ࡱ>  + Xbjbj ^ܪDhܪDhPl: : 8 /Btn4n4n4n48j=$?AAAAAAA$CYFA?7|8??An4n44AJAJAJA?Fn4n4AJA?AJAJAJAn4`-4:@FJAAA0/BJAF@FJAFJAP??JA?????AAJA???/B????F?????????: X : LESSON 4 COTERMINAL ANGLES Topics in this lesson: 1.  HYPERLINK \l "Topic1" THE DEFINITION AND EXAMPLES OF COTERMINAL ANGLES 2.  HYPERLINK \l "Topic2" FINDING COTERMINAL ANGLES 3.  HYPERLINK \l "Topic3" TRIGONOMETRIC FUNCTIONS OF COTERMINAL ANGLES 1. THE DEFINITION AND EXAMPLES OF COTERMINAL ANGLES Definition Two angles are said to be coterminal if their terminal sides are the same. Examples Here are some examples of coterminal angles. 1. The two angles of  EMBED Equation.3  and  EMBED Equation.3  are coterminal angles.   EMBED Equation.3   EMBED Equation.3  HYPERLINK "http://www.math.utoledo.edu/~janders/1330/Lectures/Lesson4/CotAng140andNeg220Deg.gif"Animation of the making of these two coterminal angles. 2. The two angles of  EMBED Equation.3  and  EMBED Equation.3  are coterminal angles.   EMBED Equation.3   EMBED Equation.3  HYPERLINK "http://www.math.utoledo.edu/~janders/1330/Lectures/Lesson4/CotAng7Pi6and31Pi6.gif"Animation of the making of these two coterminal angles. 3. The two angles of  EMBED Equation.3  and  EMBED Equation.3  are coterminal angles.   EMBED Equation.3   EMBED Equation.3  HYPERLINK "http://www.math.utoledo.edu/~janders/1330/Lectures/Lesson4/CotAngNeg1845and1395Deg.gif"Animation of the making of these two coterminal angles.  HYPERLINK \l "TopicsList" Back to Topics List 2. FINDING COTERMINAL ANGLES Theorem The difference between two coterminal angles is a multiple (positive or negative) of  EMBED Equation.3  or  EMBED Equation.3 . Examples Find three positive and three negative angles that are coterminal with the following angles. 1.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  NOTE:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  NOTE:  EMBED Equation.3   EMBED Equation.3  NOTE:  EMBED Equation.3   EMBED Equation.3  NOTE:  EMBED Equation.3  NOTE: There are other answers for this problem. If the difference between the angle of  EMBED Equation.3  and another angle is a multiple of  EMBED Equation.3 , then this second angle is an answer to problem. 2.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  NOTE:  EMBED Equation.3   EMBED Equation.3  NOTE:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  NOTE:  EMBED Equation.3   EMBED Equation.3  NOTE:  EMBED Equation.3  NOTE: There are also other answers for this problem. If the difference between the angle of  EMBED Equation.3  and another angle is a multiple of  EMBED Equation.3 , then this second angle is an answer to problem. Lets see how the angle  EMBED Equation.3  is made. We will need the following property of arithmetic, which comes from the check for long division.  EMBED Equation.3  Dividing both sides of the equation  EMBED Equation.3  by b, we obtain the equation  EMBED Equation.3 . For the work that we will do in order to find one particular coterminal angle of a given angle in radians, we will want the number q above to be an even number. Now, consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3  given above.  EMBED Equation.3  Thus,  EMBED Equation.3 . The number 17 is an odd number. However, we may write 17 as  EMBED Equation.3 , where the number 16 is an even number. Thus, we have that  EMBED Equation.3  Thus,  EMBED Equation.3 . Now, multiply both sides of this equation by  EMBED Equation.3  to obtain that  EMBED Equation.3  This equation tells us how to make the angle  EMBED Equation.3 . In order to make the angle  EMBED Equation.3 , it will take eight complete revolutions and an additional rotation of  EMBED Equation.3  going in the counterclockwise direction in order to make the angle  EMBED Equation.3 . Thus, the two angles  EMBED Equation.3  and  EMBED Equation.3  are coterminal angles. Notice that  EMBED Equation.3 . HYPERLINK "http://www.math.utoledo.edu/~janders/1330/Lectures/Lesson4/CotAng87Pi5and7Pi5.gif"Animation of the making of these two coterminal angles. Another HYPERLINK "http://www.math.utoledo.edu/~janders/1330/Lectures/Lesson4/CotAng87Pi5and7Pi5Delay.gif"animation of the making of these two coterminal angles. Examples Find the angle between 0 and  EMBED Equation.3  or the angle between  EMBED Equation.3  and 0 that is coterminal with the following angles. 