Questions: Solve sin(x)=0.1 on 0 ≤ x<2π There are two solutions, A and B, with A<B A= B= Give your answers accurate to 3 decimal places

Transcript text: Solve $\sin (x)=0.1$ on $0 \leq x<2 \pi$ There are two solutions, A and B , with $\mathrm{A}<\mathrm{B}$ \[ \begin{array}{l} A= \\ B= \end{array} \] Give your answers accurate to 3 decimal places

Solution

Solution Steps

To solve the equation $\sin(x) = 0.1$ on the interval $0 \leq x < 2\pi$, we need to find the angles $x$ for which the sine value is 0.1. The sine function is periodic with a period of $2\pi$, and within one period, it will have two solutions for this equation. We can use the inverse sine function to find the principal value and then determine the second solution using the properties of the sine function.

Solution Approach

Use the inverse sine function to find the principal solution $A$.
Use the property of the sine function that $\sin(\pi - x) = \sin(x)$ to find the second solution $B$.
Ensure both solutions are within the interval $[0, 2\pi)$.

Step 1: Find the Principal Solution

To solve the equation $\sin(x) = 0.1$, we first find the principal solution using the inverse sine function: \[ A = \arcsin(0.1) \approx 0.1002 \]

Step 2: Find the Second Solution

The sine function has a property that allows us to find the second solution in the interval $[0, 2\pi)$: \[ B = \pi - A \approx 3.0414 \]

Step 3: Round the Solutions

We round both solutions to three decimal places: \[ A \approx 0.100 \quad \text{and} \quad B \approx 3.041 \]