rotate around itself multiple times terminal Initial

6.3: Angles and Radian Measure An angle in the coordinate plane is said to be in standard position if its vertex is at the origin (0,0) and its initial side is on the positive x-axis (going to the right) Angles formed by rotations that have the same initial and terminal sides are called coterminal. 0º and 360º angles are coterminal, because their ending points are the same as their starting points. Example 1: Coterminal angles Find three angles coterminal with an angle of 60º 60º + 360º = 420º 60º - 360º = -300º 60º + 2(360º) = 780º

6.3: Angles and Radian Measure Arc Length The length of the arc of a circle is equal to the central angle created: Arc length can be calculated by considering an arc as a fraction of the entire circle. Since there are 360º in a full circle, the arc is of the circle. Since the circumference is 2r, the length of the arc is arc 

6.3: Angles and Radian Measure Arc Length Arc length: Example 2: Finding an Angle Given an Arc Length An arc in a circle has an arc length l which is equal to the radius r. Find the measure of the central angle that the arc intercepts.

6.3: Angles and Radian Measure There are 360º in a circle The circumference of a circle = 2r. So if the radius of a circle were 1, then there a circle would contain 2 radians. This gives us our conversion factor: 360º = 2 radians Note: Dividing both sides by 2 gives: 180º =  radians 1 revolution around a circle = 2 radians 3/4 revolution: 3/4 * 2 = 3/2 radians 1/2 revolution: 1/2 * 2 =  radians 1/4 revolution: 1/4 * 2 = /2 radians

6.3: Angles and Radian Measure Radian Measure of Special Angles We’ve dealt with three special angles so far: 30°, 45°, & 60° 30° = 360/12, so 30° = 1/12 of a circle * 2 radians = /6 radians 45° = 360/8, so 45° = 1/8 of a circle * 2 radians = /4 radians 60° = 360/6, so 60° = 1/6 of a circle * 2 radians = /3 radians You won’t be required for homework, but it’s advisable you copy Figure 6.3-9 on page 437 to have reference to many special angles.

6.3: Angles and Radian Measure Converting Between Degree and Radians Use the conversion factor:  = 180° Convert the following radian measurements to degrees /5 /5 * 180°/ = 180/5 = 36° 4/9 4/9 * 180°/ = 720/9 = 80° 6 6 * 180°/ = 1080° Convert the following degree measurements to radians 75° 75° * /180° = 75/180 = 5/12 220° 220° * /180° = 220/180 = 11/9 400° 400° * /180° = 400/180 = 20/9

6.3: Angles and Radian Measure Assignment Page 441 1 - 45, odd problems

Chapter 6: Trigonometry 6.3: Angles and Radian Measure Day 2 Essential Questions: How many degrees are in a circle? How many radians are in a circle?

6.3: Angles and Radian Measure Arc Length An arc with central angle measure θ radians has length: l = r • θ The arc length is the radius times the radian measure of the central angle of the arc. Example 5: The second hand on a clock is 6 inches long. How far does the tip of the second hand move in 15 seconds? Answer The second hand makes a full revolution every 60 seconds, so 60 seconds = 2π radians. Use conversion factors. l = r • θ = (6 in)(π/2 rad) = 3π ≈ 9.4 inches

6.3: Angles and Radian Measure Central Angle Measure Example 6: Find the central angle measure (in radians) of an arc of length 5 cm on a circle with a radius of 3 cm. Answer: Use the formula l = r • θ l = 5cm, r = 3cm 5cm = 3cm • θ 5/3 = θ

6.3: Angles and Radian Measure Linear and Angular Speed Linear speed = arc length / time = Angular speed = angle / time = Linear/Angular speed are simply the rate at which it takes to move from one point on the circle to another. Linear speed calculates the time to move from the end of the radius; angular speed calculates the time to change along the angle. Note that the only difference between the two is the “r”, which means that linear speed totally depends on the distance the point is away from the center of the circle.

6.3: Angles and Radian Measure Linear and Angular Speed Linear speed = arc length / time = Angular speed = angle / time = Example 7: A merry-go-round makes 8 revolutions per minute What is the angular speed of the merry go round in radians per minute? If one revolution = 2π radians, then 8 revolutions = 8 • 2π = 16π radians How fast is a horse 12 feet from the center traveling? Use the linear speed formula r • angular speed = 12 feet • 16π = 192π feet/minute How fast is a horse 4 feet from the center traveling? r • angular speed = 4 feet • 16π = 64π feet/minute

6.3: Angles and Radian Measure Assignment Page 441 47 - 63, odd problems