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Trigonometry Right Triangle and Trigonometric Functions.

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Presentation on theme: "Trigonometry Right Triangle and Trigonometric Functions."— Presentation transcript:

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2 Trigonometry Right Triangle and Trigonometric Functions

3 Overview This module begins with right triangle trigonometry (trig), followed by trigonometric functions and their inverses. Trigonometry (trig) means the measurement of triangles. Trig is used in many applications.

4 Topics Right Triangle Trig Pythagorean Theorem Special Right Triangles Sine, Cosine, Tangent Ratios Reciprocal Ratios: Cosecant, Secant, Cotangent Problem Solving Trig Functions Degree Radian Measure of Angles Unit Circle and Trig Ratios Trig Functions and Their Characteristic Graphs of Sine, Cosine, Tangent Functions Transformation of Trig Functions Inverse Functions Problem Solving

5 Pythagorean Theorem Website The Pythagorean Theorem relates the lengths of the sides of a right triangle. The first page of the website explains the Pythagorean Theorem and the second page includes uses of the theorem to solve problems. Pythagorean Theorem (review)

6 Practice Problems Website Practice Problems Video Practice Problems WebsitePractice Problems Video Practice Problems Website Practice Problems Video Practice Problems WebsitePractice Problems Video Now, it is your turn to solve problems using the Pythagorean Theorem. The website has application problems to solve with explanations provided. Pythagorean Theorem: Practice Problems

7 30˚-60˚-90˚ Website30˚-60˚-90˚ Website 45 ˚- 45˚- 90˚ Website45 ˚- 45˚- 90˚ Website30˚-60˚-90˚ Website30˚-60˚-90˚ Website 45 ˚- 45˚- 90˚ Website45 ˚- 45˚- 90˚ Website Two types of right triangles are very important to the study of trigonometry. They are called special right triangles: 30⁰ - 60⁰- 90˚ right triangle 45⁰ - 45⁰ - 90˚right triangle The 2 websites explain the relationship that exists between the length of the legs and hypotenuse for both types. Special Right Triangles

8 Special Right Triangle Practice Website The website provides practice problems using the relationships of the sides of the special right triangles. Special Right Triangles Practice

9 Right Triangle Trig VideoRight Triangle Trig Video Trig Ratios WebsiteTrig Ratios WebsiteRight Triangle Trig VideoRight Triangle Trig Video Trig Ratios WebsiteTrig Ratios Website As you learned in the previous resources, the ratios of the sides of the special right triangles are always constant, regardless of the size of the triangles. This is also true of any right triangle. In this video and website, you will learn about these ratios and the common memory device for remembering the ratios: SOH CAH TOA Right Triangle Trig Ratios: Sine, Cosine, Tangent

10 Practice Problems Website On this website you will read example problems as well as review questions. Solve # 1-9. Then check your work by looking at the review answers to questions # 1-9. Right Triangle Trig Practice Problems

11 Six Trig Ratios Video Three additional trig ratios exist, the reciprocal ratios: The reciprocal of the sine ratio is the cosecant (csc) = hypotenuse/opposite The reciprocal of the cosine ratio is the secant (sec) = hypotenuse/adjacent The reciprocal of the tangent ratio is the cotangent (cot) = adjacent/opposite The video reviews all six trig functions. Three Additional Trig Ratios: Cotanget, Cosecant, Secant (Reciprocals)

12 Practice Problems Website On this website you will find practice problems using the Pythagorean Theorem and/or right triangle trig. Try to solve the problems. The answers and an explanation are provided. Right Triangle Trig Practice Problems

13 Radian PDF Radian VideoRadian PDF Radian Video The measure of an angle is determined by the amount of rotation from the initial side (starting side) to the terminal side (position after rotation). Degree is one unit used to measure the size of an angle. Another way to measure angles is the unit radian. In the study of trig angles are measured in either degrees or radians. On the PDF focus on sections 1 – 5. Angles: Degree and Radian Measure

14 Radian VideoRadian Video Radians PDFRadians PDFRadian VideoRadian Video Radians PDFRadians PDF On the PDF stop after problem #31. On the video stop when the presenter defines arc length (writes s =) Degree/Radian Additional help

15 Unit Circle PDFUnit Circle PDF Standard Position/Coterminal Angle Video Unit Circle Video Standard Position/Coterminal Angle VideoUnit Circle VideoUnit Circle PDFUnit Circle PDF Standard Position/Coterminal Angle Video Unit Circle Video Standard Position/Coterminal Angle VideoUnit Circle Video Right triangles only allow for trig ratios to be defined for positive acute angles (angles less than 90˚) The unit circle is used to extend trig ratio definitions to angle measures ≥ 90˚ and negative angles. The unit circle is used to evaluate trig ratios for any size angle. The PDF defines all six trig functions on the unit circle (radius of one unit). Also explained is the meaning of an angle to be in standard position as does the first video along with what it means for angles to be conterminal. The second video focuses on the 3 primary trig ratios: sin, cos and tan Trig ratios for Angles ≥ 90˚

