Trigonometry Standard Position and Radians

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Piers Prosper Spencer
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1 MHF 4UI Unit 6 Day 1 Tigonomety Standad Position and Radians A. Standad Position of an Angle teminal am initial am Angle is in standad position when the initial am is the positive x-axis and the vetex is at the oigin. A positive angle otates in the counte-clockwise diection. So fa we have only measued angles in degees. We will now measue angles in adians B. Radian Measue y O B A x In the cicle, O is the cente. is the angle subtended at the cente of the cicle by an ac AB. ac length θ adius a y When a = x θ a 1 adian y When a = x θ a adians y When a = 3 x θ a 3 3 adians

2 MHF 4UI Unit 6 Day 1 What is the adian measue of one complete evolution? Convesion facto: 1 ad o 1 1. Convet each of the following to degees. π a) ad 5π b) ad 6 c).53. Convet each of the following to adians. Give exact values, then ound to 4 decimals. a) 45 b) 10 c) 31

3 MHF 4UI Unit 6 Day 1 Sine Function sin sin

4 MHF 4UI Unit 6 Day 1 Cosine Function cos cos

5 MHF 4UI Unit 6 Day Gaphing Recipocal Tigonometic Functions The function sketched below is fx = Gaph the ecipocal of the function shown below: Clealy indicate any vetical asymptotes. Clealy mak any value(s) which ae the same on fx and the ecipocal of fx. Using the big / little popety, sketch the ecipocal of fx y x If gx is the ecipocal of fx, wite its equation two diffeent ways. gx = and gx = Popeties of Vetical Asymptotes: Domain: Range: Peiod:

6 MHF 4UI Unit 6 Day The function sketched below is fx = Gaph the ecipocal of the function shown below: Clealy indicate any vetical asymptotes. Clealy mak any value(s) which ae the same on fx and the ecipocal of Using the big / little popety, sketch the ecipocal of fx fx. y x If gx is the ecipocal of fx, wite its equation two diffeent ways. gx = and gx = Popeties of Vetical Asymptotes: Domain: Range: Peiod:

7 MHF 4UI Unit 6 Day The function sketched below is fx = Gaph the ecipocal of the function shown below: Clealy indicate any vetical asymptotes. Clealy mak any value(s) which ae the same on fx and the ecipocal of Using the big / little popety, sketch the ecipocal of fx fx. y x If gx is the ecipocal of fx, wite its equation two diffeent ways. gx = and gx = Popeties of Vetical Asymptotes: Domain: Range: Peiod:

8 MHF 4UI Unit 6 Day Tangent Function tan tan

9 MHF 4UI Unit 6 Day Popeties of Tigonometic Functions peiod y-intecept peiod y-intecept peiod y-intecept zeoes sine chaacteistics zeoes cosine chaacteistics zeoes tangent chaacteistics minimum: minimum: minimum: maximum: maximum: maximum: asymptotes: asymptotes: asymptotes: peiod y-intecept peiod y-intecept peiod y-intecept zeoes cosecant chaacteistics zeoes secant chaacteistics zeoes cotangent chaacteistics minimum: maximum: asymptotes: minimum: maximum: asymptotes: minimum: maximum: asymptotes:

10 MHF 4UI Unit 6 Day 3 Tansfomations of Sine and Cosine Functions Basic tansfomations: y = a sin [ k (x p) ] + q y = a cos [ k (x p) ] + q Notes: amplitude = a ; half the distance between the min and max values amplitude = max - min 360 peiod k π k ad vetical shift = max min ; if q > 0, shift up q units if q < 0, shift down q units phase shift: if p > 0, shift ight p units; (x p) if p < 0, shift left p units; (x + p) eflections: if a < 0, eflect in the x axis; (vetical eflection) if k < 0, eflect in the y axis; (hoizontal eflection)

11 MHF 4UI Unit 6 Day 3 1. Sketch one peiod of each of the following. Also state the domain and ange. Plot a minimum of 5 odeed pais of the function. a) y = 3cos(x) a = peiod = v.s. = p.s. = D = R = b) y = -sin(3x) + 4 a = peiod = v.s. = p.s. = D = R =

12 MHF 4UI Unit 6 Day 3 c) y = 1.5cos(x - 3 π ) + a = peiod = v.s. = p.s. = D = R =

13 MHF 4UI Unit 6 Day 4 Moe Tansfomations of Sine and Cosine Functions 1. Sketch one peiod of each of the following. Plot a minimum of 5 odeed pais. a) y = -4 sin [ (x - 3 π )] + 1 a = peiod = v.s. = p.s. = b) Detemine the equation of a cosine function with a maximum value of 0, amplitude 8, π peiod 3π and a phase shift. 4

14 MHF 4UI Unit 6 Day 4 π c) A sine function on the inteval x [0, ) has its fist maximum point at (, 4) and 4 7π its fist minimum point at (, -). Detemine a possible sine equation. 1 d) Repeat c) with a cosine function.

15 MHF 4UI Unit 6 Day 5 Tigonometic Ratios P(x,y) x y Pimay tig atios Recipocal tig atios 1. P 6,3 is a point on the teminal am of an angle in standad position whee Detemine the exact values of sin, cos and tan. Include a clealy labelled sketch. 0. The CAST ule confims the sign of ou answes.

