Chapter 2 Trigonometric Functions of Real Numbers Section 2.1 The Unit Circle y The Unit Circle x2 y 2 1 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. x 1 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. t x0 , y0 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. t 2 t 1 1 t 3 2 t t 4 1 1 1 t 2 3 4 1 5 t 4 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 t 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: / / 6 3 2 or / 4 4 2 2 ; (0,1) t 1 3 ; , 3 2 2 t 2 2 ; , 4 2 2 t / 3 1 ; , 6 2 2 t 0; (1,0) 0 1 Reference Number A reference number is similar to a reference angle. The reference number for a given number is the shortest distance you would need to travel to get to the x-axis on either the positive or negative side. The reference numbers can be used to calculate the coordinates of various points on the unit circle. The reference number for t is usually denoted t y t 3 P t y 2 3 t 5 4 x x t 4 P The reference number for: t is the number: t 3 The point P has coordinates given by: 2 3 The reference number for: t is the number: 1 3 , 2 2 t 5 4 4 The point P has coordinates given by: 2 2 2 , 2 Example 1 Example 2 Find the point on the unit circle whose radian measure is 5 . A point in the third quadrant has an x-coordinate of -⅓. Find its ycoordinate. 3 I begin by drawing a circle. The reference angle is: 3 The triangle that is made is : The coordinates of P are: y / P : , y / 6 3 2 1 3 , 2 2 Take the negative root because y is negative in the third quadrant. 1 9 y2 1 y2 y y 5 t 3 x 1 3 t 3 P 31 2 y 2 1 1 3 P x 8 9 8 3