Question

Where do the graphs of f of x equals cosine of the quantity x over 2 and g of x equals square root of 2 minus cosine of the quantity x over 2 intersect on the interval [0, 360°)?

Answer 1

Given the functions f(x) and g(x) defined as:

`\begin{gathered} f(x)=\cos ((x)/(2)) \\ g(x)=\sqrt[]{2}-\cos ((x)/(2)) \end{gathered}`

The intersection of the graphs will be at f(x) = g(x):

`\begin{gathered} \cos ((x)/(2))=\sqrt[]{2}-\cos ((x)/(2)) \\ 2\cos ((x)/(2))=\sqrt[]{2} \\ \cos ((x)/(2))=\frac{\sqrt[]{2}}{2}=\frac{1}{\sqrt[]{2}} \end{gathered}`

This is true for:

`\begin{gathered} (x)/(2)=45\degree\Rightarrow x=90\degree \\ (x)/(2)=315\degree\Rightarrow x=630\degree \end{gathered}`

But only x = 90° is within the interval [0°, 360°)

Answer: 90°