Question

find all solutions in the interval $0,2\pi)$.
$\cos ^{2}\theta - 2\cos \theta - 1 = 0$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. $x =$ (type your answer in radians. round to four decimal places as needed. use a comma to separate answers as needed.)
b. there is no solution.

Explanation:

Step1: Let \( t = \cos\theta \)

The equation \( \cos^{2}\theta-2\cos\theta - 1=0 \) becomes \( t^{2}-2t - 1=0 \)

Step2: Solve the quadratic equation \( t^{2}-2t - 1=0 \)

Using the quadratic formula \( t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \), where \( a = 1 \), \( b=-2 \), \( c=-1 \)

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Step3: Analyze the values of \( t \)

Since \( - 1\leqslant\cos\theta\leqslant1 \)
For \( t = 1+\sqrt{2}\approx1 + 1.4142=2.4142>1 \) (rejected)
For \( t=1-\sqrt{2}\approx1-1.4142=-0.4142 \)

Step4: Find \( \theta \) when \( \cos\theta=1 - \sqrt{2} \)

\( \theta=\cos^{-1}(1-\sqrt{2})\) and \( \theta = 2\pi-\cos^{-1}(1-\sqrt{2}) \)
\( \cos^{-1}(1 - \sqrt{2})\approx2.0344 \) radians
\( 2\pi-\cos^{-1}(1-\sqrt{2})\approx2\pi - 2.0344\approx4.2488 \) radians

Answer:

A. \( x = 2.0344,4.2488 \)