Transcript Trigonometry

Trigonometry
Right Triangle and Trigonometric Functions
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.
Topics
• Right Triangle Trig
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Pythagorean Theorem
Special Right Triangles
Sine, Cosine, Tangent Ratios
Reciprocal Ratios: Cosecant, Secant, Cotangent
Problem Solving
• Trig Functions
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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
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.
Pythagorean Theorem:
Practice Problems
Practice Problems Website Practice Problems Video
• Now, it is your turn to solve problems using the Pythagorean
Theorem.
• The website has application problems to solve with
explanations provided.
Special Right Triangles
30˚-60˚-90˚ Website
45 ˚- 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
Practice
Special Right Triangle Practice Website
• The website provides practice problems using the
relationships of the sides of the special right triangles.
Right Triangle Trig Ratios:
Sine, Cosine, Tangent
Right Triangle Trig Video Trig 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 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.
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.
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.
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.
Degree/Radian Additional help
Radian Video Radians PDF
• On the PDF stop after problem #31.
• On the video stop when the presenter defines arc length
(writes s =)
Trig ratios for Angles ≥ 90˚
Unit Circle PDF Standard Position/Coterminal Angle Video Unit 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
Unit Circle Definition of Trig
Ratios
Use of Unit Circle Video Unit 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.
Patterns in the Unit Circle:
Reference Angles
Reference Angles Website Patterns 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.
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 ycoordinate of the point on the circle; and the cos ratio is the xcoordinate.
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.
Graphs of Sin and Cos
Functions
Sin and Cos Website
Quadrantal 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 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
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 x has
the parameters: a = 1; b = 1; c = 0; d = 0
• y = 1 sin 1(x – 0) + 0
Properties of the Standard Sin and Cos
Functions: Domain and Range; Period
Sin Domain /Range Video Sin, 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 Standard Sin and Cos Function:
Period, Frequency, Amplitude
Period, Amplitude, Frequency Website Amplitude 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
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.
Properties of the Standard Tan Function
Graph: Domain and Range, Period
Tan Function Graph and Properties Video Tan 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)
Transformations: Basics
Sin Cos Vertical & Horizontal Shift Video
Video
Sin Cos Period, Amplitude
• 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.
Transformations of Trig Functions
Trig Graph Transformation Video Tan Cot Transformation Video Illuminations 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.
Trig Transformation Practice
Transformation Review PDF Transformed Trig Function Video Transformation
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)
1
2
• y = 3 cos ( x) + 4
Other Trig Functions:
Reciprocals
Secant, Cosecant, Cotangent Trig Ratios PDF
Secant 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.
Trig Applications Problems
Trig Application Problems Website
Applications Video
Sin Application Problem Video Trig
• 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.
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.
Inverse Trig Functions
Resources
Inverse Trig Function Video
Inverse 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
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.