Unit 3 – Polynomial Functions, Equations and...

Chapter 6 – Trigonometry

Name: _______________

Chapter 6 – Trigonometry ● 1

6.1 Trigonometric Ratios for Any Angle in Standard Position Date:

Standard Position: An angle is in standard position when the initial arm starts at (0, 0) and lies on the positive x-axis; the terminal arm is rotated about (0, 0).

Angles in standard position are graphed on the same plane as functions, and we label the four sections as quadrants:

Example 1: For each angle, draw the angle in standard position:a) b)

Suppose the ray continues to rotate. When the ray first reaches its starting position it forms an angle of ______ (if rotating counter-clockwise) and form an angle of ______ (if rotating clockwise). After rotating all around until a full cycle, you will end on the same position as standard position, this new angle will be either greater than 360° or less than −360°.

Example 2: For each angle in example 1, determine the angle after a full cycle around.

When a ray rotates more than 1 revolution, it forms angles greater than 720° (when rotating counter-clockwise) and form angles less than −720° (when rotating clock-wise).

Example 3: For each angle in example 1, determine the angle after 2 full cycles around.

You can continuously adding or subtracting 360° and you will end up with the same terminal forever. These values are called coterminal angles.

Coterminal Angles: Two or more angles in standard position, when the position of P is the same for each angle. When θ represents any angle, then any angle coterminal with θ is represented by these expressions, where n is any integer.

, where (natural number)

Principles of Mathematics 12

III

III IV

2 ● Chapter 6 – Trigonometry

You Try…Identify the angles coterminal with 210° that satisfy the domain −720° ≤ θ ≤ 720°. Express the angles coterminal with 210° in general form.

Reference AnglesDefinition: Consider the angle θ in standard position. The reference angle of θ is ______________

_______________________________________________________.

Sketch the reference angle, θr, of each angle θ.


y

xθ

y

x

θ

y

xθ

y

x

θ


Unit Circle

The unit circle is simply a circle whose radius equals _______.

For our purposes, we will place the centre of a unit circle at the origin of the Cartesian plane.

Relating tan θ to sin θ and cos θSince and for the unit circle. Then

Special Angles:

Recall the two special triangles:

Definition of Secant, Cosecant, and Cotangent

In mathematics, the ratios , , (reciprocal) are used so often that we give them special names.

We define the secant, cosecant, and cotangent functions are follows:


x

y

P(x, y)

x

y


Secant of θ Cosecant of θ Cotangent of θ

Example 4: Determine the trigonometric ratio for each degree

θ in degrees 0° 30° 60° 90° 45° 180° 270° 360°

sin θ

cos θ

tan θ

sec θ

csc θ

cot θ

Example 5: Determine the exact value of the six trigonometric ratios of 495°.


y

x

5

–5–10 105

10

–10

–5


You Try… Determine the exact value of the six trigonometric ratios of −150°.

Determining Trigonometric Ratios of non-special triangles.Example 6: Determine the exact value of the six trigonometric ratios of 200°.

Determining All Trigonometric Ratios and Angle Measure given one Trigonometric Ratio.

Example 7: a) Suppose . Determine the exact values for the other trigonometric ratios for

.

b) To the nearest degree, determine the possible θ in the domain .



You Try… Suppose . Determine the exact values for the other trigonometric ratios for . To the nearest degree, determine the possible θ in the domain .

Determining All Trigonometric Ratios and Angle Measure given a point.Example 7: The terminal arm of an angle θ contains the point P(–2, –1).

a. Sketch θ (and its reference angle θr) on the grid below.

y

x

1

–1–2 21

2

–2

–1

Note that by Pythagoras

, ,

, ,

b. Compute the sine and cosine of θ as well as the sine, cosine and tangent of θr.

c. Find θ to the nearest tenth of a degree.

Assignment: page 474- #3-11; page 480 #1-2



6.3 Radian Measure & 6.2 Angles in Standard Position and Arc Length Date:

A radian is a unit for measuring angles. Radians are different than degrees and in many situations, more useful.

Radian Measure

In a circle with radius r:

A central angle of 1 radian is subtended by an arc length r.

The relationship between Radians and Degrees

______ radians is equal to ______ degrees.

Converting Radians to Degrees Converting Degrees to Radians

Important: any angle measure given without a unit is assumed to be in radians.

