Download Apply Trigonometric Methods in Solving Problems

AS 91575
Apply Trigonometric Methods
in Solving Problems
Horowhenua College: K. Staniford
AS 91575
Apply Trigonometric
Methods in Solving
Problems
Chapter 1: Introduction
Course: Calculus
Teacher: K. Staniford
Credits: 4
Assessment: Internal
SECTION 1
PUTTING QUANTITATIVE CONCEPTS IN
CONTEXT
•
Geologic structures and mapping
•
Compass direction, strike and dip
An understanding of angles and the ability to manipulate trigonometric
•
Ocean and seismic waves
expressions is essential for an introductory geoscience student.
•
Dunes, wind direction and currents
•
Subduction zones
Geoscientists use both of these concepts in numerous applications, from
•
Slope stability
•
Planetary applications (parallax)
coastal geology to mineralogy to geologic mapping. Helping students to
develop the tools they need to understand these important concepts can
be challenging. Most college students have learned about trigonometry at
some point in their school career (remember "soh, cah, toa? sin = opposite/
hypotenuse..."); however, moving beyond the abstract mathematics to
application is often difficult for the student.
This unit in the Calculus course will help you to progress your
understanding of Trigonometry and application of the core skills to a level
useful in a variety of different degree courses including Geosciences.
Waves refracting as they approach shore.
Details
2
Section 2
AS 91575: Apply trigonometric methods in solving problems
What will I learn in this unit?
• use radians and degrees
• solve problems involving trigonometric functions of the form: Asin B (x + C) + D
• Rewrite trigonometric expressions in terms of a single trigonometric function
http://ncea.tki.org.nz/content/download/3470/11127/file/maths3_3_int_sep12.doc
What do I need to know for each level of attainment?
and Acos B (x + C) + D, where C, D may be zero. The solution of the problems
could require knowledge of amplitude, period, and frequency.
Achieved
Apply trigonometric methods in solving problems involves:
• solve equations of the form Asin Bx = K, Asin B(x + C) = K
• solve equations of the form Acos Bx = K, Acos B(x +C) = K
• solve equations of the form Atan Bx = K, Atan B(x+C) = K (giving all solutions
within a specified domain, in radians or degrees)
selecting and using methods
demonstrating knowledge of concepts and terms
communicating using appropriate representations.
Merit
Relational thinking involves one or more of:
• Form an equation using trigonometric functions to model a situation and use the
model to solve problems.
• The equations will be of the form:
y = Asin B(x + C) + D
y = Acos B(x + C) + D y = Atan B(x + C) + D
selecting and carrying out a logical sequence of steps
connecting different concepts or representations
demonstrating understanding of concepts
forming and using a model;
relating findings to a context, or communicating thinking using appropriate
• prove trigonometric identities
• Use Reciprocal Relationships / Pythagorean Identities / Compound Angle
Formulae / Double Angle Formulae
• Evaluate models and solve problems that require a proof
mathematical statements.
Excellence
Extended abstract thinking involves one or more of:
devising a strategy to investigate or solve a problem
identifying relevant concepts in context
• Solve 3D trigonometry problems
developing a chain of logical reasoning, or proof
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Section 3
Radian Measure
RADIANS
EXERCISE 1
What is a radian?
1. Convert the following to radians:
If a circle has a radius of r units and the arc length is
also r units, then the angle of the sector is one radian.
If the arc length changes to any other size (a) then the
angle of the sector is a/r radians.
Angle α= a/r
360 degrees = 2π radians
How do I convert between degrees and radians?
• 90°
• 270°
• 180°
• 45°
• 60°
• 120°
2. Convert the following radian measurements to degrees:
• π
•
•
•
3. A sector of a circle of radius 10cm has an angle of 40°. Convert this to radians and the
arc length of the sector.
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Section 4
Exact Values
Consider the graphs below:
• In the clip, Sheldon says the value of sin30° is ½. How does Sheldon
know this exact value?
