Download The Tangent Ratio - Distribution Access

TEACHER’S GUIDE
TEACHER’S GUIDE
TEACHER’S GUIDE
Extension Activities
• Have students use the Internet and other reference material to learn
about applications of trigonometry in construction, engineering,
science, and art. Have them present their findings to the class.Their
work should include a written summary as well as diagrams and
drawings.
• Challenge small groups to build their own clinometers and measure an
object on the school grounds (flagpole, height of gymnasium, etc.).
Simple instructions can be found at the following Web site:
www.zip.com.au/~elanora/tclinom.html
Suggested Internet Resources
Periodically, Internet Resources are updated on our Web site at
www.LibraryVideo.com
• www.apsu.edu/hamelt/pictures/Department/clinometer.html
The Tangent Ratio
www.nrm.qld.gov.au/education/modules/junior_secondary/
secondary_greenhouse/clinometer.pdf
Grades 8 & up
These pages contain lessons that include instructions on building a
simple clinometer to calculate the height of objects.
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
• astrosun.tn.cornell.edu/courses/astro201/
famous_people.htm
This site contains interesting information on famous ancient scientists
who used angles to study the Earth and moon.
• 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.
Suggested Print Resources
• Blum, Raymond. Classic MatheMagic. Barnes & Noble Books, New
York, NY; 2002.
• Gardner, Martin. The Colossal Book of Mathematics: Classic Puzzles,
Paradoxes, and Problems. W.W. Norton & Company; 2001.
• Niederman, Derrick. Math Puzzles for the Clever Mind. Sterling
Publishing Company, New York, NY; 2001.
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 2002 Channel Four Television Corporation
All rights reserved.
5
B2117
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
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 was 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 Earth and heavens.
‘Trigon’ comes from Greek roots meaning ‘three sides’ and is another name
for a triangle,‘-metry’ is a suffix meaning ‘measurement,’ and the full term
roughly translates to ‘triangle measurement.’ The Greeks developed trigonometry into an ordered science by analyzing the arcs of circles.
Each type of trigonometric calculation (finding an angle, finding the opposite
and finding the adjacent) is demonstrated in detail with the necessary
rearrangements of the basic formula (Tan A = opposite/adjacent) discussed.
Some preparatory work on similar triangles and constant ratios would be
helpful prior to viewing. Students will probably need considerable classroom
reinforcement of the techniques covered. Knowledge of the other ratios is
not assumed.Therefore, the program could be viewed by students who have
only been introduced to the tangent or used to provide enrichment and
context alongside teaching of all three ratios.
The roller coaster item used opposite = 30 or 20 and adjacent = 33 or 22 to
find the angle of 42º.
The height (20m) of the Angel of the North was calculated from a distance of
23.8 m with an angle of elevation to the top of the head of 40º.
At the lighthouse, Katie was 41.6 m above sea level and she viewed Jamie at
an angle of depression of 10º, making his distance from the rocks 236 m.
Learning Objectives
• Understand that the ratio ‘Tan A = opposite/adjacent’ is constant for a
given angle in a right-angled triangle.
• Use the tangent ratio to calculate an angle in a right triangle.
• Use the tangent ratio to calculate either of the perpendicular sides in a
right triangle.
• Work with angles of elevation and depression.
• Draw an appropriate right 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 triangle.
• Calculate the length of the hypotenuse.
Vocabulary
adjacent — The triangle leg next to the angle being analyzed in a right
triangle.
Program Overview
Katie points out that the tan in question is the tangent ratio! She goes on to
explain some right-angled triangle terminology and introduces the tangent as
the opposite over the adjacent. Jamie demonstrates how the tangent
increases as the angle increases.
On most roller coasters, the first drop is the steepest. How steep is the first
drop on the ‘Ultimate’ roller coaster at Lightwater Valley? Jamie uses the
tangent ratio to work out the angle of the first drop, drawing similar triangles
to get the measurements he needs.
The ‘Angel of the North’ sculpture stands impressively beside the A1 and is as
high as four double-decker buses. Katie sets out to calculate exactly how high
that is, with the aid of a clinometer and the tangent ratio.
Jamie takes to the high seas in a bid to illustrate a context that’s a textbook
favorite. Katie is at the top of a lighthouse looking down towards Jamie, and
the task is to work out how far his boat is from the rocks at the foot of the
cliff. Katie needs to apply the tangent ratio to the problem quickly as Jamie is
beginning to feel seasick!
degree — A unit equal to 1/360 of a circle.
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.The sum of the
angles of a right triangle are 180 degrees.
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 A = opposite / hypotenuse).
tangent, tan — In a right triangle, the ratio between the leg opposite the
angle and the adjacent leg (tan A = opposite/adjacent).
trigonometry — The study of how the sides and angles of a triangle are
related to each other.
(Continued)
clinometer — A device that measures the angular distance in degrees
from the horizon to an object above the horizon.
Questions
1. Complete the general equation for the tangent ratio.
2.What is a clinometer? What is a clinometer used for?
3. Jerome stood 500 cm from the base of a tree. His “eye” height was
136 cm. Through the clinometer, he read a 14% angle which has a
tangent of .25. Calculate the height of the tree.
Answers
1. tan A = opposite / adjacent
2.An instrument for measuring angles of elevation or inclination.
Clinometers are used by scientists studying forest growth, geologists,
astronomers, jewelers and engineers.
3. First, multiply 500 x .25 = 125.Then add Jerome’s eye height of 136 to
arrive at 261 cm.Then convert that figure to meters.The height of the
tree is 2.61 meters.
Key Facts and Exam Tips
• A right-angled triangle contains one angle of 90°.
• 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 tangent ratio is constant for any given angle. Remember the
formula ‘Tan A = opposite/adjacent.’ This formula can be used to calculate the length of a side or the size of an angle. Practice rearranging the
formula to find the length of the opposite side or of the adjacent side.
• Learn how to operate the tangent function on your calculator. Make
sure you know the order in which you need to enter values on your
machine.
• To find an angle, you need to use the inverse tangent function (tan-1).
Again, check that you can do this on your calculator.
• Make sure you know how to check whether your calculator is working
in degrees and how to correct it if it isn't.
(Continued)
2
3
4