TRIGONOMETRY

INTRODUCTION

Trigonometry is a branch of mathematics that

studies triangles and the relationships between the lengths of their sides and the angles between those

sides.

HISTORY OF TRIGONOMETRY

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian

mathematics and Babylonian mathematics. Systematic study of

trigonometric functions began in Hellenistic mathematics, reaching

India as part of Hellenistic astronomy.

Uses of trigonometry

Scientific fields that make use of trigonometry include:

 architecture, astronomy, civil engineering, geophysics, electrical

engineering, electronics, land surveying and  many physical sciences, mechanical

engineering, oceanography, optics, pharmacology, probability

theory, seismology, statistics.

TRIGONOMETRY AND TRIANGLES

Using Trigonometry we can find the relationships between the lengths of sides of the triangle and the angles between those

sides.

TRIGONOMETRY AND TRIANGLES

Angles add to 180°• The angles of a triangle always add up to

180°44°

68° 68°

44°68°

+ 68°

180°

20°

130°30°

20°30°

180°

+ 130°

Right triangles• We only care about right triangles.

– A right triangle is one in which one of the angles is 90°– Here’s a right triangle:

• We call the longest side the hypotenuse• We pick one of the other angles--not the right angle• We name the other two sides relative to that angle

Here’s theright angle

hypotenuse

Here’s the anglewe are looking at

adjacent

op

posi

te

TRIGONOMETRY AND TRIANGLES

The Pythagorean Theorem

If you square the length of the two shorter sides and add them, you get the square of the length of the hypotenuse

adj2 + opp2 = hyp2

32 + 42 = 52, or 9 + 16 = 25

TRIGONOMETRY AND TRIANGLES

The Pythagorean Theorem

• There are few triangles with integer sides that satisfy the Pythagorean formula

• 3-4-5 and itsmultiples (6-8-10, etc.)are the best known

• 5-12-13 and its multiples form another set.

• 25 + 144 = 169

TRIGONOMETRY AND TRIANGLES

hypadj

opp

Ratios• Since a triangle has three

sides, there are six ways to divide the lengths of the sides

• Each of these six ratios has a name (and an abbreviation)

• Three ratios are most used:– sine = sin = opp / hyp– cosine = cos = adj / hyp– tangent = tan = opp / adj

• The other three ratios are redundant with these and can be ignored

TRIGONOMETRY AND TRIANGLES

hypotenuse

adjacent

op

posi

te

• The ratios depend on the shape of the triangle (the angles) but not on the size

Using the ratios• With these functions, if you know an angle (in

addition to the right angle) and the length of a side, you can compute all other angles and lengths of sides

• If you know the angle marked in blue (call it A) and you know the length of the adjacent side, then– tan A = opp / adj, so length of opposite side is given by

opp = adj * tan A– cos A = adj / hyp, so length of hypotenuse is given by

hyp = adj / cos A

hypotenuse

adjacent

op

posi

te

TRIGONOMETRY AND TRIANGLES

Important Formulas• The formulas for right-triangle

trigonometric functions are:– Sine = Opposite / Hypotenuse– Cosine = Adjacent / Hypotenuse– Tangent = Opposite / Adjacent

• Mnemonics for those formulas are:– Some Old Horse Caught Another Horse Taking Oats Away

– Saints On High Can Always Have Tea Or Alcohol

TRIGONOMETRY AND TRIANGLES

THANK YOU

By – Remin Rajesh