By K. S. Snell, J. B. Morgan, W. J. Langford and E. A. Maxwell (Auth.)
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Additional resources for Elementary Analysis. Volume 2
Sample text
Verify the formulae for sin(ö ± Φ) and cos(ö ± φ) when φ = 0°, (ii) θ = φ = 45°. (i) θ = 90°, Multiple angles If w e put ^ = Ö in the addition formulae, w e obtain s o m e very important results. Thus (7) sin 2Θ = sin (Θ + θ)=2 sin θ cos Θ, (8) cos 2Θ = cos (θ + θ) = cos2 θ - sin2 θ = 2 cos2 θ - 1 = l - 2 s i n 2 Θ, (9) t a n 2 e = t a n ( ^ + ö)=j4:^. Ahernative forms of (8) which will prove very useful are (10) cos2e = Kl + cos2e> (11) sin2 e = K l - c o s 2 e ) E x . 2. Put ^ = Ö in formula (3).
3. Express in factors: (i) sin 60° + sin 30°, (iii) cos 4 0 ° + cos 80°, (v) sin(0 + h) - sin 0, 4. Simplify: (i) 2 sin 15° cos 15°, (ii) sin 50 - sin 30, (iv) cos 2x - cos 4A:, (vi) cos(0 + h) - cos 0. (ii) cos2 15° - (iii) ] 4 l a ¿ T 5 ' ° ' (ν) 1 - ^ '''' 2 sin 20, (vi) 2 cos2 sin2 15°, """"^ - 1. 5. If sin 0 = 5/13 and 90° < 0 < 180°, find the values of tan 0 and cos 20' 6. Sketch the graphs of —(1 + cos 0) and cos 20 in the same diagram for 0° < 0 < 360°. Solve the equation cos 20 + cos 0 + 1 = 0 in the given domain.
The relation between radians and degrees is therefore fnradians = 9 0 ° , or TT radians = 180°. E x . 2. Give simpler forms for sin (in + 0), cos (π — 0), tan (f π + 0), sec {i-rr — 0), cosec (π + 0), cot(2π — 0), the angles being measured m radians. E x . 3 . 4 shows a sector Ρ Ο β of a circle with centre O and radius r ; POQ = 0 radians. Show that, for the sector FOQ, arc PQ Ρ Ο β = ir20. = r0, area TRIGONOMETRIC FUNCTIONS 47 Q Fig. 4 The advantages of the radian as the unit of angle should now be clear.