trigonometry standard position and radians
TRANSCRIPT
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
MHF 4UI Unit 6 Day 1
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°
MHF 4UI Unit 6 Day 1
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
MHF 4UI Unit 6 Day 1
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
MHF 4UI Unit 6 Day 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: _________________
MHF 4UI Unit 6 Day 2
x
y
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: _________________
MHF 4UI Unit 6 Day 2
x
y
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: _________________
x2
3x
2
MHF 4UI Unit 6 Day 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
MHF 4UI Unit 6 Day 2
Properties of Trigonometric Functions
period
zeroes characteristics
minimum:
maximum:
asymptotes:
sine
y-intercept period
zeroes characteristics
minimum:
maximum:
asymptotes:
cosine
y-intercept period
zeroes characteristics
minimum:
maximum:
asymptotes:
tangent
y-intercept
period
zeroes characteristics
minimum:
maximum:
asymptotes:
cosecant
y-intercept period
zeroes characteristics
minimum:
maximum:
asymptotes:
secant
y-intercept period
zeroes characteristics
minimum:
maximum:
asymptotes:
cotangent
y-intercept
MHF 4UI Unit 6 Day 3
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)
MHF 4UI Unit 6 Day 3
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
a = period = v.s. = p.s. =
D = __________ R = ____________________
MHF 4UI Unit 6 Day 3
c) y = 1.5cos(x - 3
π) + 2
a = period = v.s. = p.s. =
D = __________ R = ____________________
MHF 4UI Unit 6 Day 4
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
a = period = v.s. = p.s. =
b) Determine the equation of a cosine function with a maximum value of 20, amplitude 8,
period 3π and a phase shift 4
π .
MHF 4UI Unit 6 Day 4
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.
MHF 4UI Unit 6 Day 5
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
MHF 4UI Unit 6 Day 5
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.
MHF 4UI Unit 6 Day 5
Sketching Special Angles
2
π multiples:
4
π multiples:
3
π multiples:
6
π multiples:
0 r
0 r
0 r
0 r
MHF 4UI Unit 6 Day 6
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
MHF 4UI Unit 6 Day 6
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
MHF 4UI Unit 6 Day 6
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
MHF 4UI Unit 6 Day 7
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 ,
MHF 4UI Unit 6 Day 7
2. Solve for x, 2π x0 . State exact answers.
a) 3
1-tanx b) 1) - (x 2cos
3
2π
MHF 4UI Unit 6 Day 8
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 .
MHF 4UI Unit 6 Day 8
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.
MHF 4UI Unit 6 Day 8
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.
MHF 4UI Unit 6 Day 9
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.
MHF 4UI Unit 6 Day 10
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
MHF 4UI Unit 6 Day 10
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