Question

Which numbers complete the blanks when solving the equation cos(x+2π)=−√2/2 over the interval [0,2π]? cos(x)×blank−sin(x)×blank=−√2/2

a) 1; 0
b) 0; 1
c) -1; 0
d) 0; -1

Answer 1

To solve the equation cos(x+2π)=-√2/2, we use the periodicity of the cosine function and angle addition formulas. The correct blanks to fill in are for cos(x)×1 - sin(x)×0 = -√2/2, which simplifies to cos(x)=-√2/2, leading us to choose 1; 0 as the answer.

Step-by-step explanation:

To solve the equation cos(x+2π)=-√2/2 over the interval [0,2π], we can use the periodic properties of the cosine function. Since cosine is a periodic function with period 2π, cos(x+2π) is equivalent to cos(x). Therefore, we need to find values of x within the given interval for which cos(x) is equal to -√2/2.

The equation can be rewritten without changing its value considering the angle addition formula for cosine, which is cos(a+b) = cos(a)×cos(b) - sin(a)×sin(b). Substituting a=x and b=2π, we have cos(x)×1 - sin(x)×0 = -√2/2, because cos(2π)=1 and sin(2π)=0. This simplifies to just cos(x)=-√2/2.

The correct completion of the blanks is therefore 1; 0, which corresponds to option (a).