Download - Trigonometry
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