585 CHAPTER 5 Test Prep Trigonometric Function Values and Angle Measures 5.3 Concepts Examples Right-Triangle-Based Definitions of Trigonometric Functions Let A represent any acute angle in standard position. sin A = y r = side opposite hypotenuse csc A = r y = hypotenuse side opposite cos A = x r = side adjacent hypotenuse sec A = r x = hypotenuse side adjacent tan A = y x = side opposite side adjacent cot A = x y = side adjacent side opposite Cofunction Identities For any acute angle A, cofunction values of complementary angles are equal. sin A =cos190° −A2 cos A =sin190° −A2 sec A =csc190° −A2 csc A =sec190° −A2 tan A =cot190° −A2 cot A =tan190° −A2 Function Values of Special Angles U sin U cos U tan U cot U sec U csc U 30° 1 2 23 2 23 3 23 2 23 3 2 45° 22 2 22 2 1 1 22 22 60° 23 2 1 2 23 23 3 2 2 23 3 Reference Angle U′ for U in 10°, 360°2 U in Quadrant I II III IV U′ is u 180° - u u - 180° 360° - u Finding Trigonometric Function Values for Any Nonquadrantal Angle U Step 1 Add or subtract 360° as many times as needed to obtain an angle greater than 0° but less than 360°. Step 2 Find the reference angle u′. Step 3 Find the trigonometric function values for u′. Step 4 Determine the correct signs for the values found in Step 3. To approximate a trigonometric function value of an angle in degrees, make sure the calculator is in degree mode. Find exact values of the six trigonometric functions for angle A. A C 7 25 24 B Hypotenuse Side opposite A Side adjacent to A sin A = 7 25 cos A = 24 25 tan A = 7 24 csc A = 25 7 sec A = 25 24 cot A = 24 7 Write each function in terms of its cofunction. sin 55° = cos190° - 55°2 = cos 35° sec 48° = csc190° - 48°2 = csc 42° tan 72° = cot190° - 72°2 = cot 18° Consider each triangle. 1 2 608 308 Ë3 1 1 Ë2 458 458 30°9 60° right triangle 45°9 45° right triangle Quadrant I: For u = 25°, u′ = 25° Quadrant II: For u = 152°, u′ = 28° Quadrant III: For u = 200°, u′ = 20° Quadrant IV: For u = 320°, u′ = 40° Find sin 1050°. 1050° - 21360°2 = 330° The reference angle for 330° is u′ = 30°. sin 1050° = -sin 30° Sine is negative in quadrant IV. = - 1 2 sin 30° = 1 2 Approximate each value. cos 50° 15′ = cos 50.25° ≈0.63943900 csc 32.5° = 1 sin 32.5° ≈1.86115900 csc u = 1 sin u Coterminal angle in quadrant IV
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