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5 Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley 5.2-1 5 Trigonometric Functions 5.1 Angles 5.2 Trigonometric Functions 5.3 Evaluating Trigonometric Functions 5.4 Solving Right Triangles Copyright © 2009 Pearson Addison-Wesley 5.2-2 5.2 Trigonometric Functions Trigonometric Functions ▪ Quadrantal Angles ▪ Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities ▪ Quotient Identities Copyright © 2009 Pearson Addison-Wesley 1.1-3 5.2-3 Trigonometric Functions Let (x, y) be a point other the origin on the terminal side of an angle in standard position. The distance from the point to the origin is Copyright © 2009 Pearson Addison-Wesley 5.2-4 Trigonometric Functions The six trigonometric functions of θ are defined as follows: Copyright © 2009 Pearson Addison-Wesley 1.1-5 5.2-5 Example 1 FINDING FUNCTION VALUES OF AN ANGLE The terminal side of angle in standard position passes through the point (8, 15). Find the values of the six trigonometric functions of angle . Copyright © 2009 Pearson Addison-Wesley 1.1-6 5.2-6 Example 1 Copyright © 2009 Pearson Addison-Wesley FINDING FUNCTION VALUES OF AN ANGLE (continued) 1.1-7 5.2-7 Example 2 FINDING FUNCTION VALUES OF AN ANGLE The terminal side of angle in standard position passes through the point (–3, –4). Find the values of the six trigonometric functions of angle . Copyright © 2009 Pearson Addison-Wesley 1.1-8 5.2-8 Example 2 FINDING FUNCTION VALUES OF AN ANGLE (continued) Use the definitions of the trigonometric functions. Copyright © 2009 Pearson Addison-Wesley 1.1-9 5.2-9 Example 3 FINDING FUNCTION VALUES OF AN ANGLE Find the six trigonometric function values of the angle θ in standard position, if the terminal side of θ is defined by x + 2y = 0, x ≥ 0. We can use any point on the terminal side of to find the trigonometric function values. Choose x = 2. Copyright © 2009 Pearson Addison-Wesley 1.1-10 5.2-10 Example 3 FINDING FUNCTION VALUES OF AN ANGLE (continued) The point (2, –1) lies on the terminal side, and the corresponding value of r is Multiply by to rationalize the denominators. Copyright © 2009 Pearson Addison-Wesley 1.1-11 5.2-11 Example 4(a) FINDING FUNCTION VALUES OF QUADRANTAL ANGLES Find the values of the six trigonometric functions for an angle of 90°. The terminal side passes through (0, 1). So x = 0, y = 1, and r = 1. undefined Copyright © 2009 Pearson Addison-Wesley undefined 1.1-12 5.2-12 Example 4(b) FINDING FUNCTION VALUES OF QUADRANTAL ANGLES Find the values of the six trigonometric functions for an angle θ in standard position with terminal side through (–3, 0). x = –3, y = 0, and r = 3. undefined Copyright © 2009 Pearson Addison-Wesley undefined 1.1-13 5.2-13 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 the terminal side of a quadrantal angle lies along the x-axis, then the cotangent and cosecant functions are undefined. Copyright © 2009 Pearson Addison-Wesley 1.1-14 5.2-14 Commonly Used Function Values sin cos tan cot sec csc 0 0 1 0 undefined 1 undefined 90 1 0 undefined 0 undefined 1 180 0 1 0 undefined 1 undefined 270 1 0 undefined 0 undefined 1 360 0 1 0 undefined 1 undefined Copyright © 2009 Pearson Addison-Wesley 5.2-15 Using a Calculator A calculator in degree mode returns the correct values for sin 90° and cos 90°. The second screen shows an ERROR message for tan 90° because 90° is not in the domain of the tangent function. Copyright © 2009 Pearson Addison-Wesley 5.2-16 Caution One of the most common errors involving calculators in trigonometry occurs when the calculator is set for radian measure, rather than degree measure. Copyright © 2009 Pearson Addison-Wesley 1.1-17 5.2-17 Reciprocal Identities For all angles θ for which both functions are defined, Copyright © 2009 Pearson Addison-Wesley 1.1-18 5.2-18 Example 5(a) USING THE RECIPROCAL IDENTITIES Since cos θ is the reciprocal of sec θ, Copyright © 2009 Pearson Addison-Wesley 1.1-19 5.2-19 Example 5(b) USING THE RECIPROCAL IDENTITIES Since sin θ is the reciprocal of csc θ, Rationalize the denominator. Copyright © 2009 Pearson Addison-Wesley 1.1-20 5.2-20 Signs of Function Values in Quadrant sin cos tan cot sec csc I + + + + + + II + + III + + IV + + Copyright © 2009 Pearson Addison-Wesley 5.2-21 Signs of Function Values Copyright © 2009 Pearson Addison-Wesley 5.2-22 Example 6 IDENTIFYING THE QUADRANT OF AN ANGLE Identify the quadrant (or quadrants) of any angle that satisfies the given conditions. (a) sin > 0, tan < 0. Since sin > 0 in quadrants I and II, and tan < 0 in quadrants II and IV, both conditions are met only in quadrant II. (b) cos < 0, sec < 0 The cosine and secant functions are both negative in quadrants II and III, so could be in either of these two quadrants. Copyright © 2009 Pearson Addison-Wesley 1.1-23 5.2-23 Ranges of Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley 1.1-24 5.2-24 Example 7 DECIDING WHETHER A VALUE IS IN THE RANGE OF A TRIGONOMETRIC FUNCTION Decide whether each statement is possible or impossible. (a) sin θ = 2.5 Impossible (b) tan θ = 110.47 Possible (c) sec θ = .6 Impossible Copyright © 2009 Pearson Addison-Wesley 1.1-25 5.2-25 Pythagorean Identities For all angles θ for which the function values are defined, Copyright © 2009 Pearson Addison-Wesley 1.1-26 5.2-26 Quotient Identities For all angles θ for which the denominators are not zero, Copyright © 2009 Pearson Addison-Wesley 1.1-27 5.2-27 Example 8 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Find sin θ and cos θ, given that quadrant III. and θ is in Since θ is in quadrant III, both sin θ and cos θ are negative. Copyright © 2009 Pearson Addison-Wesley 1.1-28 5.2-28 Example 8 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) Caution It is incorrect to say that sin θ = –4 and cos θ = –3, since both sin θ and cos θ must be in the interval [–1, 1]. Copyright © 2009 Pearson Addison-Wesley 1.1-29 5.2-29 Example 8 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) Use the identity to find sec θ. Then use the reciprocal identity to find cos θ. Choose the negative square root since sec θ <0 for θ in quadrant III. Secant and cosine are reciprocals. Copyright © 2009 Pearson Addison-Wesley 1.1-30 5.2-30 Example 8 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) Choose the negative square root since sin θ <0 for θ in quadrant III. Copyright © 2009 Pearson Addison-Wesley 1.1-31 5.2-31 Example 8 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) This example can also be worked by drawing θ in standard position in quadrant III, finding r to be 5, and then using the definitions of sin θ and cos θ in terms of x, y, and r. Copyright © 2009 Pearson Addison-Wesley 1.1-32 5.2-32