Solving the Equation sin(sin(x)) 0

Understanding the trigonometric identity and solving the equation sin(sin(x)) 0 involves a multi-step process. This article explores the steps and underlying concepts, providing a detailed breakdown to ensure clarity and comprehension.

Understanding Trigonometric Identities and Zeros

To solve the equation sin(sin(x)) 0, we break it down into steps by understanding the zeros of the sine function.

Step 1: Solving sin(y) 0

The sine function equals zero at integer multiples of π:

sin(y) 0 implies y nπ for n ∈ ?.

Step 2: Applying to sin(x)

We set sin(x) nπ for y.

Step 3: Considering the Range of sin(x)

The sine function only takes values in the range [-1, 1]. Therefore, nπ must fall within this range:

-1 ≤ nπ ≤ 1.

This implies:

-1/π ≤ n ≤ 1/π.

Since n is an integer, the only possible value for n is 0.

Substituting n 0

We find:

sin(x) 0.

Solving sin(x) 0

The solutions to this equation are:

x kπ for k ∈ ?.

Final Solution

The complete solution to the equation sin(sin(x)) 0 is:

x kπ for k ∈ ?.

Graphical Interpretation

The graphs of sin(x) and sin(sin(x)) are quite similar. Both have the same frequency and phase, but the amplitude of sin(x) is 1, while the amplitude of sin(sin(x)) is approximately 0.84 when sin(x) 1 radian.

Integer Solutions

The sin function equals zero at integer multiples of π. For the equation sin(x) nπ, the integer value of n that satisfies the condition within the range [-1, 1] is n 0. Therefore, the solution is:

x 0.

Complex Solutions

Allowing x to be a complex number, n can be any integer, yielding many more solutions for sin(sin(x)) 0.