Example 1: Convert each of the following to radians:a) 30° b) 60° c) 45° d) 225°


r ___________________________

θ ___________________________

a ___________________________

r

aθ


Example 2: Convert each of the following to degrees:

e) f) g) h) 2

Revising Coterminal anglesRecall coterminal angles are angles that share the same terminal arm location. Coterminal angles can be determine with the formula , for angles in degrees. A similar formulas can be used to find coterminal angles in radians, .

Example 3: Determine all the coterminals for in the domain .

The diagram below shows the exact coordinate points on the unit circle that lie on the terminal arms of a certain SPECIAL angles in standard position, measured in degrees and radians. Note the diagram is only for degrees between 0° and 360°. For negative and other angles, you would need to find the corresponding angles that lie between 0° and 360°.



Finding Arc LengthNote the ratio: _______________ = ______________

Equation:

**Common Mistake: if θ is given in degrees, you MUST change to radians before using the equation**

Example 4: Determine the arc length in a circle of a radius 10 cm if:a) The central angle is 5 radians.

b) The central angle is 25°.

Example 5: Determine the central angle (in radians) subtended by an arc of length 3 cm in a circle of radius 10 cm.

Determining the Trigonometric Ratios of Angles in Radians

Example 6: Determine the exact values of


r

aθ


Determining Angle Measures Given a Terminal Point

Example 7: The terminal arm of an angle θ contains the point P(–4, 3). To the nearest tenth of a radian, determine the possible values of θ in the domain .

Solving a Problem Involving Radians and Trigonometric Ratios

Example 8: An approximate model of the motion of the International Space Station is that it travels 27 600 km/h in a circular orbit at an altitude of 400 km. To the nearest tenth of a radian, determine the angle between the initial and final position of the station if the station have rotated after 40 minutes.

Assignment: page 485-486 #1-4; page 494-500 #4-15; page 501 #1, 2



6.4 Graphing Trigonometric Functions Date:

Vocabulary and Definitions

All graphs of trigonometric functions are periodic; that is, they complete a cycle over an interval of the domain, and repeat that cycle indefinitely.Certain features of these graphs are given special names. Label the graph with the appropriate name:CyclePeriodAmplitude

Graphs that have the shape shown are called sinusoidal.

Define the following properties of a periodic function:

Cycle:

Period:

Amplitude:

Sinusoidal:

Investigate 1. First, complete the tables below for the functions , , and . Then sketch the three graphs on

the grids provided: Label five points on each of the graphs. Since the curves are sinusoidal, the lines joining the points are curved.



SINE GRAPH

θRad 0

Deg 0○ 30○ 60○ 90○ 120○ 150○ 180○ 210○ 240○ 270○ 300○ 330○ 360○

x

y

COSINE GRAPH

θRad 0

Deg 0○ 30○ 60○ 90○ 120○ 150○ 180○ 210○ 240○ 270○ 300○ 330○ 360○

x

y



TAGENT GRAPH

θRad 0

Deg 0○ 30○ 45○ 60○ 90○ 120○ 135○ 150○ 180○

θRad

Deg 210○ 225○ 240○ 270○ 300○ 315○ 330○ 360○

Properties of the Function y = sin x, y = cos x, and y = tan x

Function y = sin x y = cos x y = tan x

Period

Maximum

Minimum

Amplitude

Domain

Range

x-intercept

y-intercept

Assignment: page 512 #1-4



6.5 Trigonometric Functions Date:

The graphical transformations, such as shifting and stretching also apply to trigonometric graphs.

Determining the Amplitude of a Trigonometric Function

For trigonometric functions in the form y = asin x, y = acos x, and y = atan x

a is the transformation, ____________________________________________

a represents _____________________________________________________

If a is negative, the graph will be ______________________________________

If |a| > 1, there will be a _____________________________________________

If 0 < |a| < 1, there will be a __________________________________________

Example 1: Graph the following functions, and determine the amplitude, period, domain, range, asymptote , zero, maximum & minimum.

