Question

Which numbers complete the blanks when solving the equation cosine (x 2 pi) = negative startfraction startroot 2 endroot over 2 endfraction over the interval [0, 2pi]?

1) 1; 0
2) 0; 1
3) –1; 0
4) 0; –1

Answer 1

The equation cosine

`(x + 2\pi) = -(√(2))/(2)`

s solutions of x = 3π/4 and x = 5π/4, corresponding to option (4) 0; -1.

Step-by-step explanation:

To solve the equation cosine

`(x + 2\pi) = -(√(2))/(2)`

an look at the unit circle and identify the angles that have a cosine value of

`-(√(2))/(2).`

to the x-coordinate of the circle, and this value occurs at the angles of 3π/4 and 5π/4 radians within one full rotation (0 to 2π).

Adding 2π to x does not change the cosine value due to the periodicity of the cosine function. Thus, the solutions for x can be found directly as 3π/4 and 5π/4.

These angles correspond to option (4) 0; -1, since the cosine of 0 radians is 1 (positive) and we are looking for the negative cosine value.