which numbers complete the blanks when solving the equation \\( \\cos ( x + 2 \\pi ) = - \\frac { \\sqrt { 2…

which numbers complete the blanks when solving the equation \\( \\cos ( x + 2 \\pi ) = - \\frac { \\sqrt { 2 } } { 2 } \\) over the interval \\( 0,2 \\pi \\) ?\n\\( \\cos x \\)____ = \\( \\cos x \\)____= \\( - \\frac { \\sqrt { 2 } } { 2 } \\)\n1,0\n0,1\n-1,0\n0,-1

which numbers complete the blanks when solving the equation \\( \\cos ( x + 2 \\pi ) = - \\frac { \\sqrt { 2 } } { 2 } \\) over the interval \\( 0,2 \\pi \\) ?\n\\( \\cos x \\)____ = \\( \\cos x \\)____= \\( - \\frac { \\sqrt { 2 } } { 2 } \\)\n1,0\n0,1\n-1,0\n0,-1

Answer

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)