Understanding Engineering Mathematics
B. (i) sin(A−B)=sin(A+(−B)) =sinAcos(−B)+sin(−B)cosA =sinAcosB−sinBcosA on using cos(−x)=cosxand sin(−x)=−sinx (ii) The result f ...
= −(tan 45°+tan 30°) 1 −tan 45°tan 30° =− ( 1 + 1 √ 3 ) ( 1 − 1 √ 3 )= 1 + √ 3 1 − √ 3 =− 2 + √ 3 ( 21 ➤ ) D. From cos(A+B)=cosA ...
6.2.8 Trigonometric equations ➤ 173 196➤ A trigonometric equation is any equation containing ratios of an ‘unknown’ angleθ,tobe ...
wherem,nare integers. So the general solution is θ=kπ,or 2 π 3 + 2 mπ,or− 2 π 3 + 2 nπ k,m,nare integers NB – it is a common err ...
Such a conversion simplifies the solution of equations such as acosθ+bsinθ=c by conversion to the form cos(θ+α)= c r = c √ a^2 + ...
6.3 Reinforcement 6.3.1 Radian measure and the circle ➤➤ 171 173 ➤ A.Express as radians: (i) 36° (ii) 101° (iii) 120° (iv) 250° ...
(v) A= 40 °, b=5, c= 6 (vi) A= 120 °, b=3, c= 5 6.3.4 Graphs of trigonometric functions ➤➤ 172 180 ➤ Sketch the graphs of (i) 2 ...
D.Ifrcosθ=3andrsinθ=4 determine the positive value ofr, and the principal value ofθ. 6.3.7 Compound angle formulae ➤➤ 173 187 ➤ ...
(iv) sinθ=0(v)sinθ=−1(vi)sinθ= √ 3 (vii) tanθ= 0 (viii) tanθ=−1(ix)tanθ= √ 3 B.Find the general solutions of the equations (i) c ...
A 2 sin(ωt+α 2 ). When we come to complex numbers we will see how this sort of thing can be done using objects calledphasors(371 ...
6.3.2 Definitions of the trig ratios A. θ 0 π/2 π/3 π/4 π/6 sinθ 01 √ 3 2 1 √ 2 1 2 tanθ 0nd √ 31 1 √ 3 secθ 1nd2 √ 2 2 √ 3 cosθ ...
6.3.4 Graphs of trigonometric functions (i) −p 0 2 p 2 − 2 2 sin ( 2 t + (^) ) p p p 5 p t 6 3 p 3 2 3 6 2 p 3 5 p − 6 7 p − 6 − ...
(iv) √ 3 2 π 3 nπ+(− 1 )n π 3 π 6 2 nπ± π 6 (v) 1 √ 2 π 4 nπ+(− 1 )n π 4 π 4 2 nπ± π 4 (vi) − 1 √ 2 − π 4 nπ+(− 1 )n+^1 π 4 3 π ...
D. (i)^12 (sin 5x−sinx) (ii)^12 (cos 3x−cos 5x) (iii)^12 (sin 3x−sinx) (iv)^12 (cos 9x+cosx) 6.3.8 Trigonometric equations A. (i ...
7 Coordinate Geometry Coordinate Geometry turns ‘diagrammatic’ geometry of the sort discussed in Chapter 5 into algebraic and nu ...
This chapter requires very little new material and much of it relies simply on Pythagoras’ theorem. 7.1 Review 7.1.1 Coordinate ...
7.1.5 Parallel and perpendicular lines ➤214 222➤➤ A.Find the equation of the straight line parallel to the liney= 1 − 3 xand whi ...
Another two-dimensional coordinate system in common use is thepolar coordinate system, especially convenient when dealing with e ...
p 12 2 p 0 3 p 2 p 6 p 3 2 p 3 5 p 6 4 p 3 3 p 2 5 p 3 11 p 6 7 p 6 Figure 7.2Polar plot ofr= 1 +2cosθ. Solution to review quest ...
B. 123456 (i) and (ii) are both at the origin, O. p (viii) (vi) (iv) (v) (vii) (iii) (i) (ii) 2 p 0 p 2 p 3 p 6 2 p 3 5 p 6 7 p ...
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