Whenever we get one equation consisting of more than one variable or equation involves variables of different natures then the boundary conditions of trigonometric function is generally used along with | sin x | ≤ 1 ; | cos x | ≤ 1 ; | sec x | ≥ 1; | cosec x | ≥ 1 ; | tan x | ≥ 0 ; | cot x | ≥ 0. Now answer the following questions.
(i) If sin x + sin y + sin z = 3 ; x, y , z ∈ [0, 2 π ], then x 3 + y 3 – z 3 is equal to –
Text Solution
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(i)
sin x + sin y + sin z = 3
⇒ sin x = sin y = sin z = 1
∴ x 3 + y 3 – z 3 = ( π /2) 3 = π 3 /8.
(ii)
+ 
= 7,
Putting y = 
y + 10. 
= 7
⇒ y 2 – 7y + 10 = 0
⇒ y = 2, 5
But
∴ y = 
≠ 5
⇒ y = 
= 2
⇒ sin 2 x = 1
⇒ x = n π ± π /2,
Putting n = 0, 1….
Number of solution in [– π , π ] is 2.
(iii)
Sol. r 4 – 2r 2 + 3 = 2 sin θ
(r 2 – 1) 2 + 2 = 2 sin θ
For sin θ to be maximum sin θ = 1
⇒ r 2 – 1 = 0
⇒ r = –1, 1
Also, sin θ = 1
∴ θ = 2n π + π /2
If θ ∈ [0, 5 π ], θ = 
, 
, 
(r, θ ) ≡ 
Max. number of the pair (r, θ ) = 6.
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