1.  EMBED Equation.3  Consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3 .  EMBED Equation.3  Thus,  EMBED Equation.3 . The 14 is an even number. So, multiply both sides of this equation by  EMBED Equation.3  to obtain that  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take seven complete revolutions and an additional rotation of  EMBED Equation.3  going in the counterclockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. HYPERLINK "http://www.math.utoledo.edu/~janders/1330/Lectures/Lesson4/CotAng85Pi6andPi6.gif"Animation of the making of these two coterminal angles. Another HYPERLINK "http://www.math.utoledo.edu/~janders/1330/Lectures/Lesson4/CotAng85Pi6andPi6Delay.gif"animation of the making of these two coterminal angles. Answer:  EMBED Equation.3  2.  EMBED Equation.3  Consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3 .  EMBED Equation.3  Thus,  EMBED Equation.3 . The 22 is an even number. Now, multiply both sides of this equation by  EMBED Equation.3  to obtain that  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take eleven complete revolutions and an additional rotation of  EMBED Equation.3  going in the clockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. HYPERLINK "http://www.math.utoledo.edu/~janders/1330/Lectures/Lesson4/CotAngNeg91Pi4andNeg3Pi4Delay.gif"Animation of the making of these two coterminal angles. Answer:  EMBED Equation.3  3.  EMBED Equation.3  Consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3 .  EMBED Equation.3  Thus,  EMBED Equation.3 . The 37 is an odd number. So, we may write 37 as  EMBED Equation.3 , where the number 36 is an even number. Thus, we have that  EMBED Equation.3  Thus,  EMBED Equation.3 . Now, multiply both sides of this equation by  EMBED Equation.3  to obtain that  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take eighteen complete revolutions and an additional rotation of  EMBED Equation.3  going in the clockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. Answer:  EMBED Equation.3  4.  EMBED Equation.3  Consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3 .  EMBED Equation.3  Thus,  EMBED Equation.3 . The 29 is an odd number. So, we may write 29 as  EMBED Equation.3 , where the number 28 is an even number. Thus, we have that  EMBED Equation.3  Thus,  EMBED Equation.3 . Now, multiply both sides of this equation by  EMBED Equation.3  to obtain that  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take fourteen complete revolutions and an additional rotation of  EMBED Equation.3  going in the counterclockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. HYPERLINK "http://www.math.utoledo.edu/~janders/1330/Lectures/Lesson4/CotAng59Pi2and3Pi2Delay.gif"Animation of the making of these two coterminal angles. Answer:  EMBED Equation.3  5.  EMBED Equation.3  Consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3 .  EMBED Equation.3  Thus,  EMBED Equation.3 . The 8 is an even number. Now, multiply both sides of this equation by  EMBED Equation.3  to obtain that  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take four complete revolutions and an additional rotation of  EMBED Equation.3  going in the clockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. HYPERLINK "http://www.math.utoledo.edu/~janders/1330/Lectures/Lesson4/CotAngNeg17Pi2andNegPi2Delay.gif"Animation of the making of these two coterminal angles. Answer:  EMBED Equation.3  6.  EMBED Equation.3  The number 82 is an even number. Thus,  EMBED Equation.3 . In order to make the angle  EMBED Equation.3 , it will take forty-one complete revolutions. Thus, the angle of 0 is the coterminal angle that we are looking for. Answer: 0 7.  EMBED Equation.3  The 21 is an odd number. So, we may write 21 as  EMBED Equation.3 , where the number 20 is an even number. Now, multiply both sides of this equation by  EMBED Equation.3  to obtain that  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take ten complete revolutions and an additional rotation of  EMBED Equation.3  going in the clockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. Answer:  EMBED Equation.3  8.  