Download The Sine Ratio - Distribution Access

TEACHER’S GUIDE
• Make sure you know how to check whether your calculator is working
in degrees, and how to correct it if it isn’t.
Extension Activity
• Groups can begin with each student drawing a right triangle containing
a given angle. No other conditions are set, so a set of similar but not
identical triangles will be produced. Students then record the lengths of
the sides as accurately as possible, using the labels ‘opposite,’‘adjacent’
and ‘hypotenuse’ in relation to the given angle.The results are pooled
and entered in columns on a spreadsheet. Students should then investigate the data on the spreadsheet.This should lead them to recognize
that no matter what size the triangle, the ratios of equivalent pairs of
sides are equal. It is helpful if the spreadsheet columns are labeled with
the calculation performed to generate them — for example,‘opp/hyp.’
Finding all six possible ratios here can generate later discussion of why
only three are needed on calculators or in books of tables.The spreadsheet allows a large number of results to be processed and compared,
showing in each column the tendency to a particular value.Any mistakes in measuring or recording should stand out, and students should
appreciate that the degree of accuracy of their measurements limits that
of the results.
Suggested Internet Resources
Periodically, Internet Resources are updated on our Web site at
www.LibraryVideo.com
• aleph0.clarku.edu/~djoyce/java/trig/
A trigonometry course that includes historical background, illustrations
and Java applets that allow diagrams to be manipulated on screen.
• mathforum.org
The Math Forum at Drexel University hosts a wealth of math resources
for teachers and students.
• www.americatakingaction.com/studyhall/math.htm
This site has an enormous list of helpful math links.
• standards.nctm.org/previous/CurrEvStds/9-12s9.htm
These pages from The National Council of Teachers of Mathematics Web
site contain activities that illustrate the practical application of
trigonometry in general and the sine ratio in particular.
TEACHER’S GUIDE
TEACHER’S GUIDE
Suggested Print Resources
• Johnson,Art. Famous Problems and Their Mathematicians. Teacher
Ideas Press, Portsmouth, NH; 2000.
• Lundy, Miranda. Sacred Geometry. Walker & Company, New York, NY;
2001.
The Sine Ratio
Grades 8 & up
hese engaging programs complement traditional
lessons by encouraging mathematics discovery in
the real world. Using animated graphics, real-life
locales and vibrant young hosts, each program clearly
explains math concepts and presents students with
strategies to improve their problem-solving capabilities. Step-by-step examples of typical exam questions
are illustrated, along with common pitfalls to avoid.
T
TEACHER’S GUIDE
Paula J. Bense, M.Ed.
Curriculum Specialist, Schlessinger Media
COMPLETE LIST OF TITLES
• Area of Circles and
Composite Shapes
• Combined Probability
• Enlargement
Teacher’s Guides Included
and Available Online at:
• Loci
• The Pythagorean Theorem
• The Sine Ratio
• The Tangent Ratio
800-843-3620
Teacher’s Guide Copyright 2004 by Schlessinger Media,
a division of Library Video Company
P.O. Box 580, Wynnewood, PA 19096 • 800-843-3620
Program Copyright 2000 Channel Four Television Corporation
All rights reserved.
5
B2116
This guide is a supplement designed for teachers to
use when presenting this program and provides background information, vocabulary, practice questions
and answers, as well as Internet resources for students
and teachers to explore.
Please note that this series was produced in Great Britain,
where the decimal point is often drawn as a centered dot
(e.g., 3·1415) instead of a period (e.g., 3.1415) as is customary in the United States.
Background
Program Overview
Trigonometry is the branch of mathematics concerned with certain functions
of angles and their applications to geometry. It developed from the study of
right triangles. From the Babylonian civilization until the time of Descartes,
simple trigonometry had been used in surveying, astronomy and navigation.
Both astronomers and sailors, scanning the heavens and the seas, often
needed to calculate distances not directly measurable.They applied certain
basic rules about the relationship between the sides and angles of triangles.
The Egyptians used these relationships in land surveying and when building
the pyramids. Babylonian astronomers related angles to arcs of circles to
study the sky.
The program concentrates on the sine ratio and its applications, from
firefighters using ladders in emergency situations to young water-skiers
preparing for a competition.A champion wheelchair athlete shows how
important the relationships are between sides and angles in right triangles.
Ben explains how in practice the firefighters estimate the correct angle,
using experience and a ‘rule of thumb.’ He measures this angle with a protractor. Katie shows how her diagram is in the same proportions as the real
situation, and so the angle of her ladder is the same as Ben’s. Ben explains
that the height it reaches is the ‘opposite’ side in this case, and the
‘hypotenuse’ is the length of the ladder. He finds the value of the sine ratio
for these lengths. Katie calculates the same ratio from her scale diagram and
gets the same answer as Ben, pointing out that this is because her angle is
the same.
