UNIT 5 TRIGONOMETRIC FUNCTIONS Date Lesson Text TOPIC Homework Oct. 0 5.0 (50).0 Getting Started Pg. # - 7 Nov. 5. (5). Radian Measure Angular Velocit Pg. 0 # ( 9)doso,,, a Nov. 5 Nov. 5. (5) 5. (5).. Radian Measure and Special Angles Sine and Cosine CAST Pg. 0 # ( 8)doso, 9,,, 5 & WS 5. Graphs of Primar Trigonometric Functions Pg. # c,, 5 Nov. 7 5. (5).5 Transformations of Trigonometric Functions Amplitude, Period, Phase Shift Pg. #,, 9, 0 WS 5. Nov. 8 5.5 (55). Graphs of Reciprocal Trigonometric Functions QUIZ (5.-5.) Pg. 5 #,,,, 7 Nov. 9 5. (5).5 Solving Problems with Trigonometric Functions WS 5. Nov. 5.7 (5).5 Solving Problems with Trigonometric Functions Pg. 0 # - 8 Nov. 5.8 (58) Review for Unit 5 Test Pg. 7 # 5 Nov. 5 5.9 (59) UNIT 5 TEST
MHF U Lesson 5.0 Review of Trigonometr Ex. is a third quadrant angle such that sin 0. 7 a) Draw a sketch of angle in standard position. b) Determine the primar trig ratios for angle. c) Calculate the value of the R.A.A. and the value of (to the nearest degree) using both a counterclockwise and clockwise rotation. Ex. Find the exact value of each of the following. a) tan 0 b) sin 5 c) cos 0 Ex. Solve each of the following, where 0 0. a) b) c) sin or. cos or or. sin or.
Ex. Complete the table and sketch the graph for at least one complete ccle. sin Function 0 Amplitude (a) Period Phase Shift (ps) Vertical Shift (vs) 80 90 90 80 70 0 50 50 0 70 Function cos 80 Amplitude (a) Period Phase Shift (ps) Vertical Shift (vs) 80 90 90 80 70 0 50 50 0 70 Pg. # - 7
MHF U Lesson 5. Radian Measure Radian - one radian is the measure of an angle subtended at the centre of a circle b an arc equal in length to the radius of the circle. A B r r C radian ABC arc AC 0 circumfere nce radian r 0 r πr radians = 0 r π radians = 0 The measure of an anglein radians () lengthofarc (a) lengthofradius(r) a = r, where is in radians π radians = 80 radian = 80 57. Ex. Convert to π radians. Ex. Convert to radians correct to decimal places. a) 5 b) 5 a) 50 b) 9
Ex. Convert to the nearest degree. a) b) c). r Ex. For a circle with a radius of 5 cm, calculate the arc length of each angle. a). rad b) 7 c) 85 Ex. 5 A tire with a diameter of 8 cm rotates 0 times in 5 s. a) Determine the angular velocit,, of the tire. b) Determine the distance travelled b a rock stuck in the tire in 5 s. Pg. 0 # ( 9)doso,,, a
MHF U Lesson 5. Radian Measure and Special Angles As ou no doubt remember from last ear, the special angles are connected to the special triangles. The special angles are: 0, 5, and 0. This does not change when we use radian measure. 0 =, 5 =, 0 = The special triangles are as shown below. An angle related to a special angle has an exact value. sin cos tan or sin cos tan or or sin cos tan csc sec cot or csc sec cot csc sec cot or or Another important rule with regard to trigonometric ratios is the cast rule. The CAST rule helps us to remember in which quadrant each trig ratio is positive. Sin All Tan Cos
Ex. Determine the exact value of each of the following. a) 5 sin b) 7 cos c) cot 5 d) sec Ex. The terminal arm of an angle in standard position passes through the point (, -). a) Sketch the angle. 8 7 5 5 7 8 x
b) Determine the value of r. c) Determine the primar trig ratios for the angle. d) Calculate the radian value of, correct to the nearest hundredth in the interval [0, ]. Ex. For each of the following, determine the exact value of, for [0, ]. a) cos = b) tan = c) csc = d) sec = Pg. 0 # (, )doso, 9,,, 5 & WS 5.
The Unit Circle (cos x, sin x)
MHF U Inv. 5. Sketching the Graphs of = sin and = cos and = tan Graphing = sin x using radians. Complete the following table of values for = sin x, 0 θ. Find the approximate value, correct to the nearest thousandth. Be sure our calculator is in radian mode. Radians ( ) 0 5 π sin cos Radians ( ) 7 5 5 7 π sin cos. Use the decimal values of sin and plot the ordered pairs (, sin ) on the grid below. Join the points with a smooth curve.. a) What is the minimum value of = sin? b) A what value(s) of does the minimum occur?
. a) What is the maximum value of = sin? b) A what value(s) of does the maximum occur? 5. What is the amplitude of = sin?. Make a conjecture about the appearance of the graph = sin for the domain. 7. Use our conjecture to sketch the graph = sin for the domain. 8. a) Is the graph of = sin periodic? b) If so, what is its period? 9. How can ou verif that sin is a function? 0. For = sin, a) What is the domain? b) What is the range?. Use the decimal values of cos and plot the ordered pairs (, cos ) on the same grid as = sin. Join the points with a smooth curve using a different colour than used for = sin.. a) What is the minimum value of = cos? b) A what value(s) of does the minimum occur?. a) What is the maximum value of = cos? b) A what value(s) of does the maximum occur?. What is the amplitude of = cos? 5. Make a conjecture about the appearance of the graph = cos for the domain.. Use our conjecture to sketch the graph = cos for the domain.
