Question

Determine ONE Sine function and ONE Cosine function that satisfy the following criteria. - Sine function has one x-intercept in the domain of \(0 \leq x \leq 2\pi\). - Sine and cosine function intersect 2 or 3 times in the domain of \(0 \leq x \leq 2\pi\). Which of the following options satisfies the above criteria? a) \(f(x) = \sin(0.3x)\), \(g(x) = \cos(x) + 1\) b) \(f(x) = \sin(0.5x)\), \(g(x) = \cos(x) + 1\) c) \(f(x) = \sin(0.3x)\), \(g(x) = 3\cos(x) + 1\) d) None of the above

Answer 1

In this case, none of them satisfy the criteria that ``Sine function has one x-intercept in the domain of
`\(0 \leq x \leq 2\pi\)` and Sine and cosine function intersect 2 or 3 times in the domain of
`\(0 \leq x \leq 2\pi\)```

The answer is option ⇒d) None of the above

Explanation:

To determine the sine and cosine functions that satisfy the given criteria, let's analyze each option and see if they meet the requirements:

a)
`\(f(x) = \sin(0.3x)\), \(g(x) = \cos(x) + 1\)`:

- The sine function
`\(f(x)\)` has a coefficient of 0.3, which means it oscillates faster than the regular sine function. It does not guarantee having one x-intercept in the domain of
`\(0 \leq x \leq 2\pi\)`. This option does not satisfy the first criteria.

- The cosine function
`\(g(x)\)`is shifted upward by 1 unit, but that does not affect the intersection points. It does not guarantee having 2 or 3 intersection points with the sine function. This option does not satisfy the second criteria.

b)
`\(f(x) = \sin(0.5x)\), \(g(x) = \cos(x) + 1\)`:

- The sine function
`\(f(x)\)`has a coefficient of 0.5, which means it oscillates slower than the regular sine function. It does not guarantee having one x-intercept in the domain of
`\(0 \leq x \leq 2\pi\)`. This option does not satisfy the first criteria.

- The cosine function
`\(g(x)\)` is shifted upward by 1 unit, but that does not affect the intersection points. It does not guarantee having 2 or 3 intersection points with the sine function. This option does not satisfy the second criteria.

c)
`\(f(x) = \sin(0.3x)\), \(g(x) = 3\cos(x) + 1\)`:

- The sine function
`\(f(x)\)` has a coefficient of 0.3, which means it oscillates faster than the regular sine function. It does not guarantee having one x-intercept in the domain of
`\(0 \leq x \leq 2\pi\)`. This option does not satisfy the first criteria.

- The cosine function
`\(g(x)\)` is multiplied by 3, which stretches the graph vertically. It does not guarantee having 2 or 3 intersection points with the sine function. This option does not satisfy the second criteria.

From the given options, none of them satisfy both criteria. Therefore, the correct answer is option d) None of the above.

Remember that to satisfy the first criteria, the sine function should have one x-intercept in the given domain. To satisfy the second criteria, the sine and cosine functions should intersect 2 or 3 times in the given domain.

The answer is option ⇒d) None of the above