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This project has been made with the aim of providing basic understanding onthe subject – TRIGONOMETRY . Sincere efforts have been made to make thepresentation a unique experience to the viewer. Stress has been laid on theappearance, neatness and quality of the presentation. No effort has been sparedto make the reading and understanding of the presentation complete andinteresting. I have tried to do my best and hope that the project of mine wouldbe appreciated by all.

Supervised by – Smt Tapasi Paul ChowdhuryCreated by – Debdita Pan

Roll no :116Teacher Education DepartmentScottish Church College

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One cannot succeed alone no matter how great one’s abilities are,without the cooperation of others. This project, too, is a result of effortsof many. I would like to thank all those who helped me in making thisproject a success.

I would like to express my deep sense of gratitude to my Maths Teacher,Mrs. Tapasi Paul Chowdhury who was taking keen interest in our labactivities and discussed various methods which could be employedtowards this effect, and I really appreciate and acknowledge her paintaking efforts in this endeavour.

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Right Triangle Trigonometry

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What is TrigonometryThe word trigonometry is derivedfrom Greek words ‘tri’ (meaningthree), ‘gon’ (meaning sides) andmetron (meaning measure). In fact, The earliest known work ontrigonometry was recorded in Egyptand Babylon. Early astronomersused to find out the distance of thestars and planet s from the Earth.

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WHAT CAN YOU DO WITHTRIGONOMETRY? Historically, it was developed for astronomy and geography,

but scientists have been using it for centuries for other purposes, too. Besides other fields of mathematics, trigonometry is used in physics, engineering, and chemistry. Within mathematics, trigonometry is used primarily in calculus (which is perhaps its greatest application), linear algebra, and statistics. Since these fields are used throughout the natural and social sciences, trigonometry is a very useful subject to know.

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Some historians say that trigonometry wasinvented by Hipparchus, a Greek mathematician.He also introduced the division of a circle into360 degrees into Greece.

Hipparchus is considered the greatestastronomical observer, and by some the greatestastronomer of antiquity. He was the first Greekto develop quantitative and accurate models forthe motion of the Sun and Moon. With his solarand lunar theories and his numericaltrigonometry, he was probably the first todevelop a reliable method to predict solareclipses.

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The first use of the idea of ‘sine’ in the way we use it today was in the work ‘Aryabhatiyam’ by Aryabhatta, in A.D. 500. Aryabhatta used the word ardha-jiva for the half-chord, which shortened to jya or jiva. When it was translated into Latin, the word jiva was translated into sinus, which means curve.

Sin

Sin

Sin

SinSin

Sin

Sin

Sin

Sin

Sin

Sin

Sin

Sin

SinSin

Sin

ORIGIN OF ‘SINE’

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Foundation of COSINE & TANGENTThe origin of terms cosine and tangent was much later. The cosine function arose from the need to compute the sine of the complementary angle. Aryabhata called it kotijya. The name cosinusoriginated with Edmund Gunter. In 1674, the English mathematician Sir Jonas Moore first use the abbreviated notation cos.

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The trigonometric ratios of the angle A in the right triangle ABC see in fig.

•Sin of A =side opposite to angle A =BC

hypotenuse AC

•Cosine of A =side adjacent to angle A =AB

hypotenuse AC

•Tangent of A =side opposite to angle A =BC

side adjacent to angle A ABC

A B11

Cosecant of A = 1 = hypotenuse = AC

sin of A side opposite to angle A BC

Secant of A = 1 = hypotenuse = AC

sin of A side adjacent to angle a AB

Cotangent of A= 1 =side adjacent to angle A= AB

tangent of A side opposite to angle A BC

C

A B12

B A

C

Sin / Cosec

P (pandit)

H (har)

Cos / Sec

B (badri)

H (har)

Tan / Cot

P (prasad)

B (bole)

This is pretty easy!

BASE (B)

PERPENDICULAR (P)

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WHAT ARE TRIGONOMETRIC IDENTITIES ????

An equation involving trigonometric ratios of an angle is called a Trigonometric

Identitity, if it is true for all values of the angle(s) involved. Trigonometric identities

are ratios and relationships between certain trigonometric functions.

