Question

which numbers complete the blanks when solving the equation \\( \cos ( x + 2 \pi ) = - \frac { \sqrt { 2 } } { 2 } \\) over the interval \\( 0,2 \pi \\) ?
\\( \cos x \\)__ = \\( \cos x \\)__= \\( - \frac { \sqrt { 2 } } { 2 } \\)
1,0
0,1
-1,0
0,-1

Explanation:

Step1: Use the cosine periodicity

Since \(\cos(x + 2\pi)=\cos x\) (because the period of the cosine function \(y = \cos t\) is \(T = 2\pi\), i.e., \(\cos(t+2k\pi)=\cos t,k\in\mathbb{Z}\)). So the equation \(\cos(x + 2\pi)=-\frac{\sqrt{2}}{2}\) is equivalent to \(\cos x=-\frac{\sqrt{2}}{2}\).

Step2: Analyze the form

The left - hand side of the equation after using the periodicity property is \(\cos x\), which can be written as \(1\times\cos x+0\times\sin x\). The general form of a cosine function is \(A\cos x + B\sin x\). Here \(A = 1\) and \(B=0\).

Answer:

A. \(1,0\)