16 MHF 4UI Unit 6 Day 5. is a standad position angle in quadant III such that value of csc. Include a clealy labelled sketch. cos. Detemine the exact 3 3. is a standad position angle such that Include a clealy labelled sketch. 1 tan. Detemine the exact value of sin. 4

17 MHF 4UI Unit 6 Day 5 Sketching Special Angles π π multiples: multiples: π π multiples: multiples:

18 MHF 4UI Unit 6 Day 6 Special Tiangles and Tigonometic Ratios A. Special Tiangles Recall fom last yea: Now, using adian measue: Similaly, evaluating tig atios in degees using the CAST ule can also be accomplished in adians.

19 MHF 4UI Unit 6 Day 6 B. Evaluating Tigonometic Ratios 1. Fo each of the following: a. sketch the standad position angle b. detemine the elated acute angle c. detemine the exact value of the specified tig atio i) cos 3 ii) 7 sin 4 iii) 5 tan 6. Detemine the exact value of the following. Include a clealy labeled sketch showing the angle in standad position. 3 a) sin b) sec 3. Evaluate, accuate to fou decimal places. Be sue to set you calculato in adians. a) sin b) cot 10 5 c) sec

20 MHF 4UI Unit 6 Day 6 C. Solving Tigonometic Equations 1. Solve fo, accuate to two decimal places, whee θ [ 0, π ] a) sin θ b) tan θ Solve fo x, x [ 0, π ]. State exact answes. sin x 3

21 MHF 4UI Unit 6 Day 7 Solving Tigonometic Equations 1. Solve fo, accuate to two decimal places whee θ [ 0, π ]. a) cos θ b) 3sin (θ - ) 1 1 π Note: To extend this question to all possible angles, we can ceate co-teminal angles by adding multiples of the peiod. fo θ R, fo θ R,

22 MHF 4UI Unit 6 Day 7. Solve fo x, 0 x π. State exact answes. a) 1 π tanx - b) cos (x - ) 1 3 3

23 MHF 4UI Unit 6 Day 8 Tig Applications π 1. Given d 4.8sin (t ) 5. 8, t 0. a) Detemine the amplitude, peiod, phase shift and vetical shift. b) Detemine the maximum and when it occus, in one peiod. c) Detemine the maximum and when it occus, fo all t whee t 0.

24 MHF 4UI Unit 6 Day 8 d) Detemine the minimum and when it occus, in one peiod. e) Detemine the minimum and when it occus, fo all t whee t 0. f) Detemine d when t=13, accuate to one decimal place.

25 MHF 4UI Unit 6 Day 8. A small windmill has its cente 6 m above the gound and blades m long. In a steady wind, a point P at the tip of one blade makes a complete evolution in 1 seconds. a) Use this infomation to sketch the function ove a 1 second inteval. Assume the otation stats at the highest possible point. b) Detemine a function that gives the height of P above the gound at any time t. c) Detemine the height of the blade at 5 seconds. State the EXACT answe, then ound the answe to one decimal place.

26 MHF 4UI Unit 6 Day 9 Moe Tig Applications 1. In the Bay of Fundy, the wate aound the habou changes fom 1.5 m at low tide at 0:00 h to 15.5 m at high tide at 08:00 h. a. If the tidal cycle is sinusoidal, detemine a function to epesent the depth of the wate in the habou. b. Detemine the depth of wate in the habou at 04:30 h, coect to one decimal place.

27 MHF 4UI Unit 6 Day 10 UNIT #6 SUMMARY: Tigonomety Pat I Radians Degees Special tiangles in adians Fo y = sin, y = cos, y = tan y = csc, y = sec, y = cot State: peiod zeoes geneal fomula domain ange equation of any vetical asymptotes, (if they exist) Sketch Given a point i.e. (-1, 3) OR Given a tig atio i.e. Sketch the angle, clealy identifying the angle Find the exact value of all tig atios 7 Given an angle i.e. 6 sin 5 Sketch the angle Find the RAA Use CAST and special tiangles to find the exact value of all tig atios fo this angle Solving Tig equations Let statement intoduce new vaiable Adjust inteval Find RAA Use CAST Ae answes within inteval? Find oiginal vaiable Conclusion Geneal conclusion

28 MHF 4UI Unit 6 Day 10 Given an equation find value of x when function eaches a max imum minimum state amplitude, peiod, phase shift and vetical shift peiod k sketch max VS ampl min VS ampl state domain and ange Given infomation in wods, state equation. Infomation could be of the fom: amplitude, peiod, phase shift and vetical shift maximum point and minimum point Given sketch, state equation max min ampl max min VS peiod, solve fo k k Applications: Wind mill Feis wheel Bike pedal Tide

Name Date. Trigonometric Functions of Any Angle For use with Exploration 5.3

Name Date. Trigonometric Functions of Any Angle For use with Exploration 5.3 5.3 Tigonometic Functions of An Angle Fo use with Eploation 5.3 Essential Question How can ou use the unit cicle to define the tigonometic functions of an angle? Let be an angle in standad position with,