16 Use of Unit Circle Video Unit Circle WebsiteUse of Unit Circle VideoUnit Circle Website Use of Unit Circle Video Unit Circle WebsiteUse of Unit Circle VideoUnit Circle Website The video defines the cos and sin ratios on the unit circle and highlights when the ratios are positive and negative. The solutions to the problems on the website can be viewed by scrolling over the green blocks. Language note: saying the sine OR the sine of an angle refers to the sine ratio. Unit Circle Definition of Trig Ratios

17 Reference Angles WebsiteReference Angles Website Patterns in the Unit Circle VideoPatterns in the Unit Circle VideoReference Angles WebsiteReference Angles Website Patterns in the Unit Circle VideoPatterns in the Unit Circle Video Included on the website is an applet to practice naming the reference angle for angles in quadrants II, III and IV. Drag the “angle slider” or type in a specific angle measurement to view various angle measures in standard position and their corresponding reference angles. The video introduces how the use of standard angle measures allow for patterns to be seen in each quadrants. Patterns in the Unit Circle: Reference Angles

18 Unit Circle PDF Print out the Unit Circle PDF. Marked on the unit circle are the coordinates of the special right triangles (30/60/90 and 45/45/90). Any size angle can be located on a unit circle, however it is expected that you know the special right triangles coordinates without the aid of a calculator. Remember there are patterns among the quadrants. Use reference angles for angles not in Quadrant I. Remember on the unit circle the sin ratio of an angle is the y- coordinate of the point on the circle; and the cos ratio is the x- coordinate. Sin and Cos Ratios for Special Angles on the Unit Circle

19 Practice on Unit Circle Video As the measure of an angle varies so does the sine and cosine ratios. As the angle measure (independent variable) changes so do the trig ratios (dependent variable). For example: a 15° angle has a cosine ratio slightly less than 1; whereas an 82° angle has a cosine ratio close to 0. The video show examples of angle measures that are not all special angles [multiples of 30˚( π/6 radians) or 45˚(π/4 radians)]. These ratios can be found using a calculator. TI Calculator note: Use the MODE key to toggle between radian and degree measures. Sin and Cos of Non-Special Angles on the Unit Circle

20 Sin and Cos WebsiteSin and Cos Website Quadrantal angles videoQuadrantal angles videoSin and Cos WebsiteSin and Cos Website Quadrantal angles videoQuadrantal angles video Hit the red arrow to drag the red point around the circle. Notice how the values of x (cos) and y (sin) change and the smallest and largest each ratio can be and at what angle these occur. Note when the point is on an axis (multiples of 90° or π/2 radians). These are called quadrantal angles as the video explains. Scroll down the applet page and again hit the red arrow then drag the red dot around the circle. The resulting graphs show the relationship between varying angle measures and the sin (blue) and cos (green) ratios. Graphs of Sin and Cos Functions

21 Graph of Sin and Cos Video Trig functions relate angle measures to the various trig ratios; the independent variable is the angle measure and the dependent variable is the trig ratio. The video explains how specific points on the unit circle can be used to draw the standard sin and cos function graphs. The standard (or basic) sin and cos functions are: y = sin x y = cos x Graphs of Sin and Cos Functions Continued

22 Trig Function Parameters Website Trig functions have 4 parameters generally named: a, b, c, d outlined on the website. Think about linear functions (y = mx + b) and their 2 parameters, m and b. For example: the standard (or basic) sin function: y = sin x has the parameters: a = 1; b = 1; c = 0; d = 0 y = 1 sin 1(x – 0) + 0 Trig Function Parameters

23 Sin Domain /Range VideoSin Domain /Range Video Sin, Cos Graph Properties VideoSin, Cos Graph Properties VideoSin Domain /Range VideoSin Domain /Range Video Sin, Cos Graph Properties VideoSin, Cos Graph Properties Video The first video graphs the sin function and discusses the properties of the standard sin function: Domain and Range The second video reviews the standard sin and cos graphs as well as the domain and range for both sin and cos functions before introducing another property: Period Properties of the Standard Sin and Cos Functions: Domain and Range; Period

24 Period, Amplitude, Frequency WebsitePeriod, Amplitude, Frequency Website Amplitude Period Basics VideoAmplitude Period Basics VideoPeriod, Amplitude, Frequency WebsitePeriod, Amplitude, Frequency Website Amplitude Period Basics VideoAmplitude Period Basics Video Properties of Standard Sin and Cos Function: Period, Frequency, Amplitude The website explains properties of trig functions: Amplitude, Period, Frequency and Horizontal Shift The video shows the effect parameter changes have on the standard trig function. The example detailed is: y = -½ cos 3x Next is introduced the last of the three main trig functions: tangent

25 Tan Ratio on Unit Circle VideoTan Ratio on Unit Circle Video Tan Ratio Website Tan Ratio WebsiteTan Ratio on Unit Circle VideoTan Ratio on Unit Circle Video Tan Ratio Website Tan Ratio Website The video explains how to use the unit circle to find the value of the tan ratio. The website includes an applet where you drag the point around the circle illustrating how the tan ratio changes. One more Trig Function: Tangent (tan)

