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1 Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley 1.4-1 1 Trigonometric Functions 1.1 Angles 1.2 Angle Relationships and Similar Triangles 1.3 Trigonometric Functions 1.4 Using the Definitions of the Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley 1.4-2 1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities ▪ Quotient Identities Copyright © 2009 Pearson Addison-Wesley 1.1-3 1.4-3 Reciprocal Identities For all angles θ for which both functions are defined, Copyright © 2009 Pearson Addison-Wesley 1.1-4 1.4-4 Example 1(a) USING THE RECIPROCAL IDENTITIES Since cos θ is the reciprocal of sec θ, Copyright © 2009 Pearson Addison-Wesley 1.1-5 1.4-5 Example 1(b) USING THE RECIPROCAL IDENTITIES Since sin θ is the reciprocal of csc θ, Rationalize the denominator. Copyright © 2009 Pearson Addison-Wesley 1.1-6 1.4-6 Signs of Function Values in Quadrant sin cos tan cot sec csc I + + + + + + II + + III + + IV + + Copyright © 2009 Pearson Addison-Wesley 1.4-7 Ranges of Function Values Copyright © 2009 Pearson Addison-Wesley 1.4-8 Example 2 DETERMINING SIGNS OF FUNCTIONS OF NONQUADRANTAL ANGLES Determine the signs of the trigonometric functions of an angle in standard position with the given measure. (a) 87° The angle lies in quadrant I, so all of its trigonometric function values are positive. (b) 300° The angle lies in quadrant IV, so the cosine and secant are positive, while the sine, cosecant, tangent, and cotangent are negative. Copyright © 2009 Pearson Addison-Wesley 1.1-9 1.4-9 Example 2 DETERMINING SIGNS OF FUNCTIONS OF NONQUADRANTAL ANGLES (cont.) Determine the signs of the trigonometric functions of an angle in standard position with the given measure. (c) –200° The angle lies in quadrant II, so the sine and cosecant are positive, while the cosine, secant, tangent, and cotangent are negative. Copyright © 2009 Pearson Addison-Wesley 1.1-10 1.4-10 Example 3 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 1.1-11 1.4-11 Ranges of Trigonometric Functions Copyright © 2009 Pearson Addison-Wesley 1.1-12 1.4-12 Example 4 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 1.1-13 1.4-13 Example 5 FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Suppose that angle is in quadrant II and Find the values of the other five trigonometric functions. Choose any point on the terminal side of angle . Let r = 3. Then y = 2. Since is in quadrant II, Copyright © 2009 Pearson Addison-Wesley 1.1-14 1.4-14 Example 5 FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) Remember to rationalize the denominator. Copyright © 2009 Pearson Addison-Wesley 1.1-15 1.4-15 Example 5 Copyright © 2009 Pearson Addison-Wesley FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) 1.1-16 1.4-16 Pythagorean Identities For all angles θ for which the function values are defined, Copyright © 2009 Pearson Addison-Wesley 1.1-17 1.4-17 Quotient Identities For all angles θ for which the denominators are not zero, Copyright © 2009 Pearson Addison-Wesley 1.1-18 1.4-18 Example 6 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Choose the positive square root since sin θ >0. Copyright © 2009 Pearson Addison-Wesley 1.1-19 1.4-19 Example 6 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) To find tan θ, use the quotient identity Copyright © 2009 Pearson Addison-Wesley 1.1-20 1.4-20 Caution In problems like those in Examples 5 and 6, be careful to choose the correct sign when square roots are taken. Copyright © 2009 Pearson Addison-Wesley 1.1-21 1.4-21 Example 7 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 1.1-22 1.4-22 Example 7 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 1.1-23 1.4-23 Example 7 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 1.1-24 1.4-24 Example 7 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 1.1-25 1.4-25 Example 7 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued) This example can also be worked by drawing θ is 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 1.1-26 1.4-26