Trigonometry
- 1. TRIGONOMETRY
- 2. 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.
- 3. 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.
- 4. 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, pharmac ology, probability theory, seismology, statistics.
- 5. TRIGONOMETRY AND TRIANGLES Using Trigonometry we can find the relationships between the lengths of sides of the triangle and the angles between those sides.
- 6. TRIGONOMETRY AND TRIANGLES Angles add to 180° • The angles of a triangle always add up to 180° 20° 44° 68° 44° 68° + 68° 180° 68° 30° 130° 20° 30° + 130° 180°
- 7. TRIGONOMETRY AND TRIANGLES 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: opposite Here’s the right angle Here’s the angle we are looking at adjacent • 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
- 8. 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
- 9. TRIGONOMETRY AND TRIANGLES The Pythagorean Theorem • There are few triangles with integer sides that satisfy the Pythagorean formula • 3-4-5 and its multiples (6-8-10, etc.) are the best known • 5-12-13 and its multiples form another set. • 25 + 144 = 169 opp adj
- 10. TRIGONOMETRY AND TRIANGLES Ratios opposite • Since a triangle has three sides, there are six ways to divide the lengths of the sides adjacent • Each of these six ratios has a name (and an abbreviation) • Three ratios are most used: • The ratios depend on – sine = sin = opp / hyp the shape of the triangle – cosine = cos = adj / hyp (the angles) but not on – tangent = tan = opp / adj the size • The other three ratios are redundant with these and can be ignored
- 11. TRIGONOMETRY AND TRIANGLES Using the ratios opposite • 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 adjacent • 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
- 12. 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
- 13. THANK YOU By – Remin Rajesh