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Trigonometric Functions

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Trigonometric Functions

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    1. Chapter 2 Trigonometric Functions

    2. Standard Position An angle is in standard position if the initial side is on the x axis and vertex is at the origin

    3. QUADRANTAL ANGLES Angles in standard position with terminal sides along the x axis or y axis 90º, 180º, 270º, 360º

    4. The quadrant an angle lies in is based upon which quadrant the terminal side lies in.

    5. EXAMPLE State which quadrant or on which axis the angles with given measure in standard position. 91º 175º 310º -540º 525º

    6. Coterminal Angles Two angles in standard position with the same terminal side

    7. Determine if the following pairs of angles are coterminal. 30º & -690º 240º & -120º

    8. EXAMPLE Determine the angles of the smallest possible positive measure that are coterminal with the following angles. 900º 379º -92º

    9. Cartesian Plane

    10. Trigonometric Functions

    11. EXAMPLE The terminal side of an angle in standard position passes through the indicated point. Calculate the values of the six trigonometric functions for the angle. (2, 3)

    12. EXAMPLE The terminal side of an angle in standard position passes through the indicated point. Calculate the values of the six trigonometric functions for the angle. (-1, -3)

    13. EXAMPLE

    14. EXAMPLE Calculate the values for the six trigonometric functions of the angle ?, given in standard position, if the terminal side of ? is defined by 3y + 2x = 0, x = 0.

    15. EXAMPLE Calculate (if possible) the values for the six trigonometric functions of the angle, ?, given in standard position. ? = - 30º

    16. EXAMPLE Calculate (if possible) the values for the six trigonometric functions of the angle, ?, given in standard position. ? = -540º

    17. Quadrantal Angles

    18. 2.3 Evaluating Trigonometric functions for Any Angle

    19. RECALL…

    20. REVIEW Find the exact value of cos (?240?). Since an angle of ?240? is coterminal with an angle of ?240? + 360? = 120?, the reference angles is 180? ? 120? = 60?, as shown.

    21. All Students Take Calculus The algebraic sign of the trigonometric ratios is determined by which quadrant the terminal side lies in.

    22. EXAMPLE If cos ? = -3/5 and the terminal side of the angle lies in quadrant III, find sin?. cos? = -3/5 means that the x value is negative, so x = -3 and r = 5. Now we know that (-3)2 + y2 = 52. y2 = 25 – 9 = 16, so y = ±4. Since the angle is in quadrant III, y = -4. sin? = y/r = -4/5.

    23. Values of Quadrantal Trigonometric Functions The values of the trigonometric functions for angles along the axis are undefined for some angles. For example, along the positive y-axis, the value of x is zero, making the value of the tangent undefined.

    24. EXAMPLE Evaluate the following expressions if possible: cos 540º + sin 270º csc 90º + sec 180º

    25. Ranges of Trigonometric Functions Let us start with an angle, ?, in quadrant I and the sine function defined as the ratio sin? =___ If the measure of ? increases toward 90°, then y increases. Notice that even though the value of y approaches the value of r, they will only be equal when ? = 90°, and y can never be larger than r.

    26. Similarly… For any angle, ?, for which the trigonometric functions are defined, the six trigonometric functions have the following ranges:

    27. Example Determine if each statement is possible: sin ? =1.001 cot ? = 0 csc ? =v3

    28. Reference Angle For any angle ? in standard position whose terminal side in one of the four quadrants, there exist a reference angle,?, which is an acute angle with positive measure that is formed by the terminal side of ? and the x-axis.

    29. Example Find the reference angle for each given angle: 210º 135º

    30. Reference Triangle To form a reference angle ?, drop a perpendicular from the terminal side of the angle to the x-axis. ? is the reference angle

    31. Procedure for Evaluating Function Values for any Nonquadrantal angle Step 1: If ? < 0°, then add 360° as many times as needed to get a coterminal angle with measure between 0° and 360°. If ? > 360°, then subtract 360° as many times as needed to get a coterminal angle with measure between 0° and 360°. Step 2: Find the quadrant in which the terminal side of the angle in Step 1 lies. Step 3: Find the reference angle, a, of the coterminal angle found in Step 1. Step 4: Find the trigonometric function values for the reference angle, a. Step 5: Determine the correct algebraic signs (+/-) for the trigonometric functions based on the quadrant identified in Step 2. Step 6: Combine the trigonometric values found in Step 4 with the algebraic signs in Step 5 to evaluate the trigonometric function values of ?.

    32. EXAMPLE Find the exact value of cos 210º.

    33. Example Find the exact value of csc (-210°).

    34. Example Find all values of ?, where 0° = ? = 360°, when

    35. 2.4 Basic Trigonometric Identities

    36. Reciprocal Identities

    37. Quotient Identities

    38. Notation

    39. Pythagorean Identity

    43. EXAMPLE Find cos ? and tan ? if sin ? = 3/5 and the terminal side of ? lies in quadrant II.

    44. EXAMPLE Find sin ? and cos ? if tan ? = 3/4and if the terminal side of ? lies in quadrant III.

    45. EXAMPLE Multiply (1 – cos ?)(1 + cos ?)

    46. EXAMPLE Find the indicated expression. If cos ? = 0.1, find cos3 ?. (0.1) 3 = 0.001 If tan ? = -5, find tan2 ?. (-5)2 = 25

    47. Information from Trigonometry

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