EMBED Equation.3  Use your calculator to find the whole number of times that 360 will divide into 4110 . Since  EMBED Equation.3  and 360 times 11 equals 3960, then we have that  EMBED Equation.3  Thus,  EMBED Equation.3 . When working in degrees, the whole number of 11, in this case, does not have to be even. Thus,  EMBED Equation.3   EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take eleven complete revolutions and an additional rotation of  EMBED Equation.3  going in the counterclockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. Answer:  EMBED Equation.3  9.  EMBED Equation.3  Use your calculator to find the whole number of times that 360 will divide into 8865 . Since  EMBED Equation.3  and 360 times 24 equals 8640, then we have that  EMBED Equation.3  Thus,  EMBED Equation.3 . Multiplying both sides of this equation by  EMBED Equation.3 , we obtain that  EMBED Equation.3 . Thus,  EMBED Equation.3   EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take twenty-four complete revolutions and an additional rotation of  EMBED Equation.3  going in the clockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. Answer:  EMBED Equation.3   HYPERLINK \l "TopicsList" Back to Topics List 3. TRIGONOMETRIC FUNCTIONS OF COTERMINAL ANGLES Theorem Let  EMBED Equation.3  and  EMBED Equation.3  be two coterminal angles. Then 1.  EMBED Equation.3  4.  EMBED Equation.3  2.  EMBED Equation.3  5.  EMBED Equation.3  3.  EMBED Equation.3  6.  EMBED Equation.3  We can use this theorem to find any one of the six trigonometric functions of an angle that is numerically bigger than  EMBED Equation.3  or  EMBED Equation.3 . When given an angle that is numerically bigger than  EMBED Equation.3  or  EMBED Equation.3 , we will want to find the angle that is coterminal to it and is numerically smaller than  EMBED Equation.3  or  EMBED Equation.3 . Examples Use a coterminal angle to find the exact value of the six trigonometric functions of the following angles. 1.  EMBED Equation.3  Doing one subtraction of  EMBED Equation.3 , we obtain that  EMBED Equation.3  =  EMBED Equation.3  Thus, the two angles of  EMBED Equation.3  and  EMBED Equation.3  are coterminal. The angle  EMBED Equation.3  is in the IV quadrant and has a reference angle of  EMBED Equation.3 . Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  2.  EMBED Equation.3  Doing one addition of  EMBED Equation.3 , we obtain that  EMBED Equation.3  =  EMBED Equation.3  Doing a second addition of  EMBED Equation.3 , we obtain that  EMBED Equation.3  =  EMBED Equation.3  Thus, the two angles of  EMBED Equation.3  and  EMBED Equation.3  are coterminal. The angle  EMBED Equation.3  is in the III quadrant and has a reference angle of  EMBED Equation.3 . Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  3.  EMBED Equation.3  Consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3 .  EMBED Equation.3  Thus,  EMBED Equation.3 . This implies that  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take twelve complete revolutions and an additional rotation of  EMBED Equation.3  going in the counterclockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. The angle  EMBED Equation.3  is in the II quadrant and has a reference angle of  EMBED Equation.3 . Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  4.  EMBED Equation.3  Consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3 .  EMBED Equation.3  Thus,  EMBED Equation.3 . This implies that  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take nine complete revolutions and an additional rotation of  EMBED Equation.3  going in the counterclockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. The angle  EMBED Equation.3  is in the III quadrant and has a reference angle of  EMBED Equation.3 . Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  5.  EMBED Equation.3  Consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3 .  