Question

Use a calculator to find all the solutions to the equation 1.24− 0.5sinx=1.01
.

Assume that n is an integer.



3.62+2πn and 5.805+2πn
0.478+2πn and 2.664+2πn
2.664+2πn
0.478+2πn

Answer 1

To solve the equation \( 1.24 - 0.5 \sin x = 1.01 \), we first isolate the sine function:

\[
1.24 - 1.01 = 0.5 \sin x
\]
\[
0.23 = 0.5 \sin x
\]
\[
\sin x = \frac{0.23}{0.5} = 0.46
\]

Next, we need to find the angles \( x \) such that \( \sin x = 0.46 \). Using a calculator to find the principal value:

\[
x = \arcsin(0.46) \approx 0.478 \text{ radians}
\]

Since the sine function is positive in the first and second quadrants, we also consider the second solution:

\[
x = \pi - 0.478 \approx 2.664 \text{ radians}
\]

Thus, the general solutions for the equation can be expressed as:

\[
x = 0.478 + 2\pi n \quad \text{and} \quad x = 2.664 + 2\pi n
\]

where \( n \) is any integer.

Therefore, the complete set of solutions to the equation \( 1.24 - 0.5\sin x = 1.01 \) is:

\[
0.478 + 2\pi n \quad \text{and} \quad 2.664 + 2\pi n
\]

From the options given, the correct answer is:

**0.478 + 2πn and 2.664 + 2πn**.