Using the Definitions of the Trigonometric Functions

Transcription

1 1.4 Using the Definitions of the Trigonometric Functions Reciprocal Identities Signs and Ranges of Function Values Pythagorean Identities Quotient Identities February 1, 2013 Mrs. Poland

2 Objectives Objective #5: Students will be able to use identities to find function values. SC#8: I can use the identities to find function values given one function value. SC#9: I can use the identities to find function values given one function value and the quadrant. Objective #6: Students will be able to use the properties of the functions to solve problems. SC#10: I can use the ranges of the trig functions to solve problems. SC#11: I can use quandrant/function relationships to solve problems.

3 Reciprocal Identities For all angles θ for which both functions are defined,

4 Example ( 1(a USING THE RECIPROCAL IDENTITIES Since cos θ is the reciprocal of sec θ,

5 Example ( 1(b USING THE RECIPROCAL IDENTITIES Since sin θ is the reciprocal of csc θ, Rationalize the denominator.

6 Signs of Function Values θ in Quadrant sin θ cos θ tan θ cot θ sec θ csc θ I II III IV

7 Signs of Function Values

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. (c) 200 The angle lies in quadrant II, so the sine and cosecant are positive, while the cosine, secant, tangent, and cotangent are negative.

9 Example 3 IDENTIFYING THE QUADRANT OF AN ANGLE Identify the quadrant (or quadrants) of any angle θ that satisfies the given conditions. (a) sin θ > 0, tan θ < 0. Since sin θ > 0 in quadrants I and II, and tan θ < 0 in quadrants II and IV, both conditions are met only in quadrant II. (b) cos θ < 0, sec θ < 0 The cosine and secant functions are both negative in quadrants II and III, so θ could be in either of these two quadrants.

10 Ranges of Trigonometric Functions 1. -1<sinθ<1, -1<cosθ<1 2. tanθ and cotθ can be any Real number 3. secθ < -1 or secθ > 1, cscθ < -1 or cscθ > 1

11 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 = Possible (c) sec θ =.6 Impossible

12 Example 5 FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Suppose that angle θ is in quadrant II and Find the values of the other five trigonometric functions. Choose any point on the terminal side of angle θ. Let r = 3. Then y = 2. Since θ is in quadrant II,

13 Example 5 FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continue Remember to rationalize the denominator.

14 Pythagorean Identities For all angles θ for which the function values are defined,

15 Quotient Identities For all angles θ for which the denominators are not zero,

16 Example 6 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Choose the positive square root since sin θ >0.

17 Example 6 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (cont.) To find tan θ, use the quotient identity In problems like those in Examples 5 and 6, be careful to choose the correct sign when square roots are taken.

18 Example 7 FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT Find sin θ and cos θ, given that and θ is in quadrant III. Since θ is in quadrant III, both sin θ and cos θ are negative. 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].

19 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. Choose the negative square root since sin θ <0 for θ in quadrant III.

20 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.

21 Classwork Page #2, 12, 22, 28, 39 Page #4, 9, 15, 23, 29, 40, 42, 48