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# Chapter 6

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1. Chapter 6 Trigonometric Functions

2. Section 6.1 Angles and Their Measurement

3. Definition • An angle  is said to be in standard position if its vertex lies at the origin of a cartesian coordinate system and the initial side lies on the positive x-axis. Terminal Side y  x Vertex Initial Side

4. Definition • Let  be an angle in standard position obtained by a counterclockwise rotation. The angle  is said to have a measure of one radian if the subtended arc is equal to the radius in length.

5. Remarks • The radian measure of an angle is dimensionless. • Since the circumference of a circle of radius r is 2r, an angle that covers an entire circle has measure 2 radians. • For simplicity, we shall often use the unit circle when considering (radian) measures of angles. • If the angle  is obtained by a clockwise rotation, then we take the corresponding negative value as the radian measure.

6. Definition • One degree (1o) is defined as the measure of the angle subtended by an arc of a circle which is 1/360 of the circumference. • Also, one minute (written 1’) is 1/60 of 1o and one second (written 1’’) is 1/60 of 1’.

7. Remarks • If the measure of an angle does not have the degree symbol explicitly indicated, it is assumed to be in radians. • From the definition, we see that the degree measure of an angle which covers an entire circle is 360o. Hence 1 radian = (180/)o 1 degree = (/180) radians

8. Example 1 • Convert the ff. degree measurements to the equivalent radian measurements: (a) 445o (b) -70o (c) 45o16’48’’

9. Example 2 • Convert the ff. radian measurements to the equivalent degree measurements: (a) (b) -3

10. Definition • An angle in standard position whose terminal side lies on either the x-axis or the y-axis is called quadrantal. • Two angles are coterminal if they have the same terminal side.

11. Example • Find the radian measure of the smallest positive angle that is coterminal to the ff. angle having the given radian measure. (a) (b)

12. Section 6.2 Trigonometric Functions of General Angles

13. Definition • Let  be an angle in standard position, P(x,y) any point on the terminal side of  other than the origin, and the distance from P to the origin. Then,

14. Remark • From the definitions, • sin  and csc  are > 0 only in quadrants I and II • cos  and sec  are > 0 only in quadrants I and IV • tan  and cot  are > 0 only in quadrants I and III

15. Example 1 • Find the values of the trigonometric functions of  in standard position if P(12,-6) is on the terminal side of .

16. Example 2 • Given the ff. information, find the exact values of the remaining trigonometric functions of : (a) csc  = 4 and tan  < 0 (b) cos  = -15/17 and sin  < 0

17. Definition • An equation that is always true whenever the expression on both sides are defined is called an identity.

18. Reciprocal / Quotient Identities

19. Pythagorean Identities*

20. Sections 6.3, 6.4 Trigonometric Functions of Special Angles and Real Numbers

22. Trigonometric Functions: Multiples of /4*

23. Trigonometric Functions: Multiples of /6 and /3*

24. Example 1 • Find the exact values of the six trigonometric functions for each angle  given below: (a) (d) -90o (b) (e) 3 (c) 480o

25. Example 2 • Give the exact values of the following without using a calculator: (a) (b)

26. Theorem* • For any real number t, sin(-t) = -sin t cos(-t) = cos t tan(-t) = -tan t (if tan t is defined)

27. Theorem* • If tR and kZ, then sin(t + 2k) = sin t cos(t + 2k) = cos t tan(t + k) = tan t (if tan t is defined)

28. Example • If f(x) = csc x and f(a) = 2, find the exact value of the following: (a) f(-a) (b) f(a) + f(a + 2) + f(a + 4)

29. Remarks • The previous theorem shows that the trigonometric functions are periodic. • A function is said to be periodic if there exists a positive real number p such that for any x in the domain of f, f(x + p) = f(x). The smallest such p is called the period of f. • We can see that sin, cos, sec, and csc each have period 2, while tan and cot have period .

30. Exercise (no calculators!) • Convert the ff. angles to the corresponding radian or degree measure (whichever is applicable). Then, find the values of the six trigonometric functions for each angle. 1. 17/2 4. -17/4 2. 635/3 5. 900o 3. -210o

31. Sections 6.5, 6.6 Graphs of the Trigonometric Functions

32. Definition • The graph of a periodic function on an interval of length equal to the period is called a complete cycle, or simply a cycle, of the graph. • In this section, we shall sketch the graphs of the trigonometric functions on (at least) two cycles of their graphs.

33. Graph: Sine Function

34. Sine Function: Properties • Its domain is the set of all real numbers. • Its range is the set [-1,1]. • It is periodic with period 2. Hence, once we have drawn the graph for any interval of length 2, say from - to , then this portion will be repeated in intervals of length 2 on the x-axis. • The graph of the sine function is symmetric with respect to the origin.

35. Graph: Cosine Function

36. Cosine Function: Properties • Its domain is the set of all real numbers. • Its range is the set [-1,1]. • It is periodic with period 2. • The graph of the cosine function is symmetric with respect to the y-axis.

38. Amplitude of a Wave • The amplitude of a wave is the maximum displacement of a periodic wave. • If a function is of the form or , then the graph has amplitude |a|.

39. Examples • Find the amplitudes of the following functions and sketch their graphs. (a) f(x) = -2cos x (b) g(x) = 1/3 sin x

40. Period of a Wave • If or , then the graph has period 2/|b|. • If b < 0, we can use the identities sin(-t) = -sin t and cos(-t) = cos t to simplify the expression first.

41. Examples • Find the period and amplitude of the following functions and then sketch their graphs: (a) y = sin(2x) (b) y = -1/2 cos(3x)

42. Phase Shift • If y = sin b(x – c), then its graph can be obtained from the graph of y = sin bx by a horizontal translation of |c| units to the right if c > 0 and to the left if c < 0. • The number c is called the phase shift of the graph. • A similar method can be used to obtain the graph of y = cos b(x – c) from y = cos bx.

43. Examples • Find the phase shift, amplitude, and period of the ff. functions, and then sketch their graphs. (a) (b) (c)