A water-skier explains that the ramps used in competitions are always the
same height and angle. Ben uses that information to calculate the length of
the ramp — using the sine formula to find the hypotenuse.Then a wheelchair athlete explains that ramps are helpful for wheelchair users, but only if
they are built at an angle that makes them easy for everyone to use. She tests
a few and recommends a five degree angle to anyone thinking of building an
access ramp.
The Greeks developed trigonometry into an ordered science by analyzing the
arcs of circles.The first person known to have used trigonometric ratios was
the Greek astronomer and mathematician Hipparchus. He used them around
140 BCE to find the straight-line distances across the curved ‘heavens,’ and
constructed tables of chords at half-degree intervals for all central angles
from 0 to 180o.These trigonometric tables (corresponding to sine, cosine and
tangent) were later improved and extended by other mathematicians. But for
a long time they were used only by astronomers, sailors and cartographers.
Then the remarkable French mathematician Francis Vieta (who preceeded
Descartes by half a century) made an amazing observation. He discovered
that a trigonometric ratio could be used to solve an algebraic equation. In
effect, a series of numbers in a table could represent successive values taken
by an unknown.The statement ‘sine of angle x is y’ could also be written as
an equation ‘y = sin x’.This new way of looking at trigonometric ratios broadened the scope of trigonometry.Then Descartes developed his plotting techniques.An equation like ‘y = sin x’ could be plotted point by point to create a
curve.This curve is an endless wave, and is the exact graphic equivalent of
the ebb and flow of electric current in an ordinary AC power cable.
Although the program deals only with the sine ratio, it would still be relevant
to students who have been introduced to all three ratios. Similar triangles and
the constant ratios that they produce are mentioned. It is assumed that students will have done a fair amount of preparatory work within the topic
prior to viewing.The program would be most useful as a means of showing
students how the theory relates to the real world. It could be used to build
and consolidate understanding of the topic alongside other work. It could
also be helpful in reinforcing, or revising, the techniques needed to solve the
types of problems relating to the topic that are most often encountered.
Learning Objectives
• Draw an appropriate right-angled triangle.
• Identify and name sides in relation to the angle being worked with.
• Substitute values in the formula ‘sin = opposite / hypotenuse.’
• Calculate the length of a side opposite to a known angle in a right-angled
triangle.
• Calculate the length of the hypotenuse.
• Select appropriate calculator functions.
• Substitute lengths of sides into the formula in order to calculate an angle.
• Discover and use the sine ratio.
Vocabulary
adjacent — The triangle leg next to the angle being analyzed in a right
triangle.
hypotenuse — The longest side of a right triangle.
opposite — The triangle leg across from the angle being analyzed in a right
triangle.
triangle — A geometric figure consisting of three points or vertices which
are connected with straight line segments called sides or legs.
right angle — An angle that is 90 degrees.
right triangle — A triangle that has one 90 degree angle.
similar triangles — Triangles of different size but having the same
angles.
sine, sin — In a right triangle, the ratio between the leg opposite the
angle and the hypotenuse (sin = opposite / hypotenuse).
trigonometry — The study of how the sides and angles of a triangle are
related to each other.
Questions
1.What is the symbol often used to label the angle being worked with in
a right triangle?
2. Complete the general equation for the sine ratio.
3. Sidney places the foot of his ladder on horizontal ground and the top
against a vertical wall.The ladder is 16 feet long.The foot of the ladder
is 4 feet from the base of the wall.
(a) How high up the wall does the ladder reach?
(b) What is the angle the base of the ladder makes with the ground?
Answers
1.The theta symbol (θ)
2. sin = opposite / hypotenuse
3. (a) root (162 — 42) = 15.5 ft.
(b) cos angle = 4/16; angle = 75.5º
Key Facts and Exam Tips
• A right triangle contains one angle of 90 degrees. In a right triangle, the
longest side is opposite the right angle and is called the ‘hypotenuse.’
The other two sides are named in relation to the angle you are working
with.The side opposite this angle is called the ‘opposite.’The side next
to this angle is called the ‘adjacent.’You must identify the sides correctly
when solving problems using trigonometry.
• The sine ratio is constant for a given angle. Remember the formula ‘sin
= opposite / hypotenuse.’ This formula can be used to calculate the
length of a side or the size of an angle in a right-angled triangle. Practice
rearranging the formula to find the length of the opposite side or of the
hypotenuse.
• Learn how to operate the sine function on your calculator. Make sure
you know the order in which to enter values on your machine.To find
an angle, you need to use the inverse-sine function (sin—1).Again, check
that you can do this on your calculator.
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