Trigonometry

Sub Units:Trigonometric ratios.Inter-relation between trigonometric ratios.Trigonometric ratio’s of particular angles.Application of trigonometric ratios

Introduction:The word Trigonometry is derived from three Greek words.Trigonometry means: Tri – Three

gona- Sidesmetry-Measures

Trigonometry means the study of relation between sides and angles of a triangle. Trigonometry is not the work of any one person or nation its history spans thousands of years and has touched every major civilisation. Egyptians used this for building pyramids. Indian mathematician Aryabhatta and Bhaskara have given remarkable contribution to this field.

Following activity can be done in order to introduce this topic. Teacher can take all the students to the playground near a pole or a flag-post. Ask students to observe all the activities.

Keep a ladder leaning to the top of the pole, take another ladder longer in length and keep it in the same manner. Ask the student the following questions, refer the Figure:

AB=Pole or Flag-postAC=First LadderAD=Second LadderBC=Distance of ladder AC from pole.AD=Distance of ladder AD from pole.

What must be the height of the pole?What is the angle made by the pole with the ground?How far is the first ladder kept away from the base of the

pole?How far is the second ladder kept away from the base of the

pole?Observe the angle formed by the ladders with the ground?

Which angle is greater? If the first ladder is kept at an angle of 60 with the ground and

distance between the ladder and the base of the pole is known then how can we find the length of the pole or length of the ladder?

If the second ladder is kept at 45 ask same questions as previous.

Take all the answers and help them to draw conclusions that if the angle at which ladder is kept and distance from the pole is known, then we are able to find the length of ladder and height of the pole by 30-60-90 or 45-45-90 triangle theorem.

Explore this idea to find the height of tree, mountain, or even the height at which aeroplane is flying in the sky, as we come across the objectives of the topic Trigonometry.

Objectives:Measurement of angles, areas & sectors.Length of an arc and area of a circle.Circular motion.Trigonometric and polar representation of complex

number.De Moivre’s formula.Algebraic operations with complex number.Graphs of Trigonometric functions.

Applications:To estimate the distances of the planets and stars from

the earth.To draw maps, to determine positions of Islands in

relations to longitudes and latitudes.In the field of navigations, aerospace, engineering and

other fields of science.

Triangle:

How to apply trigonometric ratios, in the problems related with height and distance, you will know in this slide. This slide will enable you to measure, the width of river without crossing it, the height of a building without climbing it; so on many more measurements you will be able to do with the help of trigonometry. How? Let us know.

Terms, which are essential to know before the actual start. Point of Observation:

Whenever you see any object, your eye works as a point of observations. Though human beings can see an object with both eyes but we, for sake of convenience, consider it as we are watching the object with the single eye. The eye or the point from where you are watching the object is known as point of observation. In the pictorial representation of a height and distance problem this is represented by a single dot (point). There can be more than one point of observation you see an object from two or more than two points.

Line of Sight:

From the point of observation we see the object, it is supposed that we are not watching the object but only a particular point on it. It can be at the top of the object or at the bottom or anywhere else in the body of that object (according to the given problem). An imaginary line segment joining that point with the point of observation is known as line of sight or alternatively "line of vision".

Angle of Elevation:The object which is in the

consideration (in the sight) can be above or below the level of the eye (point of observation). At the level of point of observation we will draw a horizontal line. When the object is above the level of eye, then the angle which line of sight is making with this horizontal line is known as angle of elevation.

Angle of Depression: When the object is below the level

of point of observation, then the angle made by the line of sight with the horizontal line (drawn at the level of eye) is known as angle of depression.

Basic Trigonometric Ratios:

Trigonometric ratios:

Table of the values of trigonometric ratios.

Activity: 1Complete the following Table:

Activity: 2

TITLE: HEIGHT OF THE POLE:

Objective: To find the height of the pole.Pre-requisite Knowledge:1)Trigonometric ratios 2) Trigonometric table

Materials Required:1)Complete Protractor 2) A thin pipe of aluminum3) A protractor fixed on a stand 3) Measuring Pipe

ACTIVITY FOR LAB MANUAL

Procedure:Step 1. Fix a pipe of aluminum at centre of

protractor in such a way that the pipe is free to rotate.

Step 2. Rotate the pipe in such a way that the top of the pole is visible through the pipe.

Step 3. Observe the angle of elevation of the top of the pole and note it, say a=45

Step 4. Measure the horizontal distance between the pole and point of observation say, d=10m

Step 5. Measure the angle of the point of observation from the ground say h= 1m.