• What do you know that could confirm this value? How could you
check this value out?
• What other exact values might Sheldon know? How are these values
linked to each other?
• How are these exact values linked to graphs of the sin, tan and cos
functions?
These websites can be used to check answers:
•
•
Maths mistakes: Trigonometry facts
Mathwords: Trig values of special angles; exact values of
trig functions
0°
In ‘The Big Bang Theory’ Sheldon knows
sin x
some information relating to the graphs
cos x
shown.
Watch this clip: 30°
45°
60°
90°
tan x
SHELDON
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Section 5
Trigonometric Graphs
Most Trigonometry relies on three basic trig functions (sin, cos
There are more than three trigonometric functions, some of
and tan).
which are listed in the chart below
All three of the graphs continue from negative infinity to positive
infinity (domain). Sin and Cos oscillate between +1 and -1, this is
their range. Because Tan has asymptotes ever 90° it has a range
between positive and negative infinity.
Both sin and cos repeat themselves every 360° or 2π, this is the
The graph of cos x above shows both degree and radian
period. Tan repeats every 180°, therefore has a period of π.
measures.
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Section 6
Transforming Trig Graphs
1. Multiplying the function by a constant alters the amplitude or
height of the graph.
2. Adding a constant to the function translates the graph
vertically (moves the graph up and down). Look at the
diagram of y =sin x + 1 and y = sin x
In the above graph the red line is y = 2 sin x. The normal sin
graph has increased the amplitude to 2, now ranging between
negative and positive 2.
The black line is y = sin x. What is the equation of the blue line?
There is a ‘special case’ in this transformation. Investigate what
happens if the sin graph is multiplied by a negative number.
Google y = -sin x to start you off.
3. Multiplying the angle by a constant (y = cos ax) alters the
period of the graph, or how often it will repeat between 0 and
2π.
Dotted:
y = cos 2x
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4. Adding or subtracting a constant from the angle (y = sin (x ± d))
EXERCISE 2
translates the graph horizontally. An addition will move the
graph to the left, subtracting will move it to the right.
1. Some stars have a brightness that periodically increases and
decreases. The brightness (B) of one such star can be
modelled by the equation:
B = 3.9 0.33sin ( 2πt ⁄ 5.6)
where t is the time measured in days.
• What is the maximum brightness of the star?
• What is the minimum brightness of the star?
• How many days elapse between successive times of
maximum brightness?
2. Find a solution to the following trigonometric equations:
The red line above is y = cos x. The blue line is a horizontal
• tan (2x - 1) = 0.321
translation. Give at least two different equations for this line...
• 3cos 4(x - π/2) = -2
Of course it is possible to have multiple translations in the same
graph. Try to work out the equation of this graph:
3. Awarua Inlet is a tidal estuary in Abel Tasman National park.
The depth of the water in the middle of the channel changes as
the tide comes in and out. The depth can be modelled by a
cosine function of the form y = A cost Bt + C.Successive high
tides occur ever 12 1/2 hours. The maximum depth of the
water at high tide is 2.0m while the minimum depth at low tide
is 0.6m. It is safe to walk across the inlet provided the depth of
water is no more than 1m. High tide on a particular day is at
10.00am. Between what times in the afternoon of that day will
it be safe to walk across the inlet?
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Section 7
Reciprocal Trig Functions
Have a look at the website below, can you find out the domain,
Full name
secant
cosecant
cotangent
Abbreviation
sec
cosec
cot
range and period of each of the reciprocal functions? Record this
in your notes.
http://www.analyzemath.com/trigonometry/properties.html
Function
1/cos x
1/sin x
1/tan x
Work out the values below:
• sec 60°
• cot 0.56
• cosec 0
To the right are graphs of the reciprocal functions
and the standard trig functions of sin, cos and tan.
Can you plot each of these graphs on your graphics
calculator?
Investigate whether the transformation rules for
y = ±A ± B sin (Cx ± D)
are true for the reciprocal functions.