7. a) Is the graph of = cos periodic? b) If so, what is its period? 8. How can ou verif that = cos is a function? 9. For = cos, a) What is the domain? b) What is the range? 0. What would ou have to do to the graph of = cos to make it the same as the graph of = sin?. Is there more than one wa to achieve this?. If so, give more examples of what needs to be done to = cos to make it the same as of = sin. Graphing = tan using radians. Complete the table below correct to decimal places. Angle (radians) 0 5 7 π 5 5 7 π = tan. Use the results above to draw the graph of tan, on the grid below.
. a) For tan, state the (i) domain (ii) values of the x intercept (iii) values of the intercept b) Is tan a function? Wh or Wh not? c) What is the length of the period? Pg. # c,, 5
MHF U Lesson 5. Combining Transformations with Sinusoidal Functions: Summar (for the transformations a sin and a cos ): If a, there is a. If 0 a, there is a. If a 0, there is also a. a, the absolute value of a is the. To graph a sinusoidal function, ou should know: (i) amplitude (ii) period (iii) phase shift (iv) vertical shift to sketch the graph of = sin(x + ) (i) (ii) (iii) (iv) (v) Sketch the curve for at least one ccle..5.5 0.5.595.88507 9.777908.595 x 0.5.5.5
Ex. Sketch each of the following for at least ccle. a) = -cosx.5.5 0.5.595.88507 9.777908.595 x 0.5.5.5 b) sin x +.5.5 0.5.595.88507 9.777908.595 x 0.5.5.5 c).5 cos x.5.5 0.5.595.88507 9.777908.595 x 0.5.5.5
d) sin x.5.5 0.5.595.88507 9.777908.595 x 0.5.5.5 e) cosx.5.5 0.5.595.88507 9.777908.595 x 0.5.5.5 Complete the following chart: Function Amplitude (a) Phase Shift (ps) sin 0.5 Vertical Shift (vs) Period cos cos sin sin Pg. #,, 9, 0 & WS 5. Add #, 7, 8 if ou want more practice
MHF U INV 5.5 Graphs of Reciprocal Trigonometric Functions OBSERVATIONS: = csc x A. For x = 0, sin x = 0. csc x = sin x = B. For 0 x, sin x is positive and increasing in value. value. csc x = sin x is and in C. For x, sin x =. csc x = sin x = D. For x sin x is and in value. csc x = sin x is and in value. E. For x = π, sin x = 0 csc x = sin x is. F. For x sin x is and in value. csc x = sin x is and in value. G. For x H. For, sin x = csc x = sin x =. x sin x is and in value. csc x = sin x is and in value. I. For x =, sin x = 0, csc x = sin x is.
OBSERVATIONS: = sec x A. For x = 0, cos x =. sec x = co sx =. B. For 0 x, cos x is positive and decreasing in value. sec x = co sx is and in value. C. For x, cos x = 0 sec x = co sx is. D. For x cos x is and in value. sec x = co sx E. For x =, cos x = - sec x = is and in value. co sx is. F. For x cos x is and in value. sec x = co sx is and in value. G. For x, cos x = 0 sec x = co sx is. H. For x cos x is and in value. sec x = co sx is and in value. I. For x = π, cos x = sec x = co sx =.
Sketch the graph of = csc x on the graph below. = sin x is alread graphed for ou. Sketch the graph of = sec x on the axes below. = cos x is alread graphed for ou.
Complete the table below correct to decimal places. Angle (radians) 0 5 π 7 5 5 7 π = tan x = cot x Sketch the graph of = cot x on the axes below. Complete the table below for the reciprocal trig functions when 0 x. Pg. 5 #,,,, 7 = csc x = sec x = cot x Vertical Asmptotes Maximum Values Minimum Values Domain Range Period
MHF U Lesson 5. Modelling Problems with Trigonometric Functions Ex. Ba of Fund Tides
Ex. A weight is supported b a spring. The weight rests 50 cm above a tabletop. The weight is pulled down 5 cm and released at time t = 0. This creates a periodic up and down motion. It takes. s for the weight to return to the low position each time. a) Sketch a graph to represent the height of the weight assuming that its action continues indefinitel. h 75 h e i g h t (cm) 50 5 0.5.5.5 t time (s) b) Find an equation that would represent the relationship between height and time. c) Calculate the height of the weight, correct to the nearest tenth, after.0 s. d) Determine one of the times when the height of the weight, to decimal places, is 0 cm. WS 5.
MHF U Lesson 5.7 Solving Problems with Trigonometric Functions II a) Determine an equation of a sinusoidal function that best models the data below... 0.9 Proportion of Moon Visible 0.8 0.7 0. 0.5 0. 0. 0. 0. 8 0 8 0 8 0 8 0 8 50 5 5 5 x Da of the ear b) Determine the domain and range of the function c) Use our equation to determine three das on which 0% of the moon is visible. Pg. 0 # - 8