The number of solutions of the equation 2 - ^2+√2=0 in R is equal to: Answer: 1

Question

The number of solutions of the equation $2	heta - \cos^2	heta+\sqrt2=0$ in $\mathbb{R}$ is equal to:

Answer: 1

Accepted Answer

Using $\cos^2\theta = \dfrac{1+\cos2\theta}2$, the equation becomes $4\theta+2\sqrt2 = 1+\cos2\theta$. Let $y_1 = 4\theta+2\sqrt2$ (a line of slope 4) and $y_2 = 1+\cos2\theta$ (bounded in $[0,2]$).

Checking values: at $\theta=0$, $y_1=2\sqrt2\approx2.83 > y_2=2$; the line $y_1$ increases very steeply (slope 4) while $y_2$ stays bounded in $[0,2]$, so the two graphs can cross only within a narrow window where $y_1\in[0,2]$. Direct evaluation there shows exactly one crossing.

$$\text{Number of solutions} = 1$$

Solutions · 1