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5.2:Triangles and Right Triangle
Trigonometry
1.
2.
3.
4.
5.
Classifying Triangles
Using the Pythagorean Theorem
Understanding Similar Triangles
Understanding Special Right Triangles
Using Similar Triangles to Solve Applied Problems
6. Use right triangles to evaluate trigonometric functions.
7. Find function values for
8. Recognize and use fundamental identities.
9. Use equal cofunctions of complements.
10. Evaluate trigonometric functions with a calculator.
11. Use right triangle trigonometry to solve applied problem
Dr .Hayk Melikyan/ Departmen of Mathematics and CS/ [email protected]
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Classification of Triangles
Triangles can be classified according to their angles:
Acute: 3 acute angles
Obtuse: One obtuse angle
Right:
One right angle
Triangles can be classified according to their sides:
Scalene:
no congruent sides
Isosceles: two congruent sides
Equilateral: three congruent sides
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Classifying a Triangle
Classify the given triangle as acute, obtuse, right, scalene, isosceles, or
equilateral. State all that apply.
The triangle is acute because all the angles
are less than 90 degrees.
The triangle is scalene since all the sides are
different.
The Pythagorean Theorem
Given any right triangle, the sum of the squares of the lengths of the legs is
equal to the square of the length of the hypotenuse.
h
o
o2 + a2 = h2.
a
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Using the Pythagorean Theorem
Use the Pythagorean Theorem to find the length of the missing side of
the given right triangle.
a b c
2
9
2
2
92  142  c 2
14
81  196  c 2
277  c 2
277  c
what if
9
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Similar Triangles
Triangles that have the same shape but not necessarily the same size.
1. The corresponding angles have the same measure.
2. The ratio of the lengths of any two sides of one triangle is equal to
the ratio of the lengths of the corresponding sides of the other triangle.
Example: Triangles ABC and DEF are similar. Find the lengths of the missing
sides of triangle ABC.
A
D
B
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C
E
DE
F
EF
AC AB

DF DE
AC BC

DF EF
15 10

12 DE
15
8

12 EF
DE  8
DE  8
5
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Objectives:
Use right triangles to evaluate trigonometric functions.
   , 45    , and 60    .
30

Find function values for
 
 
 
6
4
3
Recognize and use fundamental identities.
Use equal cofunctions of complements.
Evaluate trigonometric functions with a calculator.
Use right triangle trigonometry to solve applied problems.
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The Six Trigonometric Functions
The six trigonometric functions are:
Function
sine
cosine
tangent
cosecant
secant
cotangent
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Abbreviation
sin
cos
tan
csc
sec
cot
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Right Triangle Definitions of Trigonometric Functions
In general, the trigonometric functions
of  depend only on the size of angle 
and not on the size of the triangle.
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Right Triangle Definitions of Trigonometric Functions(continued)
In general, the trigonometric functions
of  depend only on the size of angle
and not on the size of the triangle.
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Example: Evaluating Trigonometric Functions
Find the value of the six trigonometric functions in the figure.
We begin by finding c.
a b  c
2
2
2
c 2  32  42  9  16  25
c  25  5
3
sin  
5
4
cos 
5
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tan  
4
5
csc 
3
5
sec 
4
4
cot  
3
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Function Values for Some Special Angles
A right triangle with a 45°, or

radian, angle is isosceles –
4
that is, it has two sides of equal length.
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Function Values for Some Special Angles (continued)
A right triangle that has a 30°, or  radian, angle also has a 60°, or 
3
6
radian angle.
In a 30-60-90 triangle, the measure of the side opposite the 30° angle is onehalf the measure of the hypotenuse.
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Example: Evaluating Trigonometric Functions of 45°
Use the figure to find csc 45°, sec 45°, and cot 45°.
csc 45 
sec 45 
2
length of hypotenuse


1
length of side opposite 45
2
length of hypotenuse
2


length of side adjacent to 45
1
2
1
length of side adjacent to 45   1
cot 45 
1
length of side opposite 45
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Example: Evaluating Trigonometric Functions of 30°and 60°
Use the figure to find tan 60° and tan 30°. If a radical appears in a
denominator, rationalize the denominator.
tan 60 
length of side opposite 60
3


1
length of side adjacent to 60
tan 30 
1
1
length of side opposite 30


3
3
length of side adjacent to 30
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3
3
3

3
3
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Trigonometric Functions of Special Angles
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Fundamental Identities
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Example: Using Quotient and Reciprocal Identities
5
2
Given sin   and cos 
find the value of each of the four
3
3
remaining trigonometric functions.
sin 
tan  
cos
csc 
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2
2
3


5
3
3
3
2  2

5
5
5
5 2 5

5
5
1
1
3
 
2
sin 
2
3
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Example: Using Quotient and Reciprocal Identities (continued)

sin  
2
3
5
Given
and cos  3 find the value of each of the
four remaining trigonometric functions.
sec 
3
1  1  3

5
5
5
cos
5 3 5

5
5
3
5
1  1  5

cot  
2 5
2 5 2 5
tan 
5 5 5
5


25
2
5
5
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The Pythagorean Identities
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Example: Using a Pythagorean Identity
1
sin


Given that
and  is an acute angle, find the
2
value of cos using a trigonometric identity.
sin 2   cos 2   1
2
 1   cos 2   1
 
2
1
 cos 2   1
4
cos 2   1 
cos 2  
cos 
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1
4
3
4
3
3

4
2
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Trigonometric Functions and Complements
Two positive angles are complements if their sum is 90° or

2
.
Any pair of trigonometric functions f and g for which f ( )  g (90   )
and g ( )  f (90   ) are called cofunctions.

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Cofunction Identities
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Using Cofunction Identities
Find a cofunction with the same value as the given expression:
a.
sin 46  cos(90  46)  cos 44
b.
 
6  
5


   tan
cot  tan     tan 
12
 2 12 
 12 12 
12

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Using a Calculator to Evaluate Trigonometric Functions
To evaluate trigonometric functions, we will use the keys on a calculator
that are marked SIN, COS, and TAN. Be sure to set the mode to degrees
or radians, depending on the function that you are evaluating. You
may consult the manual for your calculator for specific directions for
evaluating trigonometric functions.
Example: Evaluating Trigonometric Functions with a Calculator
Use a calculator to find the value to four decimal places:
a. sin72.8° (hint: Be sure to set the calculator to degree mode)
sin 72.8  0.9553
b. csc1.5 (hint: Be sure to set the calculator to radian mode)
csc1.5  1.0025
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Applications: Angle of Elevation and Angle of Depression
An
angle formed by a horizontal line and the line of sight to
an object that is above the horizontal line is called the angle of
elevation. The angle formed by the horizontal line and the
line of sight to an object that is below the horizontal line is
called the angle of depression.
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Example: Problem Solving Using an Angle of Elevation
The irregular blue shape in the figure represents a lake. The distance
across the lake, a, is unknown. To find this distance, a surveyor took the
measurements shown in the figure. What is the distance across the lake?
tan 24 
a
750
a  750 tan 24
a  333.9
The distance across the lake
is approximately 333.9 yards.
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