SECTION 6.2 Trigonometric Functions: Unit Circle Approach 403 Finding the Exact Values of the Trigonometric Functions of π = ° 3 60 Find the exact values of the six trigonometric functions of 3 60 . π = ° EXAMPLE 6 Figure 27 3 60 θ π = = ° 1 608 1 2 2 21 21 1 1 y x P 5 (x, y) 5 , ( ) x2 1 y2 5 1 3 2 1 2 3 Solution Position the triangle in Figure 26(c) so that the 60° angle is in standard position. See Figure 27. The point on the unit circle that corresponds to 3 60 θ π = = ° is P 1 2 , 3 2 . = ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ Then π π π π π π = ° = = ° = = ° = = = = ° = = = ° = = = ° = = = sin 3 sin60 3 2 cos 3 cos60 1 2 csc 3 csc60 1 3 2 2 3 2 3 3 sec 3 sec60 1 1 2 2 tan 3 tan60 3 2 1 2 3 cot 3 cot60 1 2 3 2 1 3 3 3 Table 3 summarizes the information just derived for 6 30 , 4 45 , π π = ° = ° and 3 60 . π = ° Until you memorize the entries in Table 3, you should draw an appropriate diagram to determine the values given in the table. Finding the Exact Values of the Trigonometric Functions of 6 30 π = ° Find the exact values of the trigonometric functions of 6 30 . π = ° Solution EXAMPLE 7 Position the triangle in Figure 26(c) so that the 30° angle is in standard position. See Figure 28. The point on the unit circle that corresponds to 6 30 θ π = = ° is P 3 2 , 1 2 . = ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ Then sin 6 sin30 1 2 cos 6 cos30 3 2 csc 6 csc30 1 1 2 2 sec 6 sec30 1 3 2 2 3 2 3 3 tan 6 tan30 1 2 3 2 1 3 3 3 cot 6 cot30 3 2 1 2 3 π π π π π π = ° = = ° = = ° = = = ° = = = = ° = = = = ° = = θ (Radians) θ (Degrees) θ sin θ cos θ tan θ csc θ sec θ cot 6 π 30° 1 2 3 2 3 3 2 2 3 3 3 4 π 45° 2 2 2 2 1 2 2 1 3 π 60° 3 2 1 2 3 2 3 3 2 3 3 Table 3 Now Work PROBLEM 41 Figure 28 6 30 θ π = = ° 1 308 1 2 2 21 21 1 1 y x P 5 (x, y) 5 , x2 1 y2 5 1 3 2 1 2 3 ( )
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