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Trigonometric Ratio & Identities
Leonhard Euler
Euler (1707–1783) served as a medical lieutenant in the Russian navy from 1727 to 1730. Euler became professor of physics at the Academy in 1730. In 1733 Euler was appointed to senior chair of mathematics. By 1740 Euler had a very high reputation, having won the Grand Prize of the Paris Academy in 1738 and 1740. He wrote books on the calculus of variations; on the calculation of planetary orbits; on artillery and ballistics; on analysis; on shipbuilding and navigation; on the motion of the moon; lectures on the differential calculus. Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficial account of it. He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc. as functions rather than as chords. He made decisive and formative contributions to geometry, calculus and number theory. We owe to Euler the notation f(x) for a function, e for the base of natural logs, i for the square root of –1, π for pi, Σ for summation, the notation for finite differences y' and y'' and many others. Perhaps the result that brought Euler the most fame was to find a closed form for the sum of the infinite series Σ(1/n²). Euler gave the formula e^(ix) = cos x + i sin x. Analytic functions of a complex variable were investigated by Euler in a number of different contexts. Euler made substantial contributions to differential geometry, investigating the theory of surfaces and curvature of surfaces.
Introduction
The word ‘Trigonometry’ is derived from the Greek words: (i) Trigonon means a triangle, (ii) metron means a measure. Hence trigonometry means science of measuring triangles.
Angle
It may be defined as the amount of revolution undergone by a revolving line in a plane. Let a revolving line, starting from its initial position OX to the terminal position OP, then ∠XOP is said to have been traced out. Here OX is called Initial side and OP as terminal side, where O is called the vertex.
Rules for Signs of Angles
(i) The angle ∠XOP is regarded as positive if it is traced out in the anticlockwise direction.
(ii) The angle ∠XOP is regarded as negative if it is traced out in the clockwise direction.
System of Measurement of Angle
There are three systems for measurement of angles:
1. Sexagesimal system: The principal unit in this system is degree (°). One right angle = 90°, 1° = 60′, 1′ = 60″.
2. Centesimal system: One right angle = 100g, 1g = 100′, 1′ = 100″.
3. Circular system: One radian is the measure of an angle subtended at the centre of a circle by an arc of length equal to the radius of the circle.
Relation Between Radian and Degree
From the relation AB/AC = θ₁/θ₂, if θ₁ = 1c, AB = r, θ₂ = 180°, AC = πr, we get 1c = 180°/π ≈ 57°19′27″. Hence π radians = 180°.
Relation Between Three Systems
1 right angle = 90° = 100g and π radians = 180°. Hence 180° = 200g = π radians.
Length of Arc
If θ is in radians, length of an arc l = rθ, where r = radius.
Illustrations
1. Change 15/8° into degree, minute & second: 15/8° = 1°52′30″.
2. Minute hand of length 5 cm in 15 min moves through θ = 2π(15/60) = π/2 rad. Length = rθ = 5×π/2 = 5π/2 cm.
3. Railroad curve of 25° in 40 m: θ = 25×π/180 = 5π/36. Radius r = 40/θ = 288/5π ≈ 91.636 m.
Practice Exercises
1. The circular measure of an angle is 3π/10. Second angle = 70g. Find third angle in degrees.
2. Angles of triangle in A.P. Least angle (deg) : Greatest angle (rad) = 60:π. Find all angles in deg & rad.
3. In a circle of diameter 42 cm, chord length = 21 cm. Find length of minor arc.
4. A cow tied to a rope moves along a circular path of 44 m at 72°. Find rope length.
5. Kartarpur 64 km from Amritsar. Find angle subtended at Earth’s centre (Earth radius = 6400 km).
6. Angular diameter of moon = 30′. Find distance at which coin diameter 2.2 cm hides moon.
7. Eye can read letters subtending 5′ at 420 m. Find height of letters.
Answers
1. 63° 2. 30°, 60°, 90° (π/6, π/3, π/2) 3. 22 cm 4. 35 cm 5. 34′22″ approx 6. 114.5 cm 7. 61.1 cm
4.1 Trigonometric Ratios
For ∠XOA, P on terminal side, PM ⟂ x-axis, OP = r, OM = x, MP = y. Trigonometric ratios: sin θ = y/r, cos θ = x/r, tan θ = y/x, cosec θ = r/y, sec θ = r/x, cot θ = x/y.
4.2 Signs of Trigonometric Ratios
1st quadrant (x>0,y>0): all positive.
2nd quadrant (x<0,y>0): sin,cosec positive; others negative.
3rd quadrant (x<0,y<0): tan,cot positive; others negative.
4th quadrant (x>0,y<0): cos,sec positive; others negative.
4.3 Trigonometric Ratios of Standard Angles
Angle sin cos tan cot sec cosec
0° 0 1 0 ∞ 1 ∞
30° 1/2 √3/2 1/√3 √3 2/√3 2
45° 1/√2 1/√2 1 1 √2 √2
60° √3/2 1/2 √3 1/√3 2 2/√3
90° 1 0 ∞ 0 ∞ 1
4.4 Trigonometric Ratios of Allied Angles
Two angles are allied if their sum/difference is zero or multiple of 90°. If θ in degrees, allied angles = –θ, 90°±θ, 180°±θ, 360°±θ. If θ in radians, allied angles = –θ, π/2±θ, π±θ, 2π±θ.