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### Trigonometry

• 1. Trigonometry MATHS HOLIDAY HOMEWORK MADE BY SRISHTI BHOLA X-A 1
• 2. Trigonometry  Trigonometry is derived from Greek words trigonon (three angles) and metron ( measure).  Trigonometry is the branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees  Triangles on a sphere are also studied, in spherical trigonometry.  Trigonometry specifically deals with the relationships between the sides and the angles of triangles, that is, on the trigonometric functions, and with calculations based on these functions. 2
• 3. History • The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago. • Some experts believe that trigonometry was originally invented to calculate sundials, a traditional exercise in the oldest books • The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus circa 150 BC, who compiled a trigonometric table using the sine for solving triangles. • The Sulba Sutras written in India, between 800 BC and 500 BC, correctly compute the sine of π/4 (45°) as 1/√2 in a procedure for circling the square (the opposite of squaring the circle). • Many ancient mathematicians like Aryabhata, Brahmagupta,Ibn Yunus and Al-Kashi made significant contributions in this field(trigonometry). 3
• 4. Right Triangle  A triangle in which one angle is equal to 90° is called right triangle.  The side opposite to the right angle is known as hypotenuse. AB is the hypotenuse  The other two sides are known as legs. AC and BC are the legs Trigonometry deals with Right Triangles 4
• 5. Pythagoras Theorem In any right triangle, the area of the square whose side is the hypotenuse is equal to the sum of areas of the squares whose sides are the two legs. In the figure AB2 = BC2 + AC2 5
• 6. Trigonometric ratios  Sine(sin) opposite side/hypotenuse  Cosine(cos) adjacent side/hypotenuse  Tangent(tan) opposite side/adjacent side  Cosecant(cosec) hypotenuse/opposite side  Secant(sec) hypotenuse/adjacent side  Cotangent(cot) adjacent side/opposite side 6
• 7. Values of trigonometric function of Angle A sinθ = a/c cosθ = b/c tanθ = a/b cosecθ = c/a secθ = c/b cotθ = b/a 7
• 8. Values of Trigonometric function 0 30 45 60 90 Sine 0 0.5 1/√2 √3/2 1 Cosine 1 √3/2 1/√2 0.5 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 8
• 9. Calculator  This Calculates the values of trigonometric functions of different angles.  First Enter whether you want to enter the angle in radians or in degrees. Radian gives a bit more accurate value than Degree.  Then Enter the required trigonometric function in the format given below:  Enter 1 for sin.  Enter 2 for cosine.  Enter 3 for tangent.  Enter 4 for cosecant.  Enter 5 for secant.  Enter 6 for cotangent.  Then enter the magnitude of angle. 9
• 10. Trigonometric identities • sin2A + cos2A = 1 • 1 + tan2A = sec2A • 1 + cot2A = cosec2A • sin(A+B) = sinAcosB + cosAsin B • cos(A+B) = cosAcosB – sinAsinB • tan(A+B) = (tanA+tanB)/(1 – tanAtan B) • sin(A-B) = sinAcosB – cosAsinB • cos(A-B)=cosAcosB+sinAsinB • tan(A-B)=(tanA-tanB)(1+tanAtanB) • sin2A =2sinAcosA • cos2A=cos2A - sin2A • tan2A=2tanA/(1-tan2A) • sin(A/2) = ±√{(1-cosA)/2} • Cos(A/2)= ±√{(1+cosA)/2} • Tan(A/2)= ±√{(1-cosA)/(1+cosA)} 10
• 11. Relation between different Trigonometric Identities • Sine • Cosine • Tangent • Cosecant • Secant • Cotangent 11
• 12. Angles of Elevation and Depression  Line of sight: The line from our eyes to the object, we are viewing.  Angle of Elevation:The angle through which our eyes move upwards to see an object above us.  Angle of depression:The angle through which our eyes move downwards to see an object below us. 12