Module 7: Triangle

Trigonometry

1. The Six Trigonometric Ratios

2. Trigonometric Ratios of Special Angles

3. Angles of Elevation & Depression

4. Applications of Right Triangle

5. Oblique Triangles

5.1 Law of sines and its application

5.2 Law of cosines and its application

At the end of the session, the participants will be able to:

1. illustrate the six trigonometric ratios;2. apply trigonometric ratios to find the unknown

measures of the parts of a triangle3. determine the trigonometric ratios involving

special angles;4. compute the numerical values of trigonometric expressions involving special angles5. illustrate and distinguish angles of elevation

and angle of depression;

6. solve problems involving angles of elevation and depression;

7. use the trigonometric ratios in solving real-life problems involving right triangles;

8. illustrate the laws of sines and cosines; and

9. solve problems involving oblique triangles.

The word trigonometry is based on the Greek words for triangle (trigonon) and measure (metron). This study dates back more than 300 years and over the centuries has been instrumental in developing knowledge in areas such as architecture, astronomy, navigation, and surveying.

A right triangle is a triangle in which one of its angles is a right angle (90º). The longest side is called the hypotenuse. The other two sides are the legs.

B

a

C b A

c hypotenuse

If a, b and c are the length of the sides of a right triangle where c is the hypotenuse, then

c2=a2+b2

Also, if A and B are the acute angles of the right triangle, then A+B=900.

If θ is an acute angle of a right triangle, then

tosideadjacent

hypotenuse sec

toside opposite

hypotenuse csc

toside opposite

tosideadjacent cot

tosideadjacent

toside oppositetan

hypotenuse

tosideadjacent cos

hypotenuse

toside oppositesin

B

3 5

C 4 A

Find the missing part.

The trigonometric ratios can easily be remembered by the acronym

SOH-CAH-TOA. This means: SOH - Sine is Opposite over Hypotenuse

CAH - Cosine is Adjacent over Hypotenuse TOA - Tangent is Opposite over Adjacent

Activity: Special Triangles & Exact Values1. On a piece of paper, draw a square.

2. Set the length of each side to 1. 3. Cut the square along a diagonal.

What triangles are formed? 4. Indicate the right angle in the

triangle.

5. Determine the exact length of the hypotenuse using the Pythagorean theorem.

6. Label the hypotenuse and all other sides and angles.

7. Write the primary trigonometric ratios for 450 (SOH-CAH-TOA)

8. Rewrite the primary trigonometric ratios with rationalized denominators.

15

Draw an equilateral triangle as shown below.

From any vertex, draw an altitude.

16

Using Pythagorean theorem,

17

18

Solving real-life problems involving right triangles requires knowledge of some significant terms, such as line of sight, angle of elevation, and angle of depression.

The line of sight of an object P from an observation point O is the line segment from O to P.

If the object P is above the horizontal ray OA then the angle formed by the line of sight OP and the horizontal ray OA is called the angle of elevation of the object P from the observation point O.

line of sight

O A

Pangle of elevation

On the other hand, if the object P is below the horizontal ray OA then the angle formed by the line of sight OP and the horizontal ray OA is called the angle of depression of the object P from the observation point O.

line of sight

O A

P

angle of depression

Be careful when interpreting the angle of depression. Both the angle of elevation and the angle of depression are measured between the line of sight and a horizontal line.

In solving word problems, it’s better to use the method that George Polya had introduced.

4-stages of Polya’s problem solving cycle

S – see (understand the problem)

T – try (devise a plan)A – act (carry out the plan)R – relook (evaluate the

solution) Ho Weng Kin. 2012. Mathematics Education in Singapore: Current

Trends and Future Directions

Photograph by Paul Halmos

http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Polya.html

24

A student wishes to determine the height of a building. She measures a distance of 25 m from the center of the base of the building and finds that the angle of elevation to the top of the building is 47. How high is the building?

a47

25 m

The angle of depression of a boat from the top of a lighthouse is 32. If the lighthouse stands 60 ft tall., how far is the boat from the base of the lighthouse?

x

32

60

To measure the height of a tree. Two sightings both on the same side of the tree are taken a distance of 4m apart. If the first angle of elevation is 400 and second is 330, what is the height of the tree?

The bearing of a particular location P from an observer at O contains two components: the acute angle line OP makes with the north-south line and an indication if the angle formed is towards the east or west. In writing the bearing of P from O, N or S (in reference to North or South) is written first followed by the angle formed (in degrees) and then by W or E to indicate whether the angle is towards the west or east.

S

W

N

EO

P55

N55E

N

S

W EO

P 50

S50W

N

S

W EO

P 75

S75E

E

N

S

WO

P 33

N33W

The course or heading of a ship or plane is the angle formed with the initial side at the north ray, measured clockwise. If the ship is due north, then its course is 0; due east, 90; due south, 180; and due west, 270.

The course of the ship in Figure 1 and Figure 2 are 56 and 260, respectively.

N

S

W E

56

Figure 1

N

S

W EO260

Figure 2

Radar stations A and B are on east-west line, 3.7 km apart. Station A detects plane at C, on a bearing of 610. Station B simultaneously detects the same plane on a bearing of 3310. Find the distance from A to C.

The Law of Sine and its Applications

The Law of cosine and its Applications

33

An oblique triangle is a triangle which does not contain a right angle.

Two classifications:

34

Consider the triangle below

35

36

37

38

Two angles and one side are given (SAA Case)

Two sides and an angle opposite one of these sides (SSA Case)

1. a=21.3, b=18.9, A=650

2. a=8.7, A= 36.70, b=12.43. a =100, c = 125, A=670

42

Two sides and the included angle are known (SAS Case)

Three sides are known (SSS Case)

c=15.4m, b=12.9m A=42.30

If you can love your students for who they are not for what they do,

If you can offer greater dreams even to few… If you can touch a single life and improve it for one day, If you can bring encouragement through the words you

say… If you can back up with your life the truth of what you

teach, And take time to listen to the hearts you want to reach… You’ll find your classroom full each day with eager minds

to feed, And as a teacher and friend, you will always succeed.

Rebecca Barlow Jordan

The function of education is to teach one to think intensively and to think critically... Intelligence plus character - that is the goal of true education.

   Martin Luther King, Jr. quotes

Algebra and Trigonometry, Fifth Edition by Max Sobel and Norbert Lerner

College Algebra and Trigonometry by Louis Leithold

Learner’s Material for Mathematics Grade 9