384 CHAPTER 6 Trigonometric Functions When an angle θ is in standard position, either the terminal side will lie in a quadrant, in which case we say that θ lies in that quadrant, or the terminal side will lie on the x-axis or the y-axis, in which case we say that θ is a quadrantal angle. For example, the angle θ in Figure 4(a) lies in quadrant II, the angle θ in Figure 4(b) lies in quadrant IV, and the angle θ in Figure 4(c) is a quadrantal angle. FUN FACT One counterclockwise rotation was said to measure 360° because the Babylonian year had 360 days. j Figure 4 y x u (a) u lies in quadrant II y x u (b) u lies in quadrant IV (c) u is a quadrantal angle y x u II III IV I II III IV I Angles are measured by determining the amount of rotation needed for the initial side to coincide with the terminal side. The two commonly used measures for angles are degrees and radians. Degree Measure The angle formed by rotating the initial side exactly once in the counterclockwise direction until it coincides with itself (1 revolution) is said to measure 360 degrees, abbreviated 360 .° One degree, 1 ,° is 1 360 revolution. A right angle is an angle that measures 90 ,° or 1 4 revolution; a straight angle is an angle that measures 180 ,° or 1 2 revolution. See Figure 5.As Figure 5(b) shows, it is customary to indicate a right angle by using the symbol . Figure 5 (a) 1 revolution counterclockwise, 3608 Initial side Terminal side Vertex (b) right angle, revolution counterclockwise, 908 Initial side Terminal side Vertex (c) Initial side Terminal side Vertex 1 – 4 straight angle, revolution counterclockwise, 1808 1 – 2 It is also customary to refer to an angle that measures θ degrees as an angle of θ degrees. Drawing an Angle Draw each angle in standard position. (a) 45° (b) 90 − ° (c) 225° (d) 405° Solution (a) An angle of 45° is 1 2 of a right angle. See Figure 6. (b) An angle of 90 − ° is 1 4 revolution in the clockwise direction. See Figure 7. (c) An angle of 225° consists of a rotation through 180° followed by a rotation through 45 .° See Figure 8. (d) An angle of 405° consists of 1 revolution 360 ( )° followed by a rotation through 45 .° See Figure 9. EXAMPLE 1 Figure 6 ° 45 angle 45° x y Terminal side Initial side Vertex Figure 9 ° 405 angle x y Terminal side Initial side 405° Vertex Figure 7 − ° 90 angle x y Initial side Terminal side -90° Vertex Figure 8 ° 225 angle x y Terminal side Initial side 225° Vertex Now Work PROBLEM 11
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