Observations:Angle of elevation = α = 45Distance between point of observation and pole d = 10mHeight of the point of observation from the ground = h= 1m

Calculation:Let the height of pole be H.

Tan α= H – h tan 45 = H – 1 1=H - 1 d 10 10

H – 1 =10 H=11m

Conclusion:The height of the pole is 11 m.

Learning Outcome: Through this activity we have found the height of the pole which cannot be measured directly by measuring tape.

The best way to understand or to know the technique of conversion of

height and distant problem is dealing with a few problems.

Problem (I) A kite is flying at a height of

30m, from a point on the ground its angle of elevation is 60o. Find the length of string.

The pictorial representation of this problem is the ABC.

Problem (II) A man is 1.6m tall. He watches at

the top of a tower which is 20m away from him and finds the angle of elevation 45o. Find the height of the tower. The pictorial representation of this problem is the adjacent figure.

In the figure AB represents the height of the tower CD represent the height of man CE and DB represents the distance of man from the tower. Here AC represents the line of sight and ACE= 45o represent the angle of elevation.

Problem (III) A flag is mounted at the top of a

building. From a point on the ground the angle of elevation of the top and bottom of the flag staff are 60o and 45o. If the building is 20m high then find the height of flag staff.

The pictorial representation of this problem is the adjacent figure. In the figure AB represents the height of building. AD represents the flag staff.

C represents the point of observation on the ground AC is the first line of sight and CD is another. ACB = 45o and DCB = 60o are the two angles of elevations.

Problem (IV) Two cars are heading towards a tower through a straight

road. From the top of the tower the angles of depression of these cars are 30o and 30o. If the height of the tower is 30m. Then find the distance between these cars.

The pictorial representation of this problem is the adjacent figure AB represents the height of tower. Here A is point of observation C and D represents each car.

AX is the imaginary horizontal line.

AC and AD are the two lines of sight.

CAX and DAX are the two angels of depression DAX = 30o and CAX = 60o

AX and BD are the two parallel lines

CAX = ACB = 60o

and XAD = ADB = 30o [alternative angles]

Problem (V) From the deck of a ship the angle of elevation of the

top of a light house is 45o. While the angle of depression of the bottom of the light house is 30o. If the deck of the ship is 16m high then find the height of the light house.

The pictorial representation of this problem is the adjacent figure.

In the figure AB represents the light house.

CD represent the height of the deck of the ship C is the point of observation. CE the horizontal line which is situated at the level of point observation.

BD is the ground level (level of sea).

AC and CB are the two lines of sight

ACE = 45o is the angle of elevation

ECB = 30o is the angle of depression

Since CE || BD

ECB = CBD = 30o

Using Trigonometry to Find the Height of a Building To estimate the height, H, of a building, measure the distance, D, from the point of observation to the base of the building and the angle, θ (theta), shown in the diagram. The ratio of the height H to the distance D is equal to the trigonometric function tangent θ (H/D = tan θ). To calculate H, multiply tangent θ by the distance D (H = D tan θ). The angle can be roughly estimated by pointing one arm at the base of the building and the other arm at the roof and judging whether the angle formed is close to 15°, 30°, 45°, 60°, or 75°. The angle can be estimated more accurately with a protractor and a plumb bob made of a pencil hanging from a string. Hang the plumb bob from the zero point in the middle of the straight edge of the protractor. Sight along the edge of the protractor at the roof of the building. Measure the angle formed by the straight edge of the protractor and the plumb bob. Subtract this angle from 90°.

The formula for determining the height or depth y of a landform from cast shadows isy = x tan (90 - INA)

We can use trig to measure the height of a building. this will be your lab. Find the height of the classroom. Use your height from eye level, the distance to the wall and the angle between the horizontal and thecorner between the wall and the ceiling.

Suppose the students of a school are visiting Qutub minar. Now, if a student is looking at the top of the minar, a right triangle can be imagined to be made, as shown in Fig 1.Can the student find out the height of the minar, without actually measuring it?

Co-Relation:Trigonometric ratios which are studied in class IX and

class X are applied in higher education of Trigonometric functions of compound angles.

[An angle formed by the algebraic sum of the two or more angles are called Compound angles.]

(A+B)(A-B) & Trigonometric functions. Which are as follows:1)Trigonometric Equations and the solution.2)Solution of Triangles.3)Inverse trigonometric functions.4)Graphs of trigonometric functions.5)Properties of Inverse trigonometric functions.