Functions Graph InfoAmplitude

Period

Domain

Range

Maximum

Minimum

Zeros

Period

Domain

Range

Asymptote

Zeros


y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4

y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4


Hint: First consider the

graph of

Amplitude

Period

Domain

Range

Maximum

Minimum

Zeros

Determining the Period of a Trigonometric Function

For trigonometric functions in the form y = sin bx, y = cos bx, and y = tan bx

b is the transformation, ____________________________________________

b represents ____________________________________________________

If b is negative, the graph will be ______________________________________

If |b| > 1, there will be a _____________________________________________

If 0 < |b| < 1, there will be a __________________________________________

The function of y = sin bx and y = cos bx have period of ______________________

The function of y = atan x have period of __________________________________** Common Mistake: stating the period as the ‘b’ value in a trigonometric equation. **

Example 2: Without using the graphing calculator, graph the following functions for –2π ≤ x ≤ 2π, and determine the amplitude, period, domain, range, asymptote , zero, maximum & minimum.

Function Graph Info

Amplitude

Period

Domain

Range

Maximum

Minimum

Zeros


y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4

y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4


Amplitude

Period

Domain

Range

Maximum

Minimum

Zeros

Period

Domain

Range

Asymptote

Zeros

Determining the Phase Shift of Trigonometric Function

For trigonometric functions in the form y = sin(x − c), y = sin(x − c) and y = tan(x − c)

c is the transformation, ____________________________________________

c represents ____________________________________________________

If x − c, the graph will be ____________________________________________

If x + c, the graph will be ____________________________________________

c can be looked as _________________________________________________y

x

c

y = sin xy = sin (x –


y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4

y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4


Example 3: Without using the graphing calculator, graph the following functions for –2π ≤ x ≤ 2π, and determine the amplitude, period, domain, range, asymptote , zero, maximum & minimum.


Period

Domain

Range

Maximum

Minimum

ZerosAmplitude

Period

Domain

Range

Maximum

Minimum

Zeros

Period

Domain

Range

Asymptote

Zeros


y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4

y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4

y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4


Determining the Vertical Displacement of Trigonometric Function

For trigonometric functions in the form y = sin x + d, and y = cos x + d and y = tan x + d

d is the transformation, ____________________________________________

d represents _____________________________________________________

If d is positive, the graph will be ______________________________________

If d is negative, the graph will be ______________________________________

d can be looked as _________________________________________________

y

x2π–π

2π–π–3π

2 23π 2π–2π

Maximum

Minimum

SinusoidalAxis

d

Finding maximum and minimum (sin and cos only):a) The maximum value of the function = _______________________________

b) The minimum value of the function = ________________________________

Example 4: Without using the graphing calculator, graph the following functions for –2π ≤ x ≤ 2π, and determine the amplitude, period, domain, range, asymptote, maximum & minimum.


Period

Domain

Range

Maximum

Minimum


y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4


Amplitude

Period

Domain

Range

Maximum

Minimum

Period

Domain

Range

Asymptote


y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4

y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4


Assignment: page 521-526 #3-10; page 526 #1, 2

6.6 Combining Transformations of Sinusoidal Functions Date:

Example 1: (Method 1)

Sketch the graph . Describe how the graph changes with each transformation.

Complete the table below.

The graph of is shown below.On the same grid, sketch a graph of

On the second grid, sketch the graph of: (i) , (ii) , (ii)

Equation of Function

Characteristics

Period

Amplitude



Domain

Range

Phase Shift

Zeros

Example 2: (Method 2) Without using the graphing calculator, sketch a graph of

for –2π ≤ x ≤ 2π. Label 5 points on the graph.



x–1

–2

–3

–4

2π–π

2π–π–2π –3π

2 23π 2π

3

1

4

2

y

You Try…Without using the graphing calculator, sketch a graph of

for –2π ≤ x ≤ 2π. Label 5 points on the graph.

x–1

–2

–3

–4

2π–π

2π–π–2π –3π

2 23π 2π

3

1

4

2

y

Writing Equations for Sinusoidal Graphs

For sinusoidal graphs, they can be either sine or cosine functions. There are infinite possibilities. The equation will look similar, however they will differ by only a reflection and/or phase shift.

Example 3: Write one sine equation and one cosine equation to match the graph below.



y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4

You try… Write a sine equation and a cosine equation to match the graph below.



y

x–1

–2

–3

–4

2π–π

2π–π–3π

2 23π 2π–2π

1

2

3

4

Assignment: page 534-539 #3-11; page 539-540 #1, 26.7 Applications of Sinusoidal Functions Date:

Periodic data that appears to resemble a sinusoidal curve when plotted can often be modeled by a sine or cosine function. These trigonometric functions can then be used to model real-life situations.

