Trigonometry Functions
And
Solving Right Triangles
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Table OF Contents
Title Page
Table of Contents
What are the Trigonometry Functions and how are they used
What do you need to solve a right Triangle using Trig. Functions
Sine Function
Cosine Function
Tangent Function
How to find an angle using the trig. Functions
Video of a trig. Functions song ((trig) signs by the four man mathematical band)
Trig. Functions and the unit circle
The Unit Circle in degrees and the correlating points written as (CosѲ, SinѲ)
The unit circle
Video on the unit circle (Unit circle work out)
Techniques to Remember Trig. Functions
Technique on how to remember where the trig. Functions are positive in the Cartesian plane
Example I
How to solve example 1
Example 2
How to solve example 2
Problem 1
Problem 2
A few more practice Problems
References
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What are the Trigonometry Functions and
how are they used
There are six trigonometry functions which are Sine, Cosine, Tangent,
Secant, Cosecant and Cotangent. Represented respectively as SinѲ,
CosѲ, TanѲ, CscѲ, SecѲ, CotѲ.
The functions are used to determine angles, the measurement of a side
of a right triangle. The sides of the triangle can be distances, velocity,
acceleration, or any other form of measurement.
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What do you need to solve a right Triangle
using Trig. Functions
In order to solve a right triangle using trigonometry
functions you need to be given at least two sides or a
side and one of the acute angles. Later we will be able
to use equations using the trigonometry functions to
help solve problems.
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Sine Function
The Sine function is the side opposite of the angle Ѳ divided by the
Hypotenuse of the triangle.
The reciprocal of the sin function is Cosecant. CSCѲ= 1/ SinѲ= the
Hypotenuse of the triangle divided by the side opposite to the angle Ѳ
So in this Triangle CSC Ѳ= r/y
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Cosine Function
The Cosine function is the side adjacent to the angle Ѳ divided by the
Hypotenuse of the triangle. In the picture below CosѲ= x/r
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The reciprocal of Cosine is Secant. Sec Ѳ = 1/ Cos= hypotenuse of triangle /
the side adjacent to the angle Ѳ. From the picture Sec Ѳ=r/x
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Tangent Function
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The Tangent function is the side opposite the angle Ѳ divided by the side adjacent
to the angle Ѳ. Below tan Ѳ= Y/X.
The function that is the inverse of Tangent is Cotangent which is 1 divided by
tangent which equals the side adjacent to angle Ѳ divided by the side opposite to
the angle Ѳ. COT Ѳ= X/Y.
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How to find an angle using the trig. functions
In order to find the angle using the trig. Functions you need to know at least two
sides of the triangle. You use the inverse of a trig functions to find the angle. The
inverses are written Sin-1Ѳ, Cos-1Ѳ, and Tan-1Ѳ . Therefore Ѳ = Sin-1Opposite/
Hypotenuse, Cos-1Adjacent/ Hypotenuse, and Tan-1Opposite/Adjacent
In the above picture Sin-1Ѳ = y/r, Cos-1Ѳ = x/r, and Tan-1Ѳ = y/x.
So Ѳ = Sin-1y/r = Cos-1x/r = Tan-1y/x
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Trig. Functions and the unit circle
The Unit circle is a circle on the Cartesian Plane with a radius of one. The points
on the unit circle are cosine and sine of the angle. The x pint is Cosine Ѳ and the
Y point is Sine Ѳ. So (X,Y)= (Cos Ѳ, Sin Ѳ). To determine the point on the circle
you use the radius, which is one, from the center of the circle to the point and
the angle created in between the x axis and the radius. Then you solve for Cos Ѳ
and Sin Ѳ to find the point on the line. Tangent is used to determine a line that is
tangent (or perpendicular) to a point on the circle.
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The Unit Circle in degrees and the correlating points written as (CosѲ, SinѲ)
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The unit circle
On the unit circle each of the three trig functions Cosine, Sin, and Tangent are
positive and negative in specific quadrants. The reciprocal function of the trig
functions are positive and negative where ever the trig functions associated with
them is positive of negative.
Positive In Quadrant 1:
Sine
Secant
Positive In Quadrant 2:
Cosine
Cosecant
Sine and Secant
Tangent Cotangent
Positive in Quadrant 3:
Tangent and Cotangent
Positive In Quadrant 4:
Cosine and Cosecant
All of the functions are positive in the first quadrant. The sine function is
also positive in the second quadrant. The Cosine function is positive in the
fourth quadrant. Tangent is positive in the third quadrant. So Cosecant is
positive in quadrant 1 and 2, Secant is positive in 1 and 4, and Tangent is
positive in 1 and 3.
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Techniques to Remember Trig. Functions
Ways to remember the Sine, Cosine, and Tangent Functions
Use the word SohCahToa
Where S=sine O=opposite A=adjacent C=cosine T=tangent
So S=opposite/adjacent C=adjacent/ hypotenuse T=opposite/ adjacent
This is because Sohcahtoa =
S(sine)O(opposite)A(adjacent)C(cosine)A(adjacent)T(tangent)O(opposite)A(adjacent)
You can also use the sentence
Oliver Had A Headache Over Algebra
To use this sentence correlate the first two words to sine, the middle two words
to Cosine, and the last two words to Tangent.
