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H.Melikian/12001 5.2:Triangles and Right Triangle Trigonometry Dr.Hayk Melikyan/ Departmen of...

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H.Melikian/1200 5.2:Triangles and Right Triangle Trigonometry Dr .Hayk Melikyan/ Departmen of Mathematics and CS/ [email protected] 1. Classifying Triangles 2. Using the Pythagorean Theorem 3. Understanding Similar Triangles 4. Understanding Special Right Triangles 5. 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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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

H.Melikian/1200 2

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

H.Melikian/1200 3

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.

H.Melikian/1200 4

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

H.Melikian/1200 5

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

H.Melikian/1200 6

H.Melikian/1200 7

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

H.Melikian/1200 8

The Six Trigonometric Functions

The six trigonometric functions are:

Function Abbreviationsine sincosine costangent tancosecant cscsecant seccotangent cot

H.Melikian/1200 9

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.

H.Melikian/1200 10

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.

H.Melikian/1200 11

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

H.Melikian/1200 12

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

H.Melikian/1200 13

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

H.Melikian/1200 14

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

H.Melikian/1200 15

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

H.Melikian/1200 16

Trigonometric Functions of Special Angles

H.Melikian/1200 17

Fundamental Identities

H.Melikian/1200 18

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

H.Melikian/1200 19

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

H.Melikian/1200 20

The Pythagorean Identities

H.Melikian/1200 21

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

H.Melikian/1200 22

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

H.Melikian/1200 23

Cofunction Identities

H.Melikian/1200 24

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

H.Melikian/1200 25

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

H.Melikian/1200 26

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.

H.Melikian/1200 27

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.


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