Question

The number of solutions of the equation sinx = cos²x in the interval (0, 10) is ______.

Options

• 1

• 2

• 3

• 4

Answer 1

The number of solutions of the equation sinx = cos²x in the interval (0, 10) is 4.

Explanation:

Given, sinx = cos²x

⇒ sinx = 1 – sin²x

⇒ sin²x + sinx – 1 = 0

Let sin x = t

⇒ t² + t – 1 = 0

⇒ t = `(-1 +- sqrt(1^2 - 4(1)(-1)))/(2(1))`

⇒ t = `(-1 +- sqrt(5))/2`

⇒ sinx = `(-1 +- sqrt(5))/2`

As we know sin x ∈ [–1, 1]

∴ sinx = `(sqrt(5) - 1)/2`

Lets draw the graph of y = sin x and y = `(sqrt(5) - 1)/2` for x ∈ (0, 10)

∵ Both curve: intersect at 4 points for x ∈ (0, 10)

∴ Number of solution for given equation in x ∈ (0, 10) are 4.