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Section 8
Sine and Cosine Rule
REVIEW: SINE AND COSINE RULE
If you need a review of sine and cosine rule
follow this link, this link or this link!
A ruined tower is fenced off for safety
reasons.
To find the height of the tower Rashid stands
at a point A and measures the angle of
elevation as 18°.
He then walks 20 metres directly towards the
base of the tower to point B where the angle
of elevation is 31°.
Calculate the height, h, of the tower.
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Section 9
Working with Trigonometric Identities
Proving identities is a strange part of Maths...you already know the
Simple Identities
answer! You need to show that the right hand side (RHS) of the
An identity is an equation the proves to be true no matter what value
is substituted. An ‘exactly equal’ sign (
) is usually used in identities
instead of a standard = sign.
There are three simple identities from which others can be derived.
1. sin θ
cos(90°-θ) and cos θ
2. sin θ
tan θ
cos θ
sin(90°- θ)
expression matches the left hand side (LHS) or vice versa.
Try the following steps when looking at this problem:
Prove that sec x
cosec x
tan x
1. Start with the more complicated side and simplify
2. Replace other trig expressions with sin and cos where possible.
3. Simplify each side independently until equality is found.
3. sin2 θ + cos2 θ ≣ 1
Trigonometric forms of Pythagoras
1 + cot2 θ ≣ cosec2 θ
tan2 θ 1 ≣ sec2 θ
You can prove the first by dividing sin2 θ + cos2 θ 1 by sin2 θ and
simplifying. Try this, then work on finding a proof for the second.
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The Compound Angle Formulae
The Double Angle Formulae
There are six formulae for the sums and differences of angles,
because sin(A + B) DOES NOT equal sinA + sinB.
sin(A + B) = sinAcosB + cosAsinB
sin(A - B) = sinAcosB - cosAsinB
cos(A + B) = cosAcosB - sinAsinB
cos(A - B) = cosAcosB + sinAsinB
tan(A + B) =
tanA + tanB
1 - tanAtanB
tan(A - B) =
tanA - tanB
1 + tanAtanB
EXERCISE 3
1. Verify the following using the compound angle formulae:
!
cos (90°+ θ) = –sin θ
!
sin (360°– θ) = –sin θ
These formulae are recognised because sin 2A ≠ 2 sin A in the same
way that the sum of the sine of two angles is not the sum of the two
sines.
Try these:
1. Simplify cos2 6x – sin2 6x to a trigonometric function having a single
angle.
2. Verify that:
2. Rewrite the following as the sin or cos of a single angle and
evaluate where possible.
!
sin x.cos y – sin y.cos x
!
cos 2°.sin 88°+ sin 2°.cos 88°
!
cos 7x.cos 3x – sin 7x.sin 3x
!
cos 62°.cos 17° + sin 118°.cos 73°
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Changing Products to Sums
2 sin A cos B = sin (A + B) + sin (A - B)
Some useful links for further exploration:
2 cos A sin B = Sin (A + B) - sin (A - B)
2 cos A cos B = cos (A + B) + cos (A - B)
http://teachertube.com/viewVideo.php?video_id=21022
2 sin A sin B = cos (A - B) - cos (A + B)
http://www.sosmath.com/trig/Trig5/trig5/trig5.html
Try writing 2sin 45°cos 15° as a sum or difference.
http://www.khanacademy.org/math/trigonometry/v/
Changing Sums to Products
trigonometric-identities
http://www.intmath.com/analytic-trigonometry/1-trigonometricidentities.php
http://www.proprofs.com/quiz-school/story.php?
title=trigonometric-identity-quiz
Show that cos 100° + cos 20° = cos 40°
Download into your iTunes u collection library HACC
Trigonometry
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Section 10
Trig Equations
Trigonometric functions repeat themselves. When you solve an
If you solve this equation on the calculator you will find the
equation you can expect to find more than one solution within a
answer to be 30 degrees. This is called the principal solution
given range. Sometimes the solution can be found directly and
(the first positive solution). It is represented by the first green line
sometimes an identity is used in the solution.
after 0 on the graph.
