Question

the number of principal solutions for the equation sin2 theta equal to 1 upon root 2 are.....​

Answer 1

Principal Solution: When solutions for a trigonometric equation are found in the interval $[0,2\pi],$ we call then principal solutions.

Given: $sin2\theta=\frac{1}{\sqrt{2}}$

To find: the number of principal solutions

Solution:

• Given, $sin2\theta=\frac{1}{\sqrt{2}}$

• Clearly $sin2\theta$ is attaining a positive value for $2\theta\in [0,\pi]$

• Now, $sin2\theta=\frac{1}{\sqrt{2}}=sin\frac{\pi}{4}$

• $\Rightarrow 2\theta=\frac{\pi}{4}$

• $\Rightarrow \boxed{\theta=\frac{\pi}{8}}$

• Also, $sin2\theta=\frac{1}{\sqrt{2}}=sin\frac{3\pi}{4}$

• $\Rightarrow 2\theta=\frac{3\pi}{4}$

• $\Rightarrow \boxed{\theta=\frac{3\pi}{8}}$

Answer: therefore, the principal solutions are $x=\frac{\pi}{8},\:\frac{3\pi}{8}$ and this occurs only two times.