What is the y-value of the point of intersection of y equals 2 sine x minus cosine x and y = cos x over the interval 0 less than or equals x less than or equals Start Fraction pi over 2 End Fraction?

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start fraction start root 2 end root over 2 end fraction
Image with alt text: start fraction start root 2 end root over 2 end fraction

start fraction start root 3 end root over 2 end fraction
Image with alt text: start fraction start root 3 end root over 2 end fraction

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Answer 1

To find the point of intersection of the two equations $y=2\sin x-\cos x$ and $y=\cos x$, we need to set them equal to each other:

$$2\sin x-\cos x=\cos x$$

Rearranging gives us:

$$2\sin x-2\cos x=0$$

Dividing by 2 leads us to:

$$\sin x-\cos x=0$$

This implies:

$$\sin x=\cos x$$

The solutions to this equation in the interval $[0,\frac{\pi }{2}]$ occur at:

$$x=\frac{\pi }{4}$$

We can calculate the corresponding $y$-value by substituting $x=\frac{\pi }{4}$ into either original equation. Let's use $y=\cos x$:

$$y=\cos \left(\frac{\pi }{4}\right)=\frac{\sqrt{2}}{2}$$

Thus, the $y$-value of the point of intersection is:

$$\frac{\sqrt{2}}{2}$$

Given the options, the correct response is:

$$\frac{\sqrt{2}}{2}$$