The number or solutions of the equation 16^sin^2x + 16^cos^2x= 10 is
Answers
Answer:
number of possible solutions = 8.
π/6, 5π/6 , 7π/6, 11π/6, π/3, 2π/3, 4π/3, 5π/3.
Step-by-step explanation:
16^sin^2x + 16^cos^2x= 10 (∵Cos²x = 1 - Sin²x)
=> 16^sin^2x + 16^1 - sin^2x = 10
16^sin^2x + 16/16^sin^2x = 10.
Let 16^sin^2x = p
p + 16 / p = 10
p² - 10p + 16 = 0
p² - 8p -2p + 16 = 0
p(p - 8) - 2(p - 8) = 0
(p - 2 ) (p - 8) = 0
p = 2 or 8
Now substitute back the value of p.
16^sin^2x = 2 | 16^sin^2x = 8
2^4sin^2x = 2¹. | 2^4sin^2x = 2³
4sin^2x = 1 | 4sin^2x = 3
sin^2x = 1/4 | sin^2x = 3/4
sinx = ±1/2 | sinx = ±√3/2
For sinx = ±1/2
x = π/6, π - π/6, π + π/6, 2π - π/6
= π/6, 5π/6 , 7π/6, 11π/6
For sinx = ±√3/2
x = π/3, π - π/3, π + π/3, 2π - π/3
= π/3, 2π/3, 4π/3, 5π/3
Thus all possible values are π/6, 5π/6 , 7π/6, 11π/6, π/3, 2π/3, 4π/3, 5π/3.