Trigonometry is the branch of mathematics that deals with triangles particularly right triangles.

They are behind how sound and light move and are also involved in our perceptions of beauty and other facets on how our mind works.

So trigonometry turns out to be the fundamental to pretty much everything!

A trigonometric function is a ratio of certain parts of a triangle. The names of these ratios are: The sine, cosine, tangent, cosecant, secant, cotangent.

Let us look at this triangle…

a

c

b

ө A

B

C

Given the assigned letters to the sides and angles, we can determine the

following trigonometric functions.

Sinθ=

Cos θ=

Tan θ=

Side Opposite

Side Adjacent

Side AdjacentSide Opposite

Hypotenuse

Hypotenuse

=

=

= a

b

ca

b

c

11

145

2

1 45

2

145

Tan

Cos

Sin

3

130

2

3 30

2

130

Tan

Cos

Sin

1

360

2

160

2

3 60

Tan

Cos

Sin

01

00

11

1 0

01

00

Tan

Cos

Sin

dnTan

Cos

Sin

.0

190

1

090

11

1 90

One of the most ancient subjects studied by scholars all over the world, astronomers haveused trigonometry to calculate the distancefrom the earth to the planets and stars. Its also used in geography and in navigation. The knowledge of trigonometry is used to construct maps, determine the position of an island in relation to longitudes and latitudes.Trigonometry is used in almost everysphere of life around you.

Angle of depression

Angle of elevation

Angle of Elevation: In the picture below, an observer is standing at the top of a building is looking straight ahead (horizontal line). The observer must raise his eyes to see the airplane (slanting line). This is known as the angle of elevation.

Angle of elevation

Horizontal

Angle of Depression:

The angle below horizontal that an observer must look to see an object that is lower than the observer.

Angle of depression

Horizontal

Object

The angle of elevation of the top of a pole measures 45° from a point on the ground 18 ft. away from its base. Find the height of the flagpole.

Solution

Let’s first visualize the situation

Let ‘x’ be the height of the flagpole.

From triangle ABC, tan 45 ° =x/18

x = 18 × tan 45° = 18 × 1=18ft

So, the flagpole

is 18 ft. high.

45 °

A tower stands on the ground. The angle of elevationfrom a point on the ground which is 30 metres away from the footOf the tower is 30⁰. Find the height of the tower. (Take √3 = 1.732)

30⁰

30 m

h

A

BC

Let AB be the tower h metre high.

Let C be a point on the ground which is 30 m away from point B, the foot of the tower.

So BC = 30 m

Then ACB = 30⁰

Now we have to find AB i.e. height ‘h’ of the tower .

Solution .

AB

B1

√3h

30

h 30

√330 x √3

√3 x √3

30 x √3

3

10√3 m

10 x 1.732 m

17.32 m

Hence, height of the tower = 17.32 m

tan 30⁰

Now we shall find the trigonometric ratio combining AB and BC .

30⁰

30 m

h

A

BC

30 °

30 °

An airplane is flying at a height of 2 miles above the level ground.The angle of depression from the plane to the foot of a tree is 30°.Find the distance that the air plane must fly to be directly abovethe tree.

Step 1: Let ‘x’ be the distance the airplane

must fly to be directly above the tree.

Step 2: The level ground and

the horizontal are parallel, so

the alternate interior angles are equal in

measure.

Step 3: In triangle ABC, tan 30=AB/x.

Step 4: x = 2 / tan 30

Step 5: x = (2*31/2)

Step 6: x = 3.464

So, the airplane must fly about 3.464

miles to be directly above the tree.

D