Question

Find the value of x in the intervals for the equation sin2x = cos x.

Answer 1

The value of x in the intervals for the equation
`\( \sin(2x) = \cos(x) \)` is
`\( x = (\pi)/(6) + n\pi \)`, where
`\( n \)` is an integer.

Step-by-step explanation:

To find the value of
`\( x \)` in the given equation,
`\( \sin(2x) = \cos(x) \)`, we can use trigonometric identities. First, let's rewrite the equation using double-angle and basic trigonometric identities:

`\[ \sin(2x) = \cos(x) \]`

Using the double-angle identity
`\( \sin(2x) = 2\sin(x)\cos(x) \)`, we get:

`\[ 2\sin(x)\cos(x) = \cos(x) \]`

Now, we can simplify by dividing both sides by
`\( \cos(x) \) (assuming \( \cos(x) \\eq 0 \)):`

`\[ 2\sin(x) = 1 \]`

Dividing both sides by 2:

`\[ \sin(x) = (1)/(2) \]`

This gives us
`\( x = (\pi)/(6) + n\pi \)`, where
`\( n \)` is an integer. The solution is periodic because the sine function repeats every
`\( 2\pi \)` radians.

In conclusion, the solution to the equation
`\( \sin(2x) = \cos(x) \)` is
`\( x = (\pi)/(6) + n\pi \)`, where
`\( n \)` can take any integer value. This represents the intervals in which the equation holds true, considering the periodic nature of trigonometric functions.