Understanding Engineering Mathematics

(b) (i) 4 (ii) √ 119 (iii) 7 (iv) 15 (v) √ 7 B. √ 61 5.3.9 Lines and angles in a circle A. (i) 70° (ii) 140° (iii) 30° (iv) 43° ...

6 Trigonometry Trigonometry (literally: ‘the measurement of triangles’ – ‘trig’ from now on) is a fairly straightforward topic c ...

the sine and cosine rules and solution of triangles graphs of the trig functions inverse trig functions the Pythagorean identit ...

(x) sin 2 π 3 (xi) cos π 3 (xii) tan 45° (xiii) cos 30° (xiv) sin 30° (xv) tan π 3 (xvi) cos 45° (xvii) cos 3 π 2 (xviii) tan(− ...

6.1.7 Compound angle formulae ➤187 196➤➤ A.Expand sin(A+B)in terms of sine and cosine ofAandB. B. FromAderive similar expansions ...

A s q Figure 6.1Radians, arc, sectors. Solution to review question 6.1.1 A. (i) 90°= 90 180 ×π= π 2 radians (ii) − 30 °=− 30 180 ...

CB A q Figure 6.2 AB AC =sinθ(sine ofθ) AC AB =cosecθ(cosecant ofθ)= 1 sinθ BC AC =cosθ(cosine ofθ) AC BC =secθ(secant ofθ)= 1 c ...

‘positive direction’ – as shown. Note that for any angle,θ,|sinθ|and|cosθ|are both ≤1.Thenegativeofθ,−θ, means a rotation throug ...

seen that sinα=cosβ=cos( 90 °−α) cosα=sinβ=sin( 90 °−α) tanα=cotβ=cot( 90 °−α) cotα=tanβ=tan( 90 °−α) i.e. the ‘co-trig ratio’ i ...

(xx) cos ( − π 3 ) =cos (π 3 ) = 1 2 (xxi) sin( 585 °)=sin( 225 °)=−sin( 45 °)=− 1 √ 2 (xxii) cos( 225 °)=−cos( 45 °)=− 1 √ 2 (x ...

wheres=(a+b+c)/2, the semi-perimeter of the triangle. This result is very useful in surveying, for example. Note that the sine r ...

These results may be used to solve triangles given appropriate information. A triangle can be ‘solved’ for all angles and sides ...

question is given in terms of degrees/radians, then the answer should be given in the same form) f(θ+ 360 °)≡f(θ+ 2 π)=f(θ) Thus ...

− 2 p −p p 2 p q 1 − 1 0 cos q (^22) p p 3 p 2 3 p −− 2 Figure 6.9cosθ − 2 π≤θ≤ 2 π. −p 0 p 3 p p 5 p q 1 − 1 tan q 4 p 2 4 4 3 ...

Solution to review question 6.1.4 (i) − 2 p −p^0 p^2 p t 3 − 3 p 2 3 sin (t − p 2 ) 3 p 2 p 2 3 p −− 2 (ii) −p 0 2 p p 4 − 4 p p ...

6.2.5 Inverse trigonometric functions ➤ 172 195➤ The inverse function (100 ➤ )ofsinxis denoted sin−^1 x(Sometimes the notation ‘ ...

y 0 x p/2 y = tan−^1 x y − 1 1 x p p/2 0 y = cos−^1 x p − 2 Figure 6.12Inverse cosine, cos−^1 xand inverse tan,tan−^1 x. where P ...

r x y q We have x^2 +y^2 =r^2 Dividing byr^2 gives (x r ) 2 + (y r ) 2 = 1 or cos^2 θ+sin^2 θ= 1 Thisidentity should definitely ...

so cos^2 θ= 2 3 cosθ= √ 2 3 sin^2 θ= 1 −cos^2 θ= 1 − 2 3 = 1 3 and therefore sinθ= 1 √ 3 6.2.7 Compound angle formulae ➤ 173 196 ...

A Q A R P O B A ST Figure 6.13Proof of sin(A+B)=sinAcosB+cosAsinB. PuttingA=B in the compound angle identities immediately gives ...