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Reference Angles A reference angle for an angle θ in standard position is the positive acute angle between the terminal side of θ and the (closest part of the) x-axis. In these notes, the reference angle for θ will be denoted . EXAMPLE 1: A reference angle for 27 is 27 , because 27 is already an acute angle. EXAMPLE 2: A reference angle for 125 is 180 125 55 because 125 is in the second quadrant, and the closest part of the x-axis is the negative x-axis, or 180 , which is larger than 125 . EXAMPLE 3: A reference angle for 190 is 190 180 10 , because 190 is in the third quadrant. Again, the negative part of the x-axis is closest to 190 , but this time 190 is larger than 180 . EXAMPLE 4: A reference angle for 322 is 360 322 38 . This is because 322 is in the fourth quadrant, and the positive x-axis is the closest to θ. (The positive x-axis is the terminal side of both 0 and 360 .) The figure below summarizes the method for finding a reference angle for θ if 0 360 (0 2 ) We study reference angles because of the following theorem: Reference Angle Theorem If is the reference angle for , then the trigonometric functions of are the same as those for except for a possible change in sign. (That is, sin sin , cos cos , tan tan and so on.) 1 . The angles between 0 and 360 that 2 have a reference angle of 60 are 60 in Quadrant I, 120 in Quadrant II, 240 in Quadrant III, and 300 in Quadrant IV. As you can see from the figure below, x 1 x 1 cos 60 cos 300 , and cos 120 cos 240 . The values differ r 2 r 2 For example, we know that cos 60 only in sign. EXAMPLE 9: Find tan 210 . Solution: First, we find a reference angle for 210 and determine the sign of tan 210 . Since 210 is in the third quadrant, 210 180 30 , and tan 210 0 . We know that tan 30 1 3 3 3 and, therefore, tan 210 3 3 also. EXAMPLE 10: Find sec 135 . Solution: First, we note that 135 is in the second quadrant. Thus, 1 0 , and 180 135 45 . We find cos 135 1 2 sec sec 45 2 and conclude that sec 135 2 . cos 45 2 sec 135 EXAMPLE 11: Find a fourth quadrant angle in degrees such that sin Solution: First, we find the reference angle, . The acute angle such that 2 is 45 , so the fourth quadrant angle we are looking for is 2 360 45 315 . sin 2 . 2 EXAMPLE 12: Find a second quadrant angle in radians such that cos 1 . 2 1 would be 60 . The 2 angle we are looking for, then is 180 60 120 . Solution: The acute angle, , such that cos You can also find reference angles for negative angles and angles over one revolution: EXAMPLE 13: Find a reference angle for 115 . Solution: 115 is in the third quadrant. Its reference angle is ˆ 180 115 65 . EXAMPLE 14: Find a reference angle for 1790 . Solution: 1790 4(360) 350 , so 1790 is coterminal with 350 and therefore has the same reference angle. A reference angle for 350 is 360 350 10 .