If the range is given in terms of degrees, then the answer is
When you look at the graph you can see that this answer is
expected in degrees, if the range is given in radians the answer is
repeated every 360 degrees. There are also the ‘red’ answers
expected t be in radians.
which occur as the sin wave decreases. the first of these is 30
Have a look at the graph below, it represents the equation
sin x = 0.5
degrees before the first intercept with the x axis. These are mirror
images on the curve. This answer is also repeated every 360
degrees.
Name all of the answers in the range shown (e.g. between -720
and 720 degrees).
How many solutions are there in this range?
How many answers to sin x = 0.5 would there be overall?
Would the answers be repeated in the same way for cos and tan
graphs?
Solve: cos ( x -
) = 0.3 for answers between 0 and 360°
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Section 11
General Solutions to Trig Equations
It is easier to make sure that you have found all possible
solutions by using a general solution formula and applying it
across the required range.
Substituting values into the general solution:
𝜃 = 180(0) + (-1)0.17.5
𝜃 = 17.5
𝜃 = 180(1) + (-1)1.17.5
𝜃 = 162.5
The next answer when substituting n=2 is 377.5 which is out of range. Any
negative values of n would also be out of range. Therefor the solutions are 17.5°
and 162.5°.
If you need to show general solutions in radians rewrite the general solutions with
nπ for 180° or 2nπ for 360°.
Example:
Solve the equation sin x = 0.3, for all values of x in the domain [0°, 360°].
Solution:
sin-1 (0.3) = 17.5
Therefore the principal value is 17.5
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Section 12
Trig Equations in Quadratic Form
LOOK OUT FOR PYTHAGOREAN IDENTITIES!
Example:
Try these yourself:
1. 3sin2x = sin x + 2 [ 0 , 2π ]
2. cos 2x = sin x + 1
[ 0° , 180° ]
Solution:
3. sin 2x.cos x = 0
x in radians
1 - 2sin2x = sin x
4. cos (2x) - cos x -2 = 0
[ 0° , 360° ]
5. cos x = sin (2x)
[ -π , π ]
Solve the equation cos (2x) = sin x
for [0° ,360°]
2sin2x + sin x - 1 = 0
(2sin x - 1)(sin x + 1) = 0
sin x = 30 6. tan (2x) = tan x
sin x = -90
The principle values to start with are 30 and -90.
A
Have a look at these links:
Therefore;
x =180n + (-1)n x 30
and
x =180n + (-1)n x -90
1st solution set for n=0 and 1 are 30 and 150, the next is
out of range.
B
C
2nd solution set for n=1 and 2 are 270 and 270, the
other solutions are out of range.
Solutions are x = 30°, 150°, 270°
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Section 13
Double Angles and Sums and Products
Here are a list of useful links for more information on trig
Double Angle Identities:
Apply your double angle identities to the following equation:
Solve 2 sin x cos x = -1
for the domain [0, 2 π ]
Hint: use the identity sin (2x) = 2 sinxcos x, then replacing x by A.
www.sosmath.com/algebra/solve/solve0/solvtrig.html
www.purplemath.com/modules/solvtrig.htm
www.mathcentre.ac.uk/resources/uploaded/mc-ty-
Sums and Differences:
trigeqn-2009-1.pdf
Sums and differences can also be used to solve trig equations:
Solve cos (2x) = cos x = 0 for
equations:
[ -π , π ]
Here is a reminder of the identities:
www.sparknotes.com › SparkNotes › Math Study Guides
http://mathsfirst.massey.ac.nz/Trig/TrigGenSol.htm
http://www.khanacademy.org/math/trigonometry/basictrigonometry/
http://www.math.wfu.edu/Math105/Trigonometric
%20Identities%20and%20Equations.pdf
www.krysstal.com/trigequations.html
http://www.dummies.com/how-to/content/how-to-solve-atrigonometry-equation-using-the-qua.html
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