Question

If $\sin \theta ,\,\cos \theta \,$are the roots of equation $2{x^2} - 2\sqrt 2 x + 1 = 0$ then $\theta $ is equal to:
A. 15
B. 30
C. 45
D. 60

Answer 1

Hint: In order to solve this problem you need to use the concept of sum of roots and product of roots of the quadratic equations then solving you will get the right answer.

Complete Step-by-Step solution:
The given equation is $2{x^2} - 2\sqrt 2 x + 1 = 0$ and the roots of this equation is $\sin \theta ,\,\cos \theta \,$.
As we know if the quadratic equation is $a{x^2} + bx + c = 0$ then its sum of roots is $ - \dfrac{b}{a}$ and product of roots is $\dfrac{c}{a}$.
So in the given equation we can say that :
$
  \sin \theta + \cos \theta = - \dfrac{{( - 2\sqrt 2 )}}{2} = \sqrt 2 \,\,..................(1)\, \\
  \sin \theta \cos \theta = \dfrac{1}{2}\,\,\,\,\,\, \Rightarrow \cos \theta = \dfrac{1}{{2\sin \theta }}.............(2) \\
$
On putting the value obtained in (2) in the equation (1) we get the new equation as:
$
  \sin \theta + \dfrac{1}{{2\sin \theta }} = \sqrt 2 \\
  2{\sin ^2}\theta - 2\sqrt 2 \sin \theta + 1 = 0 \\
  {\text{We can write the above equation as:}} \\
  {(\sqrt 2 \sin \theta - 1)^2} = 0 \\
  \sin \theta = \dfrac{1}{{\sqrt 2 }} \\
$
So, the value of $\theta = {\text{45}}\,{\text{degrees}}$.
Hence the correct option is C.

Note: In this problem you have to use the concept of sum and product of the root of the quadratic equation with the help of that you will get an equation in terms of sin. Form there you can find the value of $\theta $ and get the right answer.