Functions in a Right Triangle

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Trigonometry

• Functions in a Right Triangle

By John Smith
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How can you tell the height of a building by
using a right triangle?
?
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Give Up?
With the help of a calculator, paper, pencil, and
Trigonometric functions it will be possible to
estimate the height of a building.
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Objectives

• Know the names of the sides of a right triangle
• Learn names of trigonometric functions
• Apply trigonometric functions to find the unknown
  in a right triangle by solving problems
• Find the height of a building by using a right
  triangle

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Names of the Sides of a Right Triangle
Knowing the names of the sides of a right
triangle is essential. These names are used to
communicate which side of a triangle you are
referring to without using a diagram. If you
misname the sides, your calculations of ratios
will be off.
Hypotenuse
Adjacent
Opposite
Click on the buttons to learn the names of the
sides. Make sure to learn each one then click
arrow to continue.
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Hypotenuse
The hypotenuse is the side directly across from
the 90º angle and is always the longest side of a
right triangle.
When you have completed learning the side, click
back.
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Adjacent
The gray shading indicates the reference angle.
The adjacent side is the side next to the
reference angle.
When you are finished learning the side, click on
button to go back.
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Opposite
The gray shading indicates the reference angle.
The opposite side is the side directly across
from the reference angle.
When you are finished learning the side click on
button to go back.
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SOH-CAH-TOA
This is a mnemonic device commonly used for
remembering the following three trigonometric
equations that will be used in this lesson.
Click on each and learn them well.
Sine
Cosine
Tangent
Next
After checking your problems click to continue
lesson
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Sine
To find angle D, we can use the relationship that
sin D side opposite b hypotenuse
c
Or solving for D, we get that
D sin-1 (b/c) or D arcsin (b/c)
Click for problems using sin
Problem
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Cosine
To find angle D, we can use the relationship that
cos D __adjacent___ a hypotenuse
c
Or solving for D, we get that
D cos-1 (a/c) or D arccos (a/c)
Click for problem using cos
Problem
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Tangent
To find angle D, we can use the relationship that
tan D side opposite b
adjacent a
Or solving for D, we get that
D tan-1 (b/a) or D arctan (b/a)
Click for problem using tan
Problems
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Problems using sin
Side b 5 side c 15 angle E 30º
1. Find angle D. Round to the nearest tenth
degree.
2. Using angle E, find side a.
Solutions
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Problems using cos
Side a 10 side c 15 angle E 30º
1. Find angle D. Round to the nearest tenth
degree.
2. Using angle E, find side b.
Solutions
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Problems using tan
Side a 10 side b 5 angle E 30º
1. Find angle D. Round to the nearest tenth
degree.
2. Using angle E, find side a.
Solutions
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Solutions using sin
Side b 11.18cm side c 15cm angle E 30º
1. To find angle D solve sin D b/c
sin D 11.18/15 D arcsin(.7453)
Angle D 48.18º
2. Using angle E, side a is sin E a/c
sin 30º a/15 15 sin 30º a a
7.5cm
Back
When you finish checking click back
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Solutions using cos
Side a 10 side c 15 angle E 30º
1. Angle D using cos is cos D a/c
cos D 10/15 D arccos(.6667) Angle
D 48.1897º
2. Using angle E side b is cos E
b/c cos 30º b/ 15 15 cos 30º
12.99 cm
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When you finish checking click
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Solutions using tan
Side a 10 side b 5 angle E 30º
1. Angle D using tan is tan D b/a
tan D 5/10 D
arctan(.5) Angle D 26.57º
2. Using the tan of angle E, side a is tan E
a/b tan 30º a/5 5 tan
30º a a 2.89cm
When you are finished click
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More Practice...
Now that you have learned how to find the missing
sides and/or angles of a right triangle using
trigonometric functions, click on the following
web page to do more problems
www.webmath.com/rtri.html
On a sheet of paper you must 1. Use each trig
function to find the angles and sides to right
triangles that you have generated 2. draw and
label at least three right triangles 3. You may
use the computer as a guide
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Now apply what youve learned...
The following problem will test what you have
learned. It is now time to estimate the height
of a building. On a sheet of paper draw a
diagram for the problem. Make sure to label all
of the parts of the triangle. Show your work
TEST
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Estimating the Height of a Building
Consider the problem of calculating the height of
a building. As a student stands outside the
building, she wonders how she will accomplish
such a task without a really long tape measure!
She notes that, at a distance of 10 feet from the
building, the angle of elevation is 75º. This is
all the information she needs to estimate the
height of the building. How will she accomplish
this?
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Credits
Advanced Mathematical concepts Pre-Calculus with
Applications 1994 pp. 269-273 Book. Grades
11th-12th. Southwestern Geometry An Integrated
Approach 1998 pp.573-580. Textbook. Grades
11th-12th. http//www.ucl.ac.hk/mathematics/geoma
th/trignb/trigmod.html. Website. Grades 9th-12th.
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