Trigonometry FunctionsAnd

Solving Right Triangles

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Title PageTable of ContentsWhat are the Trigonometry Functions and how are they usedWhat do you need to solve a right Triangle using Trig. FunctionsSine FunctionCosine FunctionTangent FunctionHow to find an angle using the trig. FunctionsVideo of a trig. Functions song ((trig) signs by the four man mathematical band)Trig. Functions and the unit circleThe Unit Circle in degrees and the correlating points written as (CosѲ, SinѲ)The unit circleVideo on the unit circle (Unit circle work out)Techniques to Remember Trig. FunctionsTechnique on how to remember where the trig. Functions are positive in the Cartesian planeExample IHow to solve example 1Example 2How to solve example 2Problem 1Problem 2A few more practice ProblemsReferences

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.

What are the Trigonometry Functions and how are they used

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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 FunctionThe 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

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

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 Next

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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.

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.

Positive In Quadrant 1:Sine SecantCosine Cosecant Tangent Cotangent

Positive In Quadrant 2:Sine and Secant

Positive in Quadrant 3:Tangent and Cotangent

Positive In Quadrant 4:Cosine and Cosecant

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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 AlgebraTo 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 useAll 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.

IALL

IIStudents

IIITake

IVCalculus

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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 angle3. What information do we know Ѳ=600 and Height= 17ft.=Opposite the angle4. What can I use to find my unknownsI know that Sin(60)=17/X , Cos(60)=Y/X, andTan(60)=17/YIn 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 X(Sin(60))=17 X= 17/Sin(60) X=19.62 ft

Tan(60)=17/YY(Tan(60))=17Y=17/Tan(60)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 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. Forboth sine and cosine I need to find the anglefirst and use the angle to find the hypotenuse.

X

Ѳ

50 ft

13 ft

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How to Solve Example 2 Continued

X

50 ft

13 ft

Ѳ5. Solve for your variables.For this problem you need to start by finding the angle.

TanѲ=50/13Ѳ=Tan-1(50/13)Ѳ= 75.42578380

Ѳ= 75.420

Now use the angle you just found for Ѳ to find X using either Sine or cosine.

Sin(75.42)=50/XX Sin(75.42)=50X=50/ Sin(75.42)X=51.66372164X= 51.6 ft

Cos(75.42)=13/XX Cos(75.42)=13X=13/Cos(75.42)X=51.64231527X=21.6 ft

The Two equations have two slightly different lengths for x because we rounded Ѳ and used the rounded Ѳ.

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Problem 1

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?

Directions: Solve the problem and then click on the correct answer to continue.

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.

XA. 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

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A few more practice Problems

Click on the links below, print the worksheets and complete them.

Unit Circle Worksheet

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Trigonometry functions Worksheet

ReferencesText 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/lessonplan1c.html•http://www.cemca.org/ciet/Trigonometry/Trigonometrymag.htm•http://ilearn.senecac.on.ca/learningobjects/MathConcepts/ApplyingTrigFunctions/main.htm•http://mathworksheetsworld.com/bytopic/trigonometry.html•www.youtube.com

Sounds fromwww.soundboard.com

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