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Trigonometry

Trigonometry. Right Triangle and Trigonometric Functions. Overview. T his 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. Topics.

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Trigonometry

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  1. Trigonometry Right Triangle and Trigonometric Functions

  2. 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.

  3. 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

  4. Pythagorean Theorem (review) 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.

  5. Pythagorean Theorem: Practice Problems 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.

  6. Special Right Triangles 30˚-60˚-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.

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

  8. Right Triangle Trig Ratios: Sine, Cosine, Tangent Right Triangle Trig VideoTrig 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

  9. Right Triangle Trig Practice Problems 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.

  10. Three Additional Trig Ratios: Cotanget, Cosecant, Secant (Reciprocals) 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.

  11. Right Triangle Trig Practice Problems 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.

  12. Angles: Degree and Radian Measure Radian 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.

  13. Degree/Radian Additional help Radian VideoRadians PDF • On the PDF stop after problem #31. • On the video stop when the presenter defines arc length (writes s =)

  14. Trig ratios for Angles ≥ 90˚ Unit Circle PDF 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

  15. Unit Circle Definition of Trig Ratios Use 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.

  16. Patterns in the Unit Circle: Reference Angles Reference Angles WebsitePatterns 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.

  17. Sin and Cos Ratios for Special Angles on the Unit Circle 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.

  18. Sin and Cos of Non-Special Angles on the Unit Circle 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.

  19. Graphs of Sin and Cos Functions Sin and Cos WebsiteQuadrantal 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.

  20. Graphs of Sin and Cos Functions Continued 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

  21. Trig Function Parameters 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 xhas the parameters: a = 1; b = 1; c = 0; d = 0 • y = 1 sin 1(x – 0) + 0

  22. Properties of the Standard Sin and Cos Functions: Domain and Range; Period Sin Domain /Range 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

  23. Properties of Standard Sin and Cos Function: Period, Frequency, Amplitude Period, Amplitude, Frequency WebsiteAmplitude Period Basics Video • 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

  24. One more Trig Function: Tangent (tan) Tan Ratio on Unit Circle Video 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.

  25. Properties of the Standard Tan Function Graph: Domain and Range, Period Tan Function Graph and Properties VideoTan 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)

  26. Transformations: Basics Sin Cos Vertical & Horizontal Shift Video Sin CosPeriod, Amplitude Video • 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.

  27. Transformations of Trig Functions Trig Graph Transformation VideoTan Cot Transformation VideoIlluminations appletGeogebra 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.

  28. Trig Transformation Practice Transformation Review PDFTransformed Trig Function VideoTransformation Practice Video • The first video shows several transformed trig graphs. • The video shows the graphing of two transformed functions using tables: • y = sin (2x – π) • y = 3 cos (x) + 4

  29. Other Trig Functions: Reciprocals Secant, Cosecant, Cotangent Trig Ratios PDFSecant 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.

  30. Trig Applications Problems Trig Application Problems WebsiteSin 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.

  31. Inverse Trigonometric Functions 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.

  32. Inverse Trig Functions Resources Inverse Trig Function VideoInverse 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 1.1-1.5

  33. Practice Problems using the Inverse Trig Functions 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.

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