<p> 3.6 Functions of Special and Quadrantal Angles</p><p>An angle whose terminal side lies on the x- or y- axis is called a quadrantal angle. y</p><p>(0,1)</p><p>(-1,0) (1,0) x</p><p>(0,-1)</p><p>Example 1: Find the values of the six trigonometric functions for each angle.</p><p>(a) </p><p>(b) 180 sin 180= csc 180=</p><p> cos 180= sec 180=</p><p> tan 180= cot 180=</p><p>(c) 630 sin 630= csc 630=</p><p> cos 630= sec 630=</p><p> tan 630= cot 630= (d) 0 sin 0= csc 0=</p><p>Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 1 cos 0= sec 0=</p><p> tan 0= cot 0=</p><p>(e) </p><p>(f) 540 sin 540= csc 540=</p><p> cos 540= sec 540=</p><p> tan 540= cot 540=</p><p>Recall from geometry the special triangles:</p><p>Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 2 45 2 1 2 60 1 45 30 1 3</p><p>Example 2: Find the exact values of the six trigonometric functions for each. </p><p>(a) 45 sin 45= csc 45=</p><p> cos 45= sec 45=</p><p> tan 45= cot 45=</p><p>(b) 30 sin 30= csc 30=</p><p> cos 30= sec 30=</p><p> tan 30= cot 30=</p><p>(c) 60 sin 60= csc 60=</p><p> cos 60= sec 60=</p><p> tan 60= cot 60=</p><p>Example 3: Find the exact values of the six trigonometric functions for each angle.</p><p>Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 3 (a) 135 sin 135= csc 135=</p><p> cos 135= sec 135=</p><p> tan 135= cot 135=</p><p>(b) 225 sin 225= csc 225=</p><p> cos 225= sec 225=</p><p> tan 225= cot 225=</p><p>(c) </p><p>(d) 315 sin 315= csc 315=</p><p> cos 315= sec 315=</p><p> tan 315= cot 315=</p><p>For any angle that is not a quadrantal angle, the reference angle ′, is the acute angle formed by the terminal side of  and the x-axis.</p><p>Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 4 y y y y</p><p>    x ' x x x ' '  '</p><p>' ' ' ' = 180 - = - 180 = 360 -  =           ' =  -  ' =  -  ' = 2 - </p><p>To find the reference angle for an angle with negative measure or for an angle greater than 360 (2), first find a coterminal angle measure whose measure is between 0 and 360 (0 and 2).</p><p>Example 4: Give the reference angle ′ for each angle in standard position. (a) 135</p><p>(b) </p><p>(c) </p><p>(d) 110</p><p>Example 5: Use the reference angle to find the sin , cos , and tan  for each value of .</p><p>(a) 150 sin 150=</p><p> cos 150=</p><p>Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 5 tan 150=</p><p>(b) 330 sin 330=</p><p> cos 330=</p><p> tan 330=</p><p>(c) </p><p>(d) -30 sin (-30)=</p><p> cos (-30)=</p><p> tan (-30)=</p><p>(e) </p><p> P(x,y) Q(x',y')</p><p>'  '</p><p>The symmetry of a circle centered at the origin provides insight into the relationship between the trigonometric values of an angle and its reference angle. The angle  and its reference angle ′ intersect the circle of radius r at P(x, y) and Q(x′, y′). By symmetry x = -x′, and y = y′. Therefore,</p><p>Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 6</p><p>For any number between -1 and 1, there are infinitely many angles  for which sin  = k. However, it is often important to determine the values of  for which sin  = k on a restricted domain, such as 0 <  < 2.</p><p>Example 6: (a) If 0 <  < 2, determine the values of  for which </p><p>(b) If 0 <  < 2, determine the values of  for which </p><p>(c) If 0 <  < 2, determine the values of  for which tan  = 1</p><p>Homework: Day 1: p. 156 => Class Exercises 1 – 8; Function Chart Day 2: p. 157 => Practice Exercises 3 – 18; 20 – 21 Day 3: pp. 157 – 158 => Practice Exercises 24-34; 36; 41-45</p><p>Advanced Math/Trigonometry: Lesson 3.6 Functions of Special and Quadrantal AnglesPage 7</p>