trigonometry standard position and radians

MHF 4UI Unit 6 Day 1

x

y

x

y

initial arm

terminal

arm

x

y

O A

B

x

y

Trigonometry – Standard Position and Radians

A. Standard Position of an Angle

Angle is in standard position when the initial arm is the

positive x-axis and the vertex is at the origin. A positive

angle rotates in the counter-clockwise direction.

So far we have only measured angles in degrees. We will

now measure angles in radians

B. Radian Measure

In the circle, O is the center. is the angle subtended

at the center of the circle by an arc AB.

r

a

radius

lengtharcθ

When a = r

radian1r

r

r

aθ

When a = 2r

radians 2r

r2

r

aθ

When a = 3r

radians3r

3r

r

aθ


What is the radian measure of one complete revolution?

Conversion factor:

rad1 or 1

1. Convert each of the following to degrees.

a) rad2

π b) rad

6

5π c) r2.53

2. Convert each of the following to radians. Give exact values, then round to 4 decimals.

a) 45° b) 210° c) 312°


Sine Function

12

6

4

3

5

12

2

7

12

2

3

3

4

5

6

11

12

sin 0.966 0.866 0.707 0.5 0.259 0

13

12

7

6

5

4

4

3

17

12

3

2

19

12

5

3

7

4

11

6

23

12

2

sin -0.259 -0.5 -0.707 -0.866 -0.966 -1 -0.966 -0.866 -0.707 -0.5 -0.259 0

1

2

-1

2

2

-2


Cosine Function

12

6

4

3

5

12

2

7

12

2

3

3

4

5

6

11

12

cos 0.966 0.866 0.707 0.5 0.259 0 -0.259 -0.5 -0.707 -0.866 -0.966 -1

13

12

7

6

5

4

4

3

17

12

3

2

19

12

5

3

7

4

11

6

23

12

2

cos -0.966 -0.866 -0.707 -0.5 -0.259 0 0.259 0.5 0.707 0.866 0.966 1

1

2

2

2


x

y

Graphing Reciprocal Trigonometric Functions

The function sketched below is f x =

Graph the reciprocal of the function shown below:

Clearly indicate any vertical asymptotes.

Clearly mark any value(s) which are the same on f x and the reciprocal of f x .

Using the big / little property, sketch the reciprocal of f x

If g x is the reciprocal of f x , write its equation two different ways.

g x = and g x =

Properties of

Vertical Asymptotes: ___________________________________

Domain: __________________________________________

Range: __________________________________________

Period: _________________


x

y







g x = and g x =

Properties of


Domain: __________________________________________

Range: __________________________________________

Period: _________________


x

y







g x = and g x =

Properties of


Domain: __________________________________________

Range: __________________________________________

Period: _________________

x2

3x

2


Tangent Function

12

6

4

3

5

12

2

7

12

2

3

3

4

5

6

11

12

tan 0.270 0.577 1.732 3.73 -3.73 -1.732 -0.577 -0.270

13

12

7

6

5

4

4

3

17

12

3

2

19

12

5

3

7

4

11

6

23

12

2

tan 0.270 0.577 1.732 3.73 -3.73 -1.732 -0.577 -0.270

1

5

-5

-3

2

2

3

4

-1

-2

-4


Properties of Trigonometric Functions

period

zeroes characteristics

minimum:

maximum:

asymptotes:

sine

y-intercept period


minimum:

maximum:

asymptotes:

cosine

y-intercept period


minimum:

maximum:

asymptotes:

tangent

y-intercept

period


minimum:

maximum:

asymptotes:

cosecant

y-intercept period


minimum:

maximum:

asymptotes:

secant

y-intercept period


minimum:

maximum:

asymptotes:

cotangent

y-intercept


Transformations of Sine and Cosine Functions

Basic transformations:

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 = 2

min - max

radk

2π

k

360period

vertical shift = 2

min max ; if q > 0, shift up q units

if q < 0, shift down q units

phase shift: if p > 0, shift right p units; (x – p)

if p < 0, shift left p units; (x + p)

reflections: if a < 0, reflect in the x – axis; (vertical reflection)

if k < 0, reflect in the y – axis; (horizontal reflection)


1. Sketch one period of each of the following. Also state the domain and range. Plot a

minimum of 5 ordered pairs of the function.

a) y = 3cos(2x)

a = period = v.s. = p.s. =

D = __________ R = ____________________

b) y = -sin(3x) + 4


D = __________ R = ____________________


c) y = 1.5cos(x - 3

π) + 2


D = __________ R = ____________________


More Transformations of Sine and Cosine Functions

1. Sketch one period of each of the following. Plot a minimum of 5 ordered pairs.

a) y = -4 sin [ 2(x -3

π)] + 1


b) Determine the equation of a cosine function with a maximum value of 20, amplitude 8,

period 3π and a phase shift 4

π .


c) A sine function on the interval ),[0x has its first maximum point at (4

π, 4) and

its first minimum point at (12

7π, -2). Determine a possible sine equation.

d) Repeat c) with a cosine function.