EMBED Equation.3  Thus,  EMBED Equation.3 . This implies that  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take twenty-one complete revolutions and an additional rotation of  EMBED Equation.3  going in the clockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. The angle  EMBED Equation.3  is in the IV quadrant and has a reference angle of  EMBED Equation.3 . Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  6.  EMBED Equation.3  Consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3 .  EMBED Equation.3  Thus,  EMBED Equation.3 . This implies that  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take forty-three complete revolutions and an additional rotation of  EMBED Equation.3  going in the clockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. The angle  EMBED Equation.3  is in the I quadrant and has a reference angle of  EMBED Equation.3 . Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  7.  EMBED Equation.3  Consider the fraction of  EMBED Equation.3  in the angle  EMBED Equation.3 . Since  EMBED Equation.3 , then  EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take four complete revolutions and an additional rotation of  EMBED Equation.3  going in the clockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. The angle  EMBED Equation.3  lies on the negative y-axis. Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3  = undefined  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  = undefined  EMBED Equation.3  8.  EMBED Equation.3  Since  EMBED Equation.3 , then in order to make the angle  EMBED Equation.3 , it will take thirty-two complete revolutions and an additional rotation of  EMBED Equation.3  going in the counterclockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. The angle  EMBED Equation.3  lies on the negative x-axis. Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  = undefined  EMBED Equation.3   EMBED Equation.3  = undefined 9.  EMBED Equation.3  Since  EMBED Equation.3 , then in order to make the angle  EMBED Equation.3 , it will take fifteen complete revolutions going in the clockwise direction. Thus, the angle of 0 is the coterminal angle that we are looking for. The angle 0 lies on the positive x-axis. Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  = undefined  EMBED Equation.3   EMBED Equation.3  = undefined 10.  EMBED Equation.3  Use your calculator to find the whole number of times that 360 will divide into 9840 . Since  EMBED Equation.3  and 360 times 27 equals 9720, then we have that  EMBED Equation.3  Thus,  EMBED Equation.3 . Thus,  EMBED Equation.3   EMBED Equation.3  In order to make the angle  EMBED Equation.3 , it will take twenty-seven complete revolutions and an additional rotation of  EMBED Equation.3  going in the counterclockwise direction. Thus, the angle of  EMBED Equation.3  is the coterminal angle that we are looking for. The angle  EMBED Equation.3  is in the II quadrant and has a reference angle of  EMBED Equation.3 . Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  11.  EMBED Equation.3  Subtracting two times  EMBED Equation.3 , we obtain that  EMBED Equation.3  =  EMBED Equation.3 . Thus, the two angles of  EMBED Equation.3  and  EMBED Equation.3  are coterminal. The angle  EMBED Equation.3  is on the negative x-axis. Thus, using Unit Circle Trigonometry, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  undefined  EMBED Equation.3   EMBED Equation.3  undefined 12.  EMBED Equation.3  Adding three times  EMBED Equation.3 , we obtain that  EMBED Equation.3  =  EMBED Equation.3 . Thus, the two angles of  EMBED Equation.3  and  EMBED Equation.3  are coterminal. The angle  EMBED Equation.3  is in the II quadrant and has a reference angle of  EMBED Equation.3 . Thus, by the theorem above, we have that  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   HYPERLINK \l "TopicsList" Back to Topics List     Copyrighted by James D. Anderson, The University of Toledo www.math.utoledo.edu/~janders/1330 478PQR   > ? @ J ҲҲҲyppcppcph_sh5>*\aJh_shaJh_shZ@5\aJh5\aJ"jh_sh_sU\aJ"joh_sh_sU\aJh_sh_s0J>*\aJ"jh_sh_sU\aJjh_sh_sU\aJh_sh_s\aJh_sh_saJh_sh5\aJ&4 ? @ 0 1 3 4 5 6 7 V W X Y Z [ \ $a$       0 1 2 3 4 5 6 7 > ? 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