h.melikian/12001 5.2:triangles and right triangle trigonometry dr.hayk melikyan/ departmen of...
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H.Melikian/1200 1
5.2:Triangles and Right Triangle Trigonometry
Dr .Hayk Melikyan/ Departmen of Mathematics and CS/ [email protected]
1. Classifying Triangles2. Using the Pythagorean Theorem3. Understanding Similar Triangles4. Understanding Special Right Triangles5. 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
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Classification of Triangles
Triangles can be classified according to their angles:Acute: 3 acute anglesObtuse: One obtuse angleRight: One right angle
Triangles can be classified according to their sides:
Scalene: no congruent sidesIsosceles: two congruent sidesEquilateral: 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 TheoremGiven 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.
o
a
h
o2 + a2 = h2.
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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.
9
14
2 2 2a b c
2 2 29 14 c
281 196 c
2277 c 277 c
what if
915
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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
BC
D
EF
AC AB
DF DE
15 10
12 DE 8DE DE
AC BC
DF EF
15 8
12 EFEF 8DE
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Objectives:
Use right triangles to evaluate trigonometric functions.
Find function values for
Recognize and use fundamental identities.
Use equal cofunctions of complements.
Evaluate trigonometric functions with a calculator.
Use right triangle trigonometry to solve applied problems.
30 ,45 , and 60 .6 4 3
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The Six Trigonometric Functions
The six trigonometric functions are:
Function Abbreviationsine sincosine costangent tancosecant cscsecant seccotangent cot
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Right Triangle Definitions of Trigonometric Functions
In general, the trigonometric functionsof depend only on the size of angleand not on the size of the triangle.
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Right Triangle Definitions of Trigonometric Functions(continued)
In general, the trigonometric functionsof depend only on the size of angleand 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.2 2 2a b c 2 2 23 4 9 16 25c
25 5c
3sin
5
4cos
5
3tan
4
5csc
3
5sec
4
4cot
3
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Function Values for Some Special Angles
A right triangle with a 45°, or radian, angle is
isosceles – that is, it has two sides of equal length.4
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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 radian angle.
In a 30-60-90 triangle, the measure of the side opposite the 30°
angle is one-half the measure of the hypotenuse.
6
3
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Example: Evaluating Trigonometric Functions of 45°Use the figure to find csc 45°, sec 45°, and cot 45°.
length of hypotenusecsc45
length of side opposite 45
2
21
length of hypotenusesec45
length of side adjacent to 45
2
21
length of side adjacent to 45cot 45
length of side opposite 45
11
1
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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.
length of side opposite 60tan 60
length of side adjacent to 60
33
1
length of side opposite 30tan30
length of side adjacent to 30
1 1 3 333 3 3
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Example: Using Quotient and Reciprocal Identities
Given and find the value of each of the
four remaining trigonometric functions.
2sin
3
5cos
3
sintan
cos
235
3
2 3 23 5 5
2 5 2 5
55 5
1csc
sin
1 3
2 23
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Example: Using Quotient and Reciprocal Identities (continued)
Given and find the value of
each of the four remaining trigonometric functions.
2sin
3 5
cos3
1sec
cos
1 3
5 53
3 5 3 555 5
1cot
tan
1 5
2 5 2 55
5 5 5 5 52 5 22 5 5
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Example: Using a Pythagorean Identity
Given that and is an acute angle,
find the value of using a trigonometric
identity.
1sin
2
cos2 2sin cos 1
221
cos 12
21cos 1
4
2 1cos 1
4
2 3cos
4
3 3cos
4 2
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Trigonometric Functions and Complements
Two positive angles are complements if their sum is 90° or
Any pair of trigonometric functions f and g for which
and are called cofunctions.
.2
( ) (90 )f g
( ) (90 )g f
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Using Cofunction Identities
Find a cofunction with the same value as the given expression:a.
b.
sin 46 cos(90 46 ) cos44
cot12 6 5
tan tan tan2 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.
Use a calculator to find the value to four decimal places:
a. sin72.8° (hint: Be sure to set the calculator to degree mode)
b. csc1.5 (hint: Be sure to set the calculator to radian mode)
Example: Evaluating Trigonometric Functions with a Calculator
sin 72.8 0.9553
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 24750a
750tan 24a
333.9a
The distance across the lake
is approximately 333.9 yards.