Question 9:
Solution:
(a)
sinθ=35cosθ=45tanθ=34
(b)
(c)
tan(360∘+θ)=tan360∘+tanθ1−tan360∘tanθ=0+tanθ1−(0)(tanθ)=tanθ=34
Given that sin θ =
35
, where θ is an acute angle, without using tables or a calculator, find the values of
(a) sin (180º + θ),
(b) cos (180º – θ),
(c) tan (360º + θ).
Solution:
(a)
sinθ=35cosθ=45tanθ=34
sin (180º + θ)
= sin 180º cos θ + cos 180º sin θ
= (0) cos θ + (– 1) sin θ
= – sin θ
=
−35
(b)
cos (180º – θ)
= cos 180º cos θ + sin 180º sin θ
= (– 1) cos θ + (0) sin θ
= – cos θ
=
−45
(c)
tan(360∘+θ)=tan360∘+tanθ1−tan360∘tanθ=0+tanθ1−(0)(tanθ)=tanθ=34
Question 10:
Prove each of the following trigonometric identities.
(a) cot2 x – cot2 x cos2x = cos2 x
(b)secxsecx−cosx=cosec2x
Solution:
(a)
LHS:cot2x−cot2xcos2x=cot2x(1−cos2x)=cot2x(sin2x)=cos2xsin2x(sin2x)=cos2x(RHS)
(b)
LHS:secxsecx−cosx=1cosx1cosx−cosx=1cosx1cosx−cos2xcosx=1cosx1−cos2xcosx=1cosx×cosx1−cos2x=11−cos2x=1sin2x=cosec2x(RHS)