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A 0 3 0 4 5 6 0 9 0

Sin A 0 1

Cos A 1 0

Tan A 0 1 Not Defined

Cosec A Not Defined

2 1

Sec A 1 2 Not Defined

Cot A Not Defined

1 0

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Sin2 + Cos2 = 1 • 1 – Cos2 = Sin2 • 1 – Sin2 = Cos2

Tan2 + 1 = Sec2 • Sec2 - Tan2 = 1• Sec2 - 1 = Tan2

Cot2 + 1 = Cosec2 • Cosec2 - Cot2 = 1• Cosec2 - 1 = Cot2

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OTHER USEFUL IDENTITIESSin θ = 1/cosec θCos θ = 1/sec θTan θ = 1/cot θCosec θ = 1/sin θSec θ = 1/cos θTan θ = 1/cot θ

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Applications Measuring inaccessible lengths

Height of a building (tree, tower, etc.) Width of a river (canyon, etc.)

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Angle of Elevation –It is the angle formed by the line of sight with the horizontal when it isabove the horizontal level, i.e., the case when we raise our head to lookat the object.

A

HORIZONTAL LEVEL

ANGLE OF ELEVATION

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Angle of Depression –It is the angle formed by the line of sight with the horizontal when it isbelow the horizontal level, i.e., the case when we lower our head tolook at the object.

A

HORIZONTAL LEVEL

ANGLE OF DEPRESSION

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Application: Height

To establish the height of a building, a person walks 120 ft away from the building.

At that point an angle of elevation of 32 is formed when looking at the top of the building.

32

120 ft

h = ?

Example 1 of 4

H = 74.98 ft21

Application: Height

An observer on top of a hill measures an angle of depression of 68 when looking at a truck parked in the valley below.

If the truck is 55 ft from the base of the hill, how high is the hill?

68

h = ?

55 ft

Example 2 of 4

H = 136.1 ft22

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?

70 ft

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Example 3 of 4

D = 52.7 ft24

h = ?

HORIZONTAL LEVEL

It is an instrument which is used to measure theheight of distant objects using trigonometricconcepts.Here, the height of the tree using T. concepts,

h = tan *(x)

‘x’ units

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The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower is 30°. Find the height of the tower

Let AB be the tower and the angle of elevation from point C (on ground) is30°.In ΔABC,

.

Therefore, the height of the tower is26

A circus artist is climbing a 20 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground level is 30 °.

Sol:- It can be observed from the figure that AB is the pole.In ΔABC,

Therefore, the height of the pole is 10 m.27

Home Assignment A ladder 15 m long just reaches the top of a vertical wall. If the

ladder makes an angle of 60°with the wall, find the height of the wall ? (7.5 √3 )

A pole 12 m high casts a shadow 4 √3 m long on the ground. Find the angle of elevation ? (60°)

The angle of elevation of the top of a tower from a point on the ground is 30° if on walking 30m towards the tower, the angle of elevation becomes 60°.Find the height of the tower ?(15√3 )

An observer 1.5m tall is 20.5m away from a tower 22m high. Determine the angle of elevation of the top of the tower from the eye of the observer ? (45°)

If the length of the shadow cast by a pole be times the length of the pole, find the angle of elevation of the sun. [ 30o ]

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h = ?

HORIZONTAL LEVEL

It is an instrument which is used to measure theheight of distant objects using trigonometricconcepts.Here, the height of the tree using T. concepts,

h = tan *(x)

‘x’ units

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Jantar Mantar observatoryFor millenia, trigonometry has played a major role in calculating distances between stellar objects and their paths.

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Trigonometry begins in the right triangle, but it doesn’t have to be restrictedto triangles. The trigonometric functions carry the ideas of triangletrigonometry into a broader world of real-valued functions and wave forms.Trig functions are the relationships amongst various sides in right triangles.The enormous number of applications of trigonometry include astronomy,geography, optics, electronics, probability theory, statistics, biology, medicalimaging (CT scans and ultrasound), pharmacy, seismology, land surveying,architecture.

I get it!

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