26 Tan Function Graph and Properties Video Tan Function Graph and Properties Video Tan Function Graph WebsiteTan Function Graph Website Tan Function Graph and Properties Video Tan Function Graph and Properties Video Tan Function Graph WebsiteTan Function Graph Website The website explores the graph of the relationship between angle measure and the tan ratio providing the graph of the standard tan function: y = tan x The video graphs y = tan x by plotting specific points as well as using a calculator. The video explains the properties of the standard tan function: Domain, Range, and Period (note: the tan function has no amplitude) Properties of the Standard Tan Function Graph: Domain and Range, Period

27 Sin Cos Vertical & Horizontal Shift Video Sin Cos Vertical & Horizontal Shift Video Sin Cos Period, Amplitude VideoSin Cos Period, Amplitude Video Sin Cos Vertical & Horizontal Shift Video Sin Cos Vertical & Horizontal Shift Video Sin Cos Period, Amplitude VideoSin Cos Period, Amplitude Video Transformations: Basics The videos discuss the 4 transformation types: amplitude, period and horizontal and vertical shifts IMPORTANT: The use of the four parameters (a, b, c, d) is not universal. The main variations are: y = a sin [b (x + c)] + d AND y = a sin (bx – c) + d In the first b is factored out and the opposite of c is the horizontal shift; in the second b is not factored out and c/b is the horizontal shift The mathispower4u video switches the naming of c and d; however what you name the parameters is not important. Pay attention to what values operate on the angle measure before the trig ratio is found; and what values operate on the trig ratio. Some trig functions name the angle variable (input) using x; and others use θ. Be careful to note how each of the resources name the general trig function.

28 Trig Graph Transformation Video Tan Cot Transformation Video Illuminations applet Geogebra Applet Trig Graph Transformation VideoTan Cot Transformation VideoIlluminations applet Geogebra Applet Trig Graph Transformation Video Tan Cot Transformation Video Illuminations applet Geogebra Applet Trig Graph Transformation VideoTan Cot Transformation VideoIlluminations applet Geogebra Applet The videos shows all 4 transformations; the first video for cos and sin; the second for tan and its reciprocal function cot. The applets isolate any one of the 4 parameters (a, b, c, d). The Geogebra applet (sin and cos) uses sliders The Illuminations applet uses pull down menus to select any of the 6 trig functions and parameters; radio buttons allow selecting degrees or radians. IMPORTANT to note the varied use in the naming of the parameters as explained in the previous slide. Transformations of Trig Functions

29 Transformation Review PDFTransformation Review PDF Transformed Trig Function Video Transformation Practice VideoTransformed Trig Function VideoTransformation Practice Video Transformation Review PDFTransformation Review PDF Transformed Trig Function Video Transformation Practice VideoTransformed Trig Function VideoTransformation Practice Video Trig Transformation Practice

30 Secant, Cosecant, Cotangent Trig Ratios PDFSecant, Cosecant, Cotangent Trig Ratios PDF Secant Cosecant Graph VideoSecant Cosecant Graph VideoSecant, Cosecant, Cotangent Trig Ratios PDFSecant, Cosecant, Cotangent Trig Ratios PDF Secant Cosecant Graph VideoSecant Cosecant Graph Video The PDF provides the graphs of the the reciprocal trig functions. There are exercises with solutions at the end of the document. The video develops the graphs of sec and csc from the graphs of cos and sin and including graphs with transformations. Other Trig Functions: Reciprocals

31 Trig Application Problems WebsiteTrig Application Problems Website Sin Application Problem Video Trig Applications VideoSin Application Problem VideoTrig Applications Video Trig Application Problems WebsiteTrig Application Problems Website Sin Application Problem Video Trig Applications VideoSin Application Problem VideoTrig Applications Video The first video uses a sin function to model the motion of a spring. The second video models various real world periodic functions and their properties. The website provides practice problems and solutions. Trig Applications Problems

32 Inverse Trig Function Notation Website You have determined the length of a side of a triangle using the trig ratios and functions. Sometimes, we know the lengths of the sides of a triangle but we want to find the angle whose sides ratios are known. To do this, we use inverse trig functions. The notation for the inverse sine function is sin -1 (x) or arcsin(x) where x represents the known sine ratio. The inverse cosine function is cos -1 (x) or arccos(x) The inverse tangent function is tan -1 (x) or arctan(x) The website explains a common notational misunderstanding. Inverse Trigonometric Functions

33 Inverse Trig Function VideoInverse Trig Function Video Inverse Trig Function PDFInverse Trig Function PDFInverse Trig Function VideoInverse Trig Function Video Inverse Trig Function PDFInverse Trig Function PDF The video starts with inverse sine. You may view videos about the inverse cosine and inverse tangent functions by clicking on links on the left side of the screen. In the following PDF resource read sections Inverse Trig Functions Resources

34 Inverse Trig Functions Problems Website This resource gives further explanation of the inverse trig functions with examples and practice problems with the answers. Scroll over the colored areas to see the problem solutions. Practice Problems using the Inverse Trig Functions


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