Which numbers complete the blanks when solving the equation [tex]$\cos (x+2 \pi)=-\frac{\sqrt{2}}{2}$[/tex] over the interval [tex]$[0,2 \pi]$[/tex]?
[tex]$\cos x \cdot \ldots-\sin x \cdot \ldots=-\frac{\sqrt{2}}{2}$[/tex]
A. 1; 0
B. [tex]$0 ; 1$[/tex]
C. -1 ; 0
D. [tex]$0 ;-1$[/tex]
Answer (2)
Apply the cosine angle sum identity: cos ( x + 2 π ) = cos x cos ( 2 π ) − sin x sin ( 2 π ) .
Evaluate cos ( 2 π ) = 1 and sin ( 2 π ) = 0 .
Substitute the values: cos ( x + 2 π ) = cos x ( 1 ) − sin x ( 0 ) = cos x .
Compare with the given equation to find the missing numbers: 1 and 0. The final answer is 1 ; 0 .
Explanation
Problem Analysis We are given the equation cos ( x + 2 π ) = − 2 2 and asked to find the values that complete the equation cos x ⋅ … − sin x ⋅ … = − 2 2 . We will use the cosine angle sum identity to expand the given equation.
Applying the Cosine Angle Sum Identity The cosine angle sum identity is cos ( a + b ) = cos a cos b − sin a sin b . Applying this to our equation, we have cos ( x + 2 π ) = cos x cos ( 2 π ) − sin x sin ( 2 π )
Evaluating cos ( 2 π ) and sin ( 2 π ) We know that cos ( 2 π ) = 1 and sin ( 2 π ) = 0 . Substituting these values into the equation, we get cos ( x + 2 π ) = cos x ( 1 ) − sin x ( 0 ) = cos x
Finding the Missing Numbers Since cos ( x + 2 π ) = cos x , we have cos x = − 2 2 . Comparing this with the equation cos x ⋅ … − sin x ⋅ … = − 2 2 , we can see that the missing numbers are 1 and 0. Therefore, the equation becomes cos x ⋅ 1 − sin x ⋅ 0 = − 2 2
Final Answer The numbers that complete the blanks are 1 and 0. Therefore, the correct option is 1; 0.
Examples
Understanding trigonometric identities like the cosine angle sum identity is crucial in fields like physics and engineering. For example, when analyzing wave interference or oscillations, these identities help simplify complex expressions and make calculations easier. In signal processing, they are used to decompose signals into their fundamental components, which is essential for filtering and analyzing audio or radio waves.
The solution to the equation cos ( x + 2 π ) = − 2 2 leads to the expressions cos x ⋅ 1 − sin x ⋅ 0 = − 2 2 . The values that complete the blanks are 1 and 0. Therefore, the correct option is 1; 0.
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