Higher Engineering Mathematics

182 GEOMETRY AND TRIGONOMETRY 18.3 Double angles (i) If, in the compound-angle formula for sin(A+B), we letB=Athen sin 2A=2 sinA ...

COMPOUND ANGLES 183 B Now try the following exercise. Exercise 82 Further problems on double angles The powerpin an electrical ...

184 GEOMETRY AND TRIGONOMETRY i.e. instantaneous power, p=25[cosπ/ 6 −cos (2ωt−π/6)] Now try the following exercise. Exercise 83 ...

COMPOUND ANGLES 185 B From equation (7), cos 6x+cos 2x=2 cos 4xcos 2x From equation (5), sin 6x+sin 2x=2 sin 4xcos 2x Hence cos ...

186 GEOMETRY AND TRIGONOMETRY resulting current isi=Imsin ( ωt− π 2 ) since current lags voltage by π 2 radians or 90◦in a purel ...

COMPOUND ANGLES 187 B p i v + 0 − π ω 2πω t (seconds) p v i Figure 18.10 Rearranging givesp=^12 VmIm(2 sinωtcosωt). Thuspower,p= ...

188 GEOMETRY AND TRIGONOMETRY − 0 + p i v v p i π ω 2 π ω t(seconds) Figure 18.11 the power curve (shown shaded) above the horiz ...

Assign-05-H8152.tex 23/6/2006 15: 8 Page 189 B Geometry and Trigonometry Assignment 5 This assignment covers the material contai ...

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Graphs C 19 Functions and their curves 19.1 Standard curves When a mathematical equation is known, co- ordinates may be calculat ...

192 GRAPHS Figure 19.3 Figure 19.4 (v) Circle(see Chapter 14, page 137) The simplest equation of a circle isx^2 +y^2 =r^2 , with ...

FUNCTIONS AND THEIR CURVES 193 C The lengthABis called themajor axisandCDthe minor axis. In the above equation, ‘a’ is the semi- ...

194 GRAPHS (xi) Polar Curves The equation of a polar curve is of the formr=f(θ). An example of a polar curve,r=asinθ, is shown i ...

FUNCTIONS AND THEIR CURVES 195 C 0 π 2 π3π 2 2π θ 1 3 y = cos θ + 2 y = cos θ (b) Figure 19.14 π 2 π 3 π 2 0 2 π x − 1 1 y π 3 y ...

196 GRAPHS Figure 19.17 (v)y=−f(x) The graph of y=−f(x) is obtained by reflect- ingy=f(x)inthex-axis. For example, graphs of y=e ...

FUNCTIONS AND THEIR CURVES 197 C Problem 1. Sketch the following graphs, showing relevant points: (a)y=(x−4)^2 (b)y=x^3 − 8 (a) ...

198 GRAPHS − 2 2 − 10 − 20 10 20 − 4 x y (c) y = −(x+2)^3 0 − 2 2 − 10 − 20 10 20 − 4 x y (d) 0 y = 5 −(x+2)^3 Figure 19.22 (Con ...

FUNCTIONS AND THEIR CURVES 199 C Now try the following exercise. Exercise 85 Further problems on simple transformations with cur ...

200 GRAPHS − 3 03 x 27 − 27 y y = x 3 (a) − 3 π/2−π −π/2 0 π/2 π 3 π/2 2 π x y y = sinx 1 − 1 (b) Figure 19.26 (a) − 10 123 x y ...

FUNCTIONS AND THEIR CURVES 201 C (a) A graph ofy=lnxis shown in Fig. 19.29(a) and the curve is neither symmetrical about the y-a ...