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1 of 5 8/6/2004 5.2 TRIGONOMETRIC FUNCTIONS 5.2 TRIGONOMETRIC FUNCTIONS Topics : • Trigonometric functions • Quadrantal angles • Reciprocal Identities • Signs and ranges of function values • Pythagorean identities • quotient identities TRIGONOMETRIC FUNCTIONS To define the six trigonometric functions, we start with an angle θ (theta) in standard position, and choose any point P having coordinates (x,y) on the terminal side of the angle θ . We find the distance, r, from P(x,y) to the origin, (0,0) using the distance formula r = x 2 + y 2 . The radius is always positive. The six trigonometric function of the angle θ are sine, cosine, tangent, cotangent, secant, and cosecant. In the following definitions, we use the customary abbreviations for the names of these functions: Homework Exercises: Sketch an angle θ ,theta, in standard position, such that θ has the smallest possible positive measure, and the given point is on the terminal side of θ ,theta 2. ( −12, −5 ) Find the values of the 6 trig functions for each angle in standard position having the given point on its terminal side. Rationalize denominators when applicable. 4. ( −4, −3) 6. (-4,0) 10. (3,-4) 12. If the terminal side of an angle θ is in QIII, what is the sign of each of the six trig functions of θ ? 1 2 of 5 8/6/2004 5.2 TRIGONOMETRIC FUNCTIONS Homework exercises 13 – 16. Suppose that the point (x,y) is in the indicated quadrant. Decide whether the given ratio is positive or negative (draw a sketch) y 14. III, r x 16. IV, y Homework exercises 17 -20. An equation of the terminal side of an angle theta in standard position is given with a restriction on x. Sketch the smallest positive such angle theta, and find the values of the 6 trig functions of theta. 18. 3x + 5 y = 0; x ≥ 0 20. −5 x − 3 y = 0; x ≤ 0 Quadrantal Angles If the terminal side of an angle in standard position lies along the y-axis, any point on this terminal side has x-coordinate 0. Similarly, an angle with terminal side on the x-axis has y-coordinate 0 for any point on the terminal side. Since the values of x and y appear in the denominators of some trigonometric function, and since a fraction is undefined if its denominator is 0, some trig function values of quadradantal angles are undefined. Homework exercises 9. Find the 6 trig functions (-2,0) 21. Find the 6 trig function values of the quadrantal angle 450° Undefined function Values If the terminal side of a quadrantal angle lies along the y-axis, then the tangent and secant functions are undefined. If it lies along the x-axis, then the cotangent and cosecant functions are undefined. See table for quadrantal angles pp 485 Reciprocal Identities: Identities are equations that are true for all values of the variable for which the expressions are defined Note: Identities can be written in different forms. For example, 1 1 sin θ = csc θ = ( sin θ )( csc θ ) = 1 cos θ sin θ 2 3 of 5 8/6/2004 5.2 TRIGONOMETRIC FUNCTIONS Homework exercises 44-49. Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. 44. 46. 48. Signs and Ranges of Function Values In the definitions of the trigonometric functions, r is the distance from the origin to the point (x,y). Distance is never negative, so r > 0. If we choose a point (x,y) in quadrant I, then both x and y will be positive. Thus, the values of all 6 functions will be positive in quadrant I. A point (x,y) in quadrant II has x <0 and y >0. This makes the values of sine and cosecant positive for quadrant II angles, while the other four functions take on negative values. Similar results can be obtained for the other quadrants, as summarized on page 487: sin θ cos θ tan θ cot θ sec θ csc θ θ in Quadrant I + + + + + + II + + III + + IV + + - Homework exercises 57 – 60. Identify the quadrant or quadrants for the angle satisfying the given conditions. 58. cos θ > 0, tan θ > 0 60. tan θ < 0, cot θ < 0 Homework exercises 61 – 66 Give the signs of the sine, cosine, and tangent functions for each angle: 62. 183 degrees 3 4 of 5 8/6/2004 5.2 TRIGONOMETRIC FUNCTIONS 64. 412 degrees 66 -121 degrees Ranges of the Trigonometric Functions For any angle θ for which the indicated functions exists: 1. −1 ≤ sin θ ≤ 1 and − 1 ≤ cos θ ≤ 1 2. tan θ and cot θ can be any real number; 3. sec θ ≤ −1 or secθ ≥ 1 and cscθ ≤ -1 or cscθ ≥ 1 (notice that sec θ and cscθ are never between -1 and 1) Homework exercises:70 -77 Decide whether each statement is possible or impossible for an angle θ . 70. sin θ = 2 72. tan θ = 0.92 74. sec θ = 1 1 76. sin θ = and cscθ =2 2 Pythagorean Identities: sin 2 θ + cos 2 θ = 1 tan 2θ + 1 = sec2 θ 1+cot 2θ = csc2 θ Quotient Identities: sin θ = tan θ cos θ cosθ = cot θ sinθ Homework exercises 78 – 86. Use identities to find each function value. Use a calculator in Exercises 84,85. 78. 80. 82. 84. Homework exercises 87 – 92. Find all trigonometric function values for each angle. Use a calculator in exercises 91 and 92. 88. 90. 4 5 of 5 8/6/2004 5.2 TRIGONOMETRIC FUNCTIONS 92. Note: do some word problems 97-100 pp 493-494 5