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Trigonometric Functions Using the Unit Circle Video Lecture Section 6.2 Course Learning Objectives: Demonstrate an understanding of trigonometric functions and their applications. Weekly Learning Objectives: 1) Find the coordinates of a terminal point. 2) Find a reference number for an angle. 3) Find the exact values of the trigonometric functions using a point on the unit circle. 4) Find the exact values of the trigonometric functions of quadrantal angles. 5) Find the exact values of the trigonometric functions of π/4 = 45⁰. 6) Find the exact values of the trigonometric functions of π/6 = 30⁰. 7) Find the exact values of the trigonometric functions of π/3 = 60⁰. 8) Find the exact values of the trigonometric functions for integer multiples of π/4 = 45⁰, π/6 = 30⁰, and π/3 = 60⁰. 9) Use a calculator to approximate the value of a trigonometric function. Trigonometric Functions Using the Unit Circle Consider a circle, radius ", center at the origin. Equation of circle: Consider a mapping where we mark off a distance, >, around the unit circle starting at the point Ð"ß !Ñ and move counterclockwise if > ! and clockwise if > !Þ The point T Bß C that we arrive at is called the terminal point. What is the terminal point of each of the following real numbers? 1 a) > œ # $1 # b) >œ c) >œ1 d) > œ '1 page 1 Find the coordinates of the terminal point if > œ 1 . % Use symmetry of the unit circle to find the terminal point for each of the following. &1 a) > œ % b) >œ (1 % c) >œ 31 4 page 2 Find the coordinates of the terminal point if > œ 1 Þ ' Use symmetry of the unit circle to find the terminal point for each of the following. (1 a) > œ ' b) >œ ""1 ' c) >œ (1 ' page 3 Use symmetry across the line C œ B and the terminal point for 1 1 >œ to find the coordinates of the terminal point for > œ Þ ' $ Use symmetry of the unit circle to find the terminal point for each of the following. (1 a) > œ $ b) >œ ""1 $ c) >œ #1 $ As you can see from the above examples the terminal point of any number relates back to the "corresponding" terminal point in the first quadrant. page 4 Definition: Let > be a real number. The reference number > associated with > is the shortest distance along the unit circle between the terminal point determined by > and the B-+B3=Þ > measures the angle between > and the closest part of the B axis. It should be between 0 and 1# Þ Find the reference numbers for each of the following: &1 a) > œ ' &1 % b) >œ c) >œ d) > œ # &1 $ To find the terminal point for any number >ß Ð1Ñ Find the reference number >Þ Ð2Ñ Find the terminal point U+ß , determined by >. Ð3Ñ The terminal point determined by > is T „ +ß „ , where the signs are chosen according to the quadrant in which the terminal point lies. page 5 It is essential that you memorize the coordinates of the special values of > given in the following table: > Terminal point determined by > ! "ß ! È$ " 1 # ß # ' 1 % 1 $ 1 # È# È# # ß # " È$ #ß # !ß " Find the coordinates for each of the following using reference numbers. a) >œ (1 $ b) >œ "$1 ' c) >œ $"1 $ d) >œ ($1 $ page 6 We can now define the six trigonometric functions: Definition of the 6 trigonometric functions: Let > be any real number and let T ÐBß CÑ be the terminal point on the unit circle determined by >Þ sin > œ C csc > œ cos > œ B " ÐC Á !Ñ C sec > œ tan > œ " ÐB Á !Ñ B C ÐB Á !Ñ B cot > œ B ÐC Á !Ñ C Find the value of each of the trigonometric functions at a) >œ &1 ' b) >œ 1 $ c) >œ 1 # page 7 The following table shows some special values of the 6 trigonometric functions. These values need to be memorized. > ! sin t ! 1 ' 1 % 1 $ 1 # " # È# # È$ # " cos > " È$ # È# # " # ! tan > ! È$ $ " È$ — csc > — # sec > " #È $ È# È# #È $ $ " # # — cot > — È$ " È$ $ ! If T ÐBß CÑ is the terminal point for any number >, then ) is the angle in standard position measured in radians whose terminal side is the ray from the origin through T . Since < œ " on a unit circle, and = œ <) with the arc length = œ >, then > œ " † ) or > œ )Þ Therefore, sin ) œ sin > csc ) œ csc > cos ) œ cos > sec ) œ sec > tan ) œ tan > cot ) œ cot > Since the values of the trig functions of an angle ) are determined by the coordinates of the terminal point T ÐBß CÑ, the units on ) are irrelevant. ) can be in degrees or radians. page 8 Find the exact value of each expression: cosÐ &%1 Ñ sin "$&° tan $"&° cscÐ '!°Ñ cotÐ &$1 Ñ sec #"!° cosÐ%)°Ñ cscÐ#"°Ñ tanÐ &"#1 Ñ cos Ð"Þ'#Ñ cot Ð"%Þ#Ñ secÐ)Þ"%Ñ page 9