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Unit 4: Trigonometry I (Day 1)
Angles in Standard Position
Consider a point P (x, y) moving around a circle drawn on a grid with center
(0,0) and radius r . We wish to measure the angle between the line joining P to
the origin and the positive x-axis. We say that this angle is in standard position.
Some conventions for angles measured in standard position:
1. The positive x axis is _____
2. Angles have their vertex at _______
3. Positive angles are measured ___________________ (ccw)
4. Negative angles are measured ________________(cw)
5. The arm on the positive x-axis is called the _______________
6. The other arm is called the _______________
Consider a circle of radius 1 (called a unit circle).
Degree Measure
The measure you are familiar with is degrees:
- One full rotation is 360
- 1
360 of a rotation has a measure of 1 degree
Draw:
i) 60 o ii) -40 o iii) -490 o
Radian Measure
Definition:
1 radian is the measure of an angle containing an arc of length r in a circle of
radius r.
A circle has 360 . The circumference of a circle is 2C r=
When 1r = → 2C =
Then one complete rotation about the unit circle is an arc length of 2 for
every 360 .
So…… 2 radians 360o = or 180o =
Draw:
iv) 3
v)
4
vi)
4
3−
vii) 6
5 viii)
6
7 ix)
3
5−
Conversion Factors:
Degrees → Radians: Multiply by o180
Radians → Degrees: Multiply by
o180
Examples:
Convert from degrees to radians:
i) 30o ii) 90o
iii) 45o iv) 110o
v) 360o v) 270o
* We often leave radians in terms of
** The units for radians are either: ‘radians’ or nothing.
If an angle measures 2 then we are working in radians.
Convert from radians to degrees.
i) 1 radian ii) 5.6 radians
iii) 6
5 iv)
4
3
Coterminal Angles:
Two or more angles in standard position with the same P position.
Find two angles coterminal with 40 o:
Find two angles coterminal with 4
:
Given any angle , any coterminal angle is represented by:
In Degrees:
In Radians:
Examples:
a) Draw the given angle
b) draw 2 coterminal angles and find their measurements
i) o150= b) 3
−
Finding the length of an arc in a circle of radius r:
1. Determine the fraction of the circle being worked with.
2. Multiply this by the circumference
Degrees: Arclength = rao
2360
=
Radians: Arclength ra
2
2=
Examples:
i) A circle has radius 8 cm. Calculate the length of the arc subtended by each
angle:
a) 2.3 radians b) 75o
ii) A circle has radius 5 cm. Find the angle at the center containing an arc length
of:
a) 6 cm (in degrees) b) 15 cm (in radians)
Homework:
Workbook p. 255 #1,4-11
WS 4-1 and 3.3
Unit 4: Trigonometry I (Day 2)
Sine and Cosine in Standard Position
Recall:
Given a right triangle, there are 3 primary trig ratios and 3 secondary
trig ratios:
Sine:
sin opposite
hypotenuse =
Cosecant:
1csc
sin
hypotenuse
opposite
= =
Cosine:
cos adjacent
hypotenuse =
Secant:
1sec
cos
hypotenuse
adjacent
= =
Tangent:
tanopposite
adjacent =
Cotangent:
1cot
tan
adjacent
opposite
= =
These definitions are for angles 0 90 . We need to expand our
definitions for 90 .
Consider P(x, y) a point on terminal side of , in a circle of radius r.
The values of ,x y and r determine the six trig ratios for angle :
sin = csc =
cos = sec =
tan = cot =
The quadrant in which is found will determine whether the trig ratio
is /+ − :
Ex:
i) In which quadrant is sin < 0 and tan > 0?
ii) In which quadrant is cos > 0 and tan <0?
Ex:
i) Determine cos if 41
4sin −= and tan 0
ii) Determine sin and cos if is an angle in standard position
whose terminal arm side is the graph 2 5 0, 0x y x+ =
iii) Given the point (1, 2)− on the terminal arm of an angle in
standard position, determine the value of all 6 trig ratios.
p. 262 # 1-3,5,6(a-d),7
WS 4-5
Unit 4: Trigonometry I (Day 3)
General and Special Angles
Learning
Intention(s): Determine the ratio of special angles
Solve a primary trig ratio for an unknown angle in both degrees and radians
Special cases: 0, 90, 180, 270, …
0 90 180 270
sin
cos
tan
There are 2 right triangles with ‘nice’ angle and side values.
45 o - 45 o - 90 o 30 o - 60 o - 90 o
0 o 30 o 45 o 60 o 90 o
sin
cos
tan
0 6
4
3
2
Reference Angles:
The acute angle formed by the terminal arm and the x-axis.
