Question

Math

Posted 4 months ago

```
Select one or more expressions that together represent all solutions to the equation. Your answer should be in radians.
Assume $\boldsymbol{n}$ is any integer.
$13 \sin (2 x)-3=3$
Choose all answers that apply:
A $-0.480+n \cdot 2 \pi$
B $-0.240+n \cdot \pi$
c $0.240+n \cdot \pi$
D $1.331+n \cdot \pi$
E $0.480+n \cdot 2 \pi$
F $2.661+n \cdot 2 \pi$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

Adding 3 to both sides of the equation gives us $13 \sin(2x) = 6$

step 3

Dividing both sides by 13 gives us $\sin(2x) = \frac{6}{13}$

step 4

To find $x$, we take the inverse sine of $\frac{6}{13}$, which gives us $2x = \sin^{-1}\left(\frac{6}{13}\right)$

step 5

The general solution for $x$ in terms of $n$ is $x = \frac{1}{2} \left(2 \pi n + \pi - \sin^{-1}\left(\frac{6}{13}\right)\right)$ and $x = \pi n + \frac{1}{2} \sin^{-1}\left(\frac{6}{13}\right)$ for $n \in \mathbb{Z}$

step 6

We calculate the value of $\frac{1}{2} \sin^{-1}\left(\frac{6}{13}\right)$ to find the specific solutions that match the given options

step 7

The value of $\sin^{-1}\left(\frac{6}{13}\right)$ is approximately $0.480$ radians. Therefore, the solutions are $x = \pi n + 0.240$ and $x = \pi n - 0.240$

step 8

Comparing these solutions to the given options, we find that option C and option B represent all solutions to the equation

B, C

Key Concept

General Solution for Trigonometric Equations

Explanation

The general solution for a trigonometric equation of the form $\sin(\theta) = a$ is $\theta = \sin^{-1}(a) + 2\pi n$ or $\theta = \pi - \sin^{-1}(a) + 2\pi n$, where $n$ is any integer. For the equation $\sin(2x) = \frac{6}{13}$, we find the specific solutions and express them in a form that matches the given multiple-choice options.

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