So:
Sine= Oliver/Had= Opposite/ Hypotenuse
Cosine=A/ Headache= Adjacent/ Hypotenuse
Tangent=Over/ Algebra= Opposite/ adjacent
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Technique on how to remember where the trig. Functions are positive in the Cartesian
plane
To recall where the trig functions are positive in the Cartesian plane use
All Students Take Calculus.
Place each word in the quadrants in order.
So: All is in the first quadrant to mean all the functions are positive
Students is in quadrant 2 to mean sine is positive
Take is in the third quadrant to mean tangent is positive
Calculus is in the fourth quadrant to mean cosine is positive.
II
Students
III
Take
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ALL
IV
Calculus
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Example I
There is a Ladder that is leaning up against a building that is 17 feet tall. If the
ladder makes a 60 degree angle with the ground. How far away from the building is
the ladder? What is the length of the ladder?
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How to solve example one.
1. First draw a picture If you do not have one or
simplify the drawing down to just a right
triangle If there is to much in the drawing.
We have a good picture already.
2. Determine what you need to find.
In this diagram we need to find X and Y.
Where x is the Hypotenuse
and Y is adjacent to the angle
3. What information do we know
Ѳ=600 and Height= 17ft.=Opposite the angle
4. What can I use to find my unknowns
I know that Sin(60)=17/X , Cos(60)=Y/X, and
Tan(60)=17/Y
In order to use Cosine in this problem I need two
unknowns so I am going to use Sine and Tangent.
5. Solve for the variables
Sin(60)=17/X
Tan(60)=17/Y
X(Sin(60))=17
Y(Tan(60))=17
X= 17/Sin(60)
Y=17/Tan(60)
X=19.62 ft
Y=9.81 ft.
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Example 2
There is a light house that is 40 ft off the shore. There is a house that is 10 ft back
on the shore. The balcony is 13 feet from the ground. A person is on the balcony
looking at the light house what is the distance from the balcony to the light
house as the person on the balcony sees the light house? Also find the angle as
you look at the light house from the balcony?
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How to Solve Example 2
Ѳ
X
13 ft
1. I draw a simpler triangle. I have drawn the
triangle under the picture on the left.
2. What Information do you know and write
the information on the right triangle.
the light house is 40 ft off shore and the
house is 10ft from the shore so the
distance between the lighthouse and shore
is 50 ft. the balcony is 13ft from the
ground. So I know the two sides of the
triangle.
3. What do I need to find.
I need to find the hypotenuse and angle at
the top of the triangle.
4. What can I use to find this information.
I can use SinѲ=50/X, CosѲ=13/X, TanѲ=50/13
I will use Tangent and either sine or cosine. For
both sine and cosine I need to find the angle
first and use the angle to find the hypotenuse.
50 ft
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How to Solve Example 2 Continued
5. Solve for your variables.
For this problem you need to start by finding the
angle.
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Ѳ
13 ft
X
TanѲ=50/13
Ѳ=Tan-1(50/13)
Ѳ= 75.42578380
Ѳ= 75.420
50 ft
Now use the angle you just found for Ѳ to find X
using either Sine or cosine.
Sin(75.42)=50/X
X Sin(75.42)=50
X=50/ Sin(75.42)
X=51.66372164
X= 51.6 ft
Cos(75.42)=13/X
X Cos(75.42)=13
X=13/Cos(75.42)
X=51.64231527
X=21.6 ft
The Two equations have two slightly
different lengths for x because we
rounded Ѳ and used the rounded Ѳ.
Problem 1
Directions: Solve the problem and then click on the correct answer to continue.
There is a person standing 10 feet west of a bush. There is a tree b ft north from the
bush. The angle from the person to the tree is 25 degree. How far away from the tree
is the person if they walk in a straight line? What is the distance between the tree
and the bush?
A. b=5ft c= 24ft
B. b=10ft c=14.14ft
C. b= 4.66ft c= 23.66 ft
D. b=23.66ft c=4.66ft
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Problem 2
There is a person standing 20 feet away from a tree. At their feet is a string
stacked into the ground, the string is attached to the top of the tree. The string
and ground makes a 45 degree angle. What is the height of the tree? What is
the length of the string? And find the angle between the string and the tree.
A. Ѳ= 450 h=20ft x=28.28ft
B. Ѳ=250 h= 28.28ft x=20ft
C. Ѳ=450 h=28.28ft x=20ft
D. Ѳ=250 h=20ft x=30ft
X
A few more practice Problems
Click on the links below, print the worksheets and complete them.
Trigonometry functions Worksheet
Unit Circle Worksheet
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References
Text Reference
Barnett, Raymond a., Michael R. Ziegler, and Karl E. Byleen.
Precalculus Graphs and Models. New York: McGraw-Hill, 2005.
The Pictures are from
•www.coolmath.com
•http://www.partnership.mmu.ac.uk/cme/Geometry/MEC/trigfacts/
TrigFacts.html
•http://www.learner.org/workshops/algebra/workshop8/lessonplan1
c.html
•http://www.cemca.org/ciet/Trigonometry/Trigonometrymag.htm
•http://ilearn.senecac.on.ca/learningobjects/MathConcepts/Applyin
gTrigFunctions/main.htm
•http://mathworksheetsworld.com/bytopic/trigonometry.html
•www.youtube.com
Sounds from
www.soundboard.com
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