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Trigonometry Functions And Solving Right Triangles Next Slide

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Table OF Contents Next Slide Title Page Table of Content s 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 (Cos Ѳ, Sin Ѳ ) 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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There are six trigonometry functions which are Sine, Cosine, Tangent, Secant, Cosecant and Cotangent. Represented respectively as Sin Ѳ, Cos Ѳ, Tan Ѳ, Csc Ѳ, Sec Ѳ, Cot Ѳ. 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 Next Slide Table of Contents

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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. Next Slide Table of Contents

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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. CSC Ѳ = 1/ Sin Ѳ = the Hypotenuse of the triangle divided by the side opposite to the angle Ѳ So in this Triangle CSC Ѳ = r/y Next Slide Table of Contents

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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 Next Slide Table of Contents

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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. Next Slide Table of Contents

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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 -1 Ѳ, Cos -1 Ѳ, and Tan -1 Ѳ. Therefore Ѳ = Sin -1 Opposite/ Hypotenuse, Cos -1 Adjacent/ Hypotenuse, and Tan -1 Opposite/Adjacent In the above picture Sin -1 Ѳ = y/r, Cos -1 Ѳ = x/r, and Tan -1 Ѳ = y/x. So Ѳ = Sin -1 y/r = Cos -1 x/r = Tan -1 y/x Next Slide Table of Contents

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Next Slide Table of Contents

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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. Next Slide Table of Contents

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The Unit Circle in degrees and the correlating points written as (CosѲ, SinѲ) Next Slide Table of Contents

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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 Secant Cosine Cosecant Tangent Cotangent Positive In Quadrant 2: Sine and Secant Positive in Quadrant 3: Tangent and Cotangent Positive In Quadrant 4: Cosine and Cosecant Next Slide Table of Contents

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Table of Contents Next Slide

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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 Next Slide Table of Contents

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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 A ll S tudents T ake C alculus. 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. I ALL II Students III Take IV Calculus Next Slide Table of Contents

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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? Next Slide Table of Contents

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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=60 0 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 X(Sin(60))=17 X= 17/Sin(60) X=19.62 ft Tan(60)=17/Y Y(Tan(60))=17 Y=17/Tan(60) Y=9.81 ft. Next Slide Table of Contents

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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? Next Slide Table of Contents

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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. For both sine and cosine I need to find the angle first and use the angle to find the hypotenuse. X Ѳ 50 ft 13 ft Next Slide

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How to Solve Example 2 Continued X 50 ft 13 ft Ѳ 5. Solve for your variables. For this problem you need to start by finding the angle. TanѲ=50/13 Ѳ=Tan -1 (50/13) Ѳ= Ѳ= 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= X= 51.6 ft Cos(75.42)=13/X X Cos(75.42)=13 X=13/Cos(75.42) X= X=21.6 ft The Two equations have two slightly different lengths for x because we rounded Ѳ and used the rounded Ѳ. Next Slide Table of Contents

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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= ft D. b=23.66ft c=4.66ft Table of Contents

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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. X A. Ѳ= 45 0 h=20ft x=28.28ft B. Ѳ=25 0 h= 28.28ft x=20ft C. Ѳ=45 0 h=28.28ft x=20ft D. Ѳ=25 0 h=20ft x=30ft Table of Contents

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A few more practice Problems Click on the links below, print the worksheets and complete them. Unit Circle Worksheet Next Slide Table of Contents Trigonometry functions 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, The Pictures are from TrigFacts.html c.html gTrigFunctions/main.htm Sounds from To Start Table of Contents

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