Trigonometric Ratios

2 2 2x y r

Primary trig ratios Reciprocal trig ratios

1. P 6,3 is a point on the terminal arm of an angle in standard position where r0 2 .

Determine the exact values of sin , cos and tan . Include a clearly labelled sketch.

The CAST rule confirms the sign of our answers.

P(x,y) r


2. is a standard position angle in quadrant III such that 2

cos3

. Determine the exact

value of csc . Include a clearly labelled sketch.

3. is a standard position angle such that 1

tan4

. Determine the exact value of sin .

Include a clearly labelled sketch.


Sketching Special Angles

2

π multiples:

4

π multiples:

3

π multiples:

6

π multiples:

0 r

0 r

0 r

0 r


Special Triangles and Trigonometric Ratios

A. Special Triangles

Recall from last year:

Now, using radian measure:

30 60 45

Similarly, evaluating trig ratios in degrees using the CAST rule can also be accomplished in

radians.

30

60

45

45

3 2

1

2

1

1

3

2

1

2

1

1


B. Evaluating Trigonometric Ratios

1. For each of the following:

a. sketch the standard position angle

b. determine the related acute angle

c. determine the exact value of the specified trig ratio

i) r2

cos3

ii)

r7sin

4

iii)

r5tan

6

2. Determine the exact value of the following. Include a clearly labeled sketch showing the

angle in standard position.

a) rsin b) r3

sec2

3. Evaluate, accurate to four decimal places. Be sure to set your calculator in radians.

a) r

sin10

b)

r2cot

5

c) rsec2


C. Solving Trigonometric Equations

1. Solve for , accurate to two decimal places, where ]2π ,0 [θ rr

a) 0.5073θ sin b) -3.9782θ tan

2. Solve for x, ]2π ,0 [x rr . State exact answers.

2

3x sin


Solving Trigonometric Equations

1. Solve for , accurate to two decimal places where ]2π ,0 [θ rr .

a) -0.2534θ cos b) 11) - (θ 3sin2

π

Note: To extend this question to all possible angles, we can create co-terminal angles by

adding multiples of the period.

for R θ , for R θ ,


2. Solve for x, 2π x0 . State exact answers.

a) 3

1-tanx b) 1) - (x 2cos

3

2π


Trig Applications

1. Given 5.22)(t8

π4.8sind

, 0t .

a) Determine the amplitude, period, phase shift and vertical shift.

b) Determine the maximum and when it occurs, in one period.

c) Determine the maximum and when it occurs, for all t where 0t .


d) Determine the minimum and when it occurs, in one period.

e) Determine the minimum and when it occurs, for all t where 0t .

f) Determine d when t=13, accurate to one decimal place.


2. A small windmill has its centre 6 m above the ground and blades 2 m long. In a steady wind, a

point P at the tip of one blade makes a complete revolution in 12 seconds.

a) Use this information to sketch the function over a 12 second interval. Assume the

rotation starts at the highest possible point.

b) Determine a function that gives the height of P above the ground at any time t.

c) Determine the height of the blade at 5 seconds. State the EXACT answer, then round

the answer to one decimal place.


More Trig Applications

1. In the Bay of Fundy, the water around the harbour changes from 1.5 m at low tide at 02:00 h

to 15.5 m at high tide at 08:00 h.

a. If the tidal cycle is sinusoidal, determine a function to represent the depth of the water

in the harbour.

b. Determine the depth of water in the harbour at 04:30 h, correct to one decimal place.


UNIT #6 SUMMARY: Trigonometry Part I

Radians Degrees

Special triangles in radians

For y = sin , y = cos , y = tan

y = csc , y = sec , y = cot

State: period

zeroes

domain

range

equation of any vertical asymptotes, (if they exist)

Sketch

Given a point i.e. (-1, 3) OR Given a trig ratio i.e. 5

2sin

Sketch the angle, clearly identifying the angle

Find the exact value of all trig ratios

Given an angle i.e. 6

7

Sketch the angle

Find the RAA

Use CAST and special triangles to find the exact value of all trig ratios for this angle

Solving Trig equations

Let statement – introduce new variable

Adjust interval

Find RAA

Use CAST

Are answers within interval?

Find original variable

Conclusion

General conclusion

general

formula


Given an equation

find value of x when function reaches a

imummin

imummax

state amplitude, period, phase shift and vertical shift

2

periodk

sketch

amplVS max

amplVS min

state domain and range

Given information in words, state equation. Information could be of the form:

amplitude, period, phase shift and vertical shift

maximum point and minimum point

Given sketch, state equation

2

minmaxampl

2

minmaxVS

2period

k

, solve for k

Applications:

Wind mill

Ferris wheel

Bike pedal

Tide

trigonometry standard position and radians

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