Higher Engineering Mathematics

162 GEOMETRY AND TRIGONOMETRY Problem 19. Use harmonic synthesis to con- struct the complex current given by: i 1 =10 sinωt+4 si ...

TRIGONOMETRIC WAVEFORMS 163 B Figure 15.34 The currenticomprises three components—a 32 mA d.c. component, a fundamental of ampli ...

164 GEOMETRY AND TRIGONOMETRY Figure 15.35 Figure 15.36 ...

TRIGONOMETRIC WAVEFORMS 165 B Now try the following exercise. Exercise 72 Further problems on harmonic synthesis with complex wa ...

Geometry and trigonometry 16 Trigonometric identities and equations 16.1 Trigonometric identities A trigonometric identityis a r ...

TRIGONOMETRIC IDENTITIES AND EQUATIONS 167 B LHS= tanx+secx secx ( 1 + tanx secx ) = sinx cosx + 1 cosx ( 1 cosx ) ⎛ ⎜ ⎝^1 + sin ...

168 GEOMETRY AND TRIGONOMETRY solutions (as shown in Chapter 15). Fig. 16.2 shows a summary for angles of any magnitude. Figure ...

TRIGONOMETRIC IDENTITIES AND EQUATIONS 169 B Problem 7. Solve 1 .5 tanx− 1. 8 = 0 for 0 ◦≤x≤ 360 ◦. 1 .5 tanx− 1. 8 =0, from whi ...

170 GEOMETRY AND TRIGONOMETRY Figure 16.6 1 2 cot (^2) y= 1 .3, from which, cot (^2) y=2(1.3)= 2 .6. Hence coty= √ 2. 6 =± 1 .61 ...

TRIGONOMETRIC IDENTITIES AND EQUATIONS 171 B 2 cosec^2 t−5 cosect=[ 12 t= 14 ◦ 29 ′, 165◦ 31 ′, 221 ◦ 49 ′or 318◦ 11 ′ ] 16.7 ...

172 GEOMETRY AND TRIGONOMETRY 16 secx− 2 =14 tan^2 x [x= 52 ◦ 56 ′or 307◦ 4 ′] 4 cot^2 A−6 cosecA+ 6 =0[A= 90 ◦] 5 sect+2 tan^2 ...

B Geometry and trigonometry 17 The relationship between trigonometric and hyperbolic functions 17.1 The relationship between tri ...

174 GEOMETRY AND TRIGONOMETRY =− 2 j ( sinjθ j ) ( cosjθ) =2 sinjAcosjA sincej^2 =− 1 i.e. sinj 2 A=2 sinjAcosjA Now try the fol ...

THE RELATIONSHIP BETWEEN TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS 175 B cosθ−cosφ=−2 sin ( θ+φ 2 ) sin ( θ−φ 2 ) (see Chapter 18, ...

Geometry and trigonometry 18 Compound angles 18.1 Compound angle formulae An electric currentimay be expressed as i = 5 sin(ωt− ...

COMPOUND ANGLES 177 B Hence tan ( x+ π 4 ) tan ( x− π 4 ) = ( 1 +tanx 1 −tanx )( tanx− 1 1 +tanx ) = tanx− 1 1 −tanx = −(1−tanx) ...

178 GEOMETRY AND TRIGONOMETRY Given cosA= 0 .42 and sinB= 0 .73 evaluate (a) sin(A−B), (b) cos(A−B), (c) tan (A+B), correct to ...

COMPOUND ANGLES 179 B Figure 18.3 Hence 3 sinωt+4 cosωt=5 sin(ωt+ 0. 927 ). A sketch of 3 sinωt, 4 cosωtand 5 sin(ωt+ 0 .927) is ...

180 GEOMETRY AND TRIGONOMETRY Hence 4 .6 sinωt− 7 .3 cosωt= 8 .628 sin(ωt− 1. 008 ). Problem 8. Express −2.7 sinωt− 4 .1 cosωt i ...

COMPOUND ANGLES 181 B Since − 15 ◦ 43 ′ is the same as− 15 ◦ 43 ′+ 360 ◦, i.e. 344◦ 17 ′, then the solutions areθ= 77 ◦ 39 ′or 3 ...