Introduction to trigonometry

Made By- Pranav Ahlawat, class X-A,

4540/RH

Mathematics Holiday Assignment

Contents Introduction What is trigonometry? History Trigonometric ratios. Easy way to learn trigonometric ratios. Ratios of some specific angles. Trigonometric ratios of complementary angles. Trigonometric identities. Applications of trigonometry. Applications of trigonometry in astronomy. Applications of trigonometry in architecture.

Introduction In this chapter, we will study some ratios of the sides of a right triangle with respect to its acute angles, called trigonometric ratios of the angle. We will also define the trigonometric ratios for the angles of measures 0⁰, 30⁰, 45⁰, 60⁰ and 90⁰. We will calculate the trigonometric ratios for some specific angles and establish some identities involving these ratios ,called trigonometric identities.

What is trigonometry? Trigonometry is a branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles. The word “trigonometry” is derived from the Greek words ‘tri’ meaning three,‘gon’ meaning sides and ‘metron’ meaning measure. The distances or heights can be easilyfound out by using ‘trigonometry’. Hipparchus is credited with compiling the first ‘trigonometric table’, and hence is known as "the father of trigonometry“.

HistoryThe 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-jya for the half-chord, which was shortened to jya or jiva in due course. The word jiva was translated into sinus, which means curve, when the Arabic version was translated into Latin. Soon the word sinus, also used as sine, became common in mathematical texts throughout Europe.

HistoryAn English Professor of astronomy Edmund Gunter (1581–1626), first used the abbreviated notation ‘sin’. The origin of the terms‘cosine’ and ‘tangent’ was much later.The cosine function arose from the need to compute the sine of the complementary angle. Aryabhatta called it kotijya. The name cosinus originated with Edmund Gunter. In 1674, the English Mathematician Sir Jonas Moore first used the abbreviated notation ‘cos’.

Trigonometric ratiosLet us consider a right triangle:Here angle A is an acute angle. Note the position of the side BC with respect to angle A. We will call the side opposite to angle A. AC is the hypotenuse of the right angled triangle and the side AB is the adjacent side in reference to angle A.The trigonometric ratios of angle A in right triangle ABC are defined as follows:

Sine of angle A= side opposite to angle A = BC hypotenuse AC

Cosine of angle A= side adjacent to angle A = AB hypotenuse AC

Tangent of angle A= side opposite to angle A = BC side adjacent to angle A AB

Trigonometric ratios Cosecant of angle A= hypotenuse = AC

side opposite to angle A BC Secant of angle A= hypotenuse = AC

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

side opposite to angle A BCThe ratios defined above are abbreviated as sin A, cos A, tan A, cosec A, sec A, and cot A respectively. Note that the ratios cosec A, sec A and cot A are respectively the reciprocals of the ratios sin A, cos A, and tan A. Also, observe that tan A= sin A/cos A and cot A = cos A /sin A . So, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.

Easy way to learn Trigonometric ratios

Ratios of some specific angles Angle 0⁰ 30⁰ 45⁰ 60⁰ 90⁰

Sine 0 1/2 1/2 3/2 1

Cosine 1 3/2 1/2 1/2 0

Tangent 0 1/3 1 3 Not defined

Cosecant Not defined

2 2 2/3 1

Secant 1 2/3 2 2 Not defined

Cotangent Not defined

3 1 1/3 0

Trigonometric ratios of complementary angles

Sin(90-A)= cos ACos(90- A)= sin A

Tan(90-a)= cot ACot(90-A)= tan A

Sec(90-A)= cosec ACosec(90-A)= sec A

Trigonometric identities

sin2A + cos2A = 1sin2A - 1 = cos2Acos2A - 1 = sin2A

1 + tan2A = sec2Asec2A - 1 = tan2Asec2A - tan2A = 1

cot2A + 1 = cosec2Acosec2A – 1 = cot2Acosec2A - cot2A = 1

Applications of trigonometryThis field of mathematics can be applied in astronomy, navigation, music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.

Applications of trigonometry in astronomy

Since ancient times trigonometry was used in astronomy.The technique of triangulation is used to measure the distance to nearby stars. In 240 B.C., a mathematician named Eratosthenes discovered the radius of the Earth using trigonometry and geometry. In 2001, a group of European astronomers did an experiment that started in 1997 about the distance of Venus from the Sun. Venus was about 105,000,000 kilometers away from the Sun .

Applications of trigonometry in architecture

Many modern buildings have beautifully curved surfaces. Making these curves out of steel, stone, concrete or glass is extremely difficult, but not impossible. One way around to address this problem is to piece the surface together out of many flat panels, each sitting at an angle to the one next to it, so that all together they create what looks like a curved surface. The more regular these shapes, the easier the building process. Regular flat shapes can be made out of triangles, and so trigonometry plays an important role in architecture.

conclusion Trigonometry is a branch of mathematics with several useful and important applications. Hence, it attracts more and more researches with several theories published year after year.