Supplemental Section: Trigonometry

1Supplemental Section: TrigonometryAngle MeasurementAngles can be measured in degrees.Counterclockwise Angles Clockwise Angles y y x xImportant Degree Relationships y 1 revolution = 3600 3 revolution = 2700 4 1 revolution = 1800 2 1 revolution = 900 4 xAngles can also be measured in radians. 2Consider the unit circle y xImportant Radian Relationships y 1 revolution = 2 radians 3 3 revolution = radians 4 2 1 revolution =  radians 2 1  revolution = radians 4 2 x 3Example 1: Draw, in standard position, the angle 3300Solution:█5 Example 2: Draw, in standard position, the angle . 4Solution:█ 4Angle Measurement ConversionsFormula for converting from degree to radian measure   Radian measure  (Degree measure)  180 Formula for converting from degree to radian measure180  Degree measure  (Radian measure)    Example 3: Convert 1200 from degrees to radians.Solution:█7 Example 4: Convert  from radians to degrees. 3Solution: Using the formula 180  Degree measure  (Radian measure)     we obtain the result7 180  1260 Degree measure         4200 3    3█ 5 6The Sine and Cosine FunctionsWe can define the cosine and sine of an angle  in radians, denoted as cos and sin , as the x and y coordinates respectively of a point P on the unit circle that is determined by the angle  .Consider the unit circle y xSine and Cosine of Basic Angle Values Degrees  Radians cos sin 0 0 30  6 45  4 60  3 90  2 180  270 3 2 360 2 7Note: If we know the cosine and sine values for an angle  in the first quadrant, we can determine the sine and cosine of related angles in other quadrants.Sign Diagram for Cosine and Sine Values y x 84 Example 5: Compute the exact values of the sine and cosine for the angle 3Solution:█11 Example 6: Compute the exact values of the sine and cosine for the angle . 4Solution:█ 9Other Trigonometric FunctionsTangent: tan  Secant: sec Cosecant: csc  Cotangent: cot 11 Example 7: Find the exact values of the six trigonometric functions for the angle . 6Solution:█ 10 11Note: sin 2   (sin )2 , cos2   (cos )2Basic Trigonometric IdentitiesPythagorean Identities Double Angle Formula 1. sin 2   cos2   1 sin 2  2sin cos 2. tan 2  1  sec2  3. cot 2  1  csc2 Example 8: Find the values of x in the interval [0,2 ] that satisfy the equation sin 2x  sin x .Solution: We solve the equation using the following steps: sin 2x  sin x 2sin x cos x  sin x (Re place left hand side with sin 2x  2sin x cos x) 2sin x cos x  sin x  0 (Subtract sin x from both sides) sin x(2cos x  1)  0 (Factor sin x) sin x  0, 2cos x  1  0 (Set each factor equal to zero) sin x  0, 2cos x  1 (Solve for cos x) 1 sin x  0, cos x  21 In the interval [0,2 ], sin x  0 when x  0,  , 2 . In the interval [0,2 ], cos x  when 2  5 x  , . Thus the five solutions are 3 3 5 x  0, , , , 2 3 3█ 12Graphs of Sine and CosinePeriod – distance on the x axis required for a trigonometric function to repeat its output values.The period of f (x)  sin x and f (x)  cos x is 2 .Example 9: Graph f (x)  sin xSolution: The graph can be plotted using the following Maple command:> plot(sin(x), x = -4*Pi..4*Pi, y = -2..2, color = red, thickness = 2);█ 13Example 10: Graph f (x)  cos xSolution: The graph can be plotted using the following Maple command:> plot(cos(x), x = -4*Pi..4*Pi, y = -2..2, color = red, thickness = 2);█ 14Graph of the Tangent FunctionPeriod of the tangent function is  radians. The vertical asymptotes of the tangent function sin x  f (x)  are the values of x where cos x  0 , that is the odd multiples of , cos x 2 3   3   , , , , . 2 2 2 2Example 11: Graph f (x)  tan x .Solution: The graph is given by the following:█ 15Practice Problems1. Convert from degrees to radians. a. 2100 c. 90 b.  3150 d. 3602. Convert from radians to degrees. 3 a. 4 c.  8 7 8 b.  d. 2 33. Draw, in standard position, the angle whose measure is given. 3 a.  3150 c.  rad 4 7 b. rad d.  3 rad 34. Find the exact trigonometric ratios for the angle whose radian measure is given. 3 4 a. c. 4 3 11 b.  d. 65. Find all values of x in the interval [0, 2 ] that satisfy the equation. a. 2cos x 1  0 b. sin 2x  cos xSelected Answers7 7 1. a. , b.  6 4 2. a. 7200 , b.  63003 2 3 2 3 3 3 3 4. a. cos   ,sin  , tan  1, sec   2 , csc  2 , cot  1 4 2 4 2 4 4 4 4 b. cos  1,sin  0 , tan  0 , sec  1, csc : undefined , cot : undefined 5 5. x  , 3 3

Supplemental Section: Trigonometry

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