In many real-life applications the data appears periodic and repeats after 12 months.

a) Find the value of b in the equation y = sin bx if the period equals 12.

b) Use the value for b found in part a) and insert it in the equation y = sin bx: _______________



c) If the equation in part b) is written in the form, , ___ has a value of ____. This value will

also be the __________ for the graph of ____________.

Summary

If b (in y = sin bx and y = cos bx) is in the form , then the parameter ‘p’ will represent the _______ of the

function.

Example 1: Predict the period of the following trigonometric function. Graph the functions, and use the fact

that the period equals to check.

Trig Function Period Grapha)

x–1–2

–1

–3–4–5

1

1 3 4 52

y

b)

x–1–2

–1

–3–4–5

1

1 3 4 52

y

Example 2: Write a trigonometric equation to match the graph below:



x–1–2

–1

–3–4–5

1

1 3 4 52

y

Example 3: List the values of each property of the trigonometric function and graph

the function.

Reflection

x–1

6

8

4

2

1 3 4 52 6 8 9 10 117

y

Amplitude

Period

Phase Shift

Vertical Displacement

SummaryThe general formulas for sine and cosine functions used to model real-life situations now be expanded to included transformations. They are:

and where

Amplitude:

Period:

Phase shift:

Vertical displacement:



Tidal Waves and Temperature (where maximum and minimum is given)Using the general sinusoidal equation: Where a is the amplitude, b is 2π divided by the period, c is the phase shift, and d is the vertical displacement. Problems that model periodic data can be solved.

Example 4: The average monthly temperature of a location in Northern BC as a function of the month number can be modelled using the equation y = acosb(x – c) + d. The highest average monthly temperature is 20°C and the lowest average monthly temperature is –10°C, which occurs in January (month 0) with an annually repeating pattern. The sketch of the graph that models this relationship is shown below.

a) Write the simplest equation that models this relationship.

x

20

0

10

–10

1 123 4 52 6 8 9 10 117

y



Ferris wheels (where time for one or more rotations is given)Example 5: A Ferris wheel has a radius of 10m and the maximum height the chair reaches is 22m. The wheel takes 90 seconds to complete one rotation.

a) Sketch a graph that represents the height h in meters, of the bottom chair, as a function of time t, in seconds. Sketch one complete cycle.

Time (t)

8

6

4

2

16

12

10

20

22

18

14

Hei

ght(

h)

6040 8020 10

b) Write an equation in terms of cosine, that expresses the height h of the bottom chair, as a function of time in seconds. That is h = acosb(t – c) + d.

c) What is the lowest height that any chair is above the ground?

d) Estimate the height the chair is above the ground after 20 seconds.

e) Estimate one of the times when the chair is 19m above the ground.



Using Graphing CalculatorExample 6: In the West Kootenay town the sunrise for a non-leap year, can be found using the following formula:

Where t is the time in hours after midnight, and d is the number of the day in the year.

a) Use the formula to determine to the nearest minute when the sun rose on February 14, the 45th day of the year.

b) Use the formula to determine the first day of the year when the sun rose before 6:30 a.m., and the last date of the year when the sun rose before 6:30 a.m.

c) For how many days does the sun rise before 6:30 a.m.?



Using a Sinusoidal Functions to Model Given DataExample 7: The following data show the predicted tide heights every 2 hours, starting at midnight, for St. Andrews, PEI, on March 9, 2011:(00, 4.6), (02, 6.5), (04, 5.7), (06, 3.5), (08, 1.4), (10, 1.7), (12, 4.0), (14, 6.1), (16, 5.8), (18, 3.9), (20, 1.8), (22, 1.7)a) Graph the data, and then write an equation of a sinusoidal function that models that data.

NOTE: the graph is NOT a perfect cosine function, but it is close enough to make a sinusoidal model reasonable.

b) Use technology to graph the function in part a. Estimate the tide height at 17:00 (5:00 pm). Give the answer to the nearest tenth of a metre.

Assignment: page 548-555 #3-11; page 555-556 #1, 2Chapter Review: page 506-509 #112; page 560-566 #1-13; page 567-570 #1-7


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