Section 6.1 Radian Measure Chapter 6 Section 6.1 Radian Measure

Terminal Points and Radian Measure The Unit Circle The unit circle is a circle of radius 1 with it center at the origin. The equation of the unit circle is x2+y2=1. Any point on the unit circle will have the sum of the squares of its x and y coordinates equal to 1. 1 x y Terminal Points and Radian Measure A terminal point on the unit circle is a point on the unit circle that forms an angle with the positive x-axis. The distance you travel on the unit circle starting from the point (1,0) on the positive x-axis and ending at the terminal point (x0,y0) is the radian measure of the angle. (Remember the measure is positive if you move counterclockwise and negative if you move clockwise.) The radian angle measure we usually denote with the letter t. 1 The length of the red arc above is the radian measure of the angle in standard position with the point (x0,y0) on it terminal side.

The symbol  used in radian measure stands for the number   3 The symbol  used in radian measure stands for the number   3.1415926…. This number is irrational (i.e. its decimal expansion will never end or repeat). The reason that radian measure is used more often in mathematics, physics, engineering and other disciplines is that the angle measure is the length of arc (part of a circle) of a unit circle ( a circle of radius 1). Radian measure represents a physical distance. x y 1/8 circle 1/8 · 2 = /4 1 The part of the unit circle marked in red is called an arc. The length of this (if you straighten it out) is /4 or the measure of the angle in radians. This can be related to a number by the following calculation: For an arc that is part of a circle of radius r its length often we use the symbol (s) can be found by taking s=·r where  is the measure of the angle in radians. r  24= 2𝜋 3 ∙𝑟 𝑟= 36 𝜋 ≈11.4591 An arc of radian measure 2𝜋 3 is of length 24. What is the radius of the corresponding circle?

Radian Measure The radian measurement of an angle is based on the radian measure of an entire circle being 2 and any fraction of the circle will be proportional (i.e. will form the same fraction). The examples below show the measure of angles in standard position. x y ½ circle ½ · 2 =  ¼ circle ¼ · 2 = /2 1/8 circle 1/8 · 2 = /4 5/8 circle 5/8 · 2 = 5/4 x y 1 circle (neg) 1 · -2 = -2 3/8 circle (neg) 3/8 · -2 = -3/4 1/12 circle (neg) 1/12 · -2 = -/6 ¾ circle (neg) ¾ · -2 = -3/2

1 The pictures above illustrate different angles on the unit circle along with their radian measure which is also the length of the red arc. What are the measures of the last 4 angles? Points and Angles There are some angles on the unit circle for which we know the coordinates of the terminal point. These come from realizing the triangle formed by the terminal point the point perpendicular on the x-axis and the origin is either a 30°-60°-90° triangle or a 45°-45°-90° triangle. In radians we would say they are: 1 or

Fraction of circle 1/12 1/8 1/6 1/4 1/3 3/8 5/12 1/2 3/4 Converting Angle Measure Both radians and degrees are based on a fraction of a circle that you are considering. Fraction of circle 1/12 1/8 1/6 1/4 1/3 3/8 5/12 1/2 3/4 Degree Measure 30 45 60 90 120 135 150 180 270 Radian Measure /6 /4 /3 /2 2/3 3/4 5/6  3/2 To convert back and forth from degrees to radians we use the proportion below where  is the measure in degrees and  is the measure in radians. 0 and 0 30 and /6 45 and /4 60 and /3 90 and /2 120 and 2/3 135 and 3/4 150 and 5/6 180 and  270 and 3/2