Understanding Engineering Mathematics

D.Ifrcosθ=3andrsinθ=4 determine the positive value ofr, and the principal value

ofθ.

6.3.7 Compound angle formulae

➤➤

173 187

➤

A.Prove the following

(i) sin 3θ=3sinθ−4sin^3 θ (ii) cos 3θ=4cos^3 θ−3cosθ

(iii)

cos 2θ

cosθ+sinθ

=cosθ−sinθ (iv) cotθ−tanθ=2cot2θ

(v) cot 2θ=

cot^2 θ− 1

2cotθ

B.Without using a calculator or tables evaluate

(i) sin 15°cos 15° (ii) sin 15° (iii) tan(π/ 12 ) (iv) cos( 11 π/ 12 )

(v) tan( 7 π/ 12 ) (vi) cos 75°

C.Evaluate

(i) sin 22. 5 ° (ii) cos 22. 5 ° (iii) tan 22. 5 °

given that cos 45°= 1 /

√

2.

D.Express the following products as sums or differences of sines and/or cosines of
multiple angles

(i) sin 2xcos 3x (ii) sinxsin 4x (iii) cos 2xsinx (iv) cos 4xcos 5x

E.Prove the following identities (Hint: putP=(A+B)/2,Q=(A−B)/2intheleft-

hand sides, expand and simplify and re-express in terms ofP andQ)

(i) sinP+sinQ≡2sin

(

P+Q

2

)

cos

(

P−Q

2

)

(ii) sinP−sinQ≡2cos

(

P+Q

2

)

sin

(

P−Q

2

)

(iii) cosP+cosQ≡2cos

(

P+Q

2

)

cos

(

P−Q

2

)

(iv) cosP−cosQ≡−2sin

(

P+Q

2

)

sin

(

P−Q

2

)

6.3.8 Trigonometric equations

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173 191

➤

A.Give the general solution to each of the equations:

(i) cosθ= 0 (ii) cosθ=− 1 (iii) cosθ=−

√

3

2