Examples: Find the reference angles for the following.
i) o220= ii) o410−=
iii) 3
2 iv)
6
5−
v) 4
7 vi)
12
7−
Determining the exact value of trig functions given angles in both degrees and radians:
1. Determine the reference angle
2. Determine the quadrant
Examples: Determine the following as exact values:
i) sin 240𝑜 ii) cos 120𝑜
iii) 4
5tan
− iv)
6
11cos
v) 13tan vi) 2
15sin
vii) 4
29tan
viii)
3
7cos
In reverse: Solve for if 3600
2
1cos = 2sin2 =
3
3tan −=
Solve for if 20
3sin2 = 2
2cos −=
Homework:
6.3 # 1-7odd, 11
Unit 4: Trigonometry (Day 4)
Graphing the Primary Trig Functions
We can graph in either degrees or radians. Radians are generally easier to count.
1. sin)( =f
0 6
2
65
67
23
611 2
sin
Properties of sin:
1. Domain 2. Range
3. Period 4. Amplitude
5. -intercepts: 6. y-intercept:
7. Maximum Value: _______ At what values of does the maximum occur? _________
8. Minimum Value: _______ At what values of does the minimum occur? __________
2. cos)( =f
0 3
2
32
34
23
35 2
cos
Properties of cos:
1. Domain 2. Range
3. Period 4. Amplitude
5. -intercepts: 6. y-intercept:
7. Maximum Value: _______ At what values of does the maximum occur? _________
8. Minimum Value: _______ At what values of does the minimum occur? __________
3. tan)( =f
0 4
2
43
45
23
47 2
tan
Properties of tan:
1. Domain:
2. Range:
3. Period:
4. -intercepts:
5. y-intercept:
6. Asymptotes:
Homework:
Practice sketching 1 period of the Sine, Cosine and Tangent functions.
Unit 4: Trigonometry (Day 5)
Graphing y = asin(x – p) + q and y = acos(x – p) + q
a : Expands/Compresses the graph vertically
p: Translates the graph left/right *also called the phase shift
q: Translates the graph up/down *also called the vertical displacement
Examples: Graph the following, then state the given properties
i) 1)sin(33+−= xy
Domain: Range:
Amplitude: Period:
ii) 2)cos(26++= xy
Domain: Range:
Amplitude: Period:
Phase Shift: Vertical Displacement:
iii) 1)sin(22−+−= xy
Domain: Range:
Amplitude: Period:
Example:
The following is a graph of a sine function. Write a possible equation.
The above could also be a cosine function. Write a possible equation.
Graphs of this shape are said to be sinusoidal. The equation of a
sinusoidal can be either sine or cosine.
Homework: WS (p. 245) #1, 3, 14, 15, 18, 19adgjmn, 25ade
Unit 4: Trigonometry (Day 6)
Graphing y = a sin b (x – p) + q
b: Expands/Compresses the graph horizontally by a factor of 1/b.
For a sin/cos graph, the b value will affect the period. Period b
T2
=
Examples: Graph the following:
i) xy21sin=
ii) xy 2cos=
iii) 3)(3sin26−−= xy
iv) 1)cos(342
1 ++−= xy
Domain: Range:
Amplitude: Period:
v) The following is a cosine function. Write a possible equation.
WS (p. 254) #1, 2, 3d, 7bc, 9bf, 11cd, 12bd, 14, 15, 17ad
(answer to 3d is 4 to the right)
Unit 4: Trigonometry (Day 7)
Graphing xyp2sin= and xy
p2cos=
Recall: The period of sine/cosine is T = b
2
Examples: Determine the period, T. Then sketch the graph
i) xy5
2sin =
ii) xy1002cos =
iii) 52sin34
)2(−=
−xy
Example:
The volume of air in one’s lungs is a sinusoidal function of time. A graph showing normal
breathing is shown below. Determine the equation for this function:
Homework:
WS (p. 265) #1-3, 5ad, 7ab, 14cd, 15ab, 20
2700
2200
Time (s) 0 5 10 15
Volume (mL)
Unit 4: Trigonometry (Day 8)
More Graphing y = a sin b (x – p) + q
Graph 1)tan(34−−= xy
Period: Asymptotes:
Domain: Range:
Unlike sine and cosine, xy tan= has a period of:
The period of bxy tan= is therefore:
Graph 1)(tan421 +−= xy
Period: Asymptotes:
Domain: Range:
Graph: )1(4
tan3 −= xy
Domain: Range:
Graph 1)cos(342
1 ++−= xy
Domain: Range:
Amplitude: Period:
The following is a cosine function. Write a possible equation.
i) Write the equation for a sine function with amplitude 6, period 8,
phase shift -5 and vertical displacement 10
ii) Write the equation of the cosine function with maximum 18,
minimum 4, period 6 and phase shift 5
iii) Given 2)3(2cos5 −+−= xy
Determine the following:
Domain Range Period
Maximum value Minimum value
and when it first occurs and when it first occurs
iv) Write the equation of the sinusoidal function with a maximum of 8 at
when x is 2 and followed by a minimum of 1 when x is 9.
Homework:
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