Math / Trigonometry

QuestionSolve $2 \sin x+1=0$ . Show solution graphically.

Studdy Solution

STEP 1

1. The equation to solve is $2 \sin x + 1 = 0$ .
2. We are looking for solutions for $x$ within the interval $[0, 2\pi]$ unless otherwise specified.
3. The solution should be shown graphically.

STEP 2

1. Isolate $\sin x$ in the equation.
2. Determine the general solutions for $x$ .
3. Identify specific solutions within the desired interval.
4. Graph the function and the solutions.

STEP 3

Isolate $\sin x$ in the equation:
$2 \sin x + 1 = 0$
Subtract 1 from both sides:
$2 \sin x = -1$
Divide both sides by 2:
$\sin x = -\frac{1}{2}$

STEP 4

Determine the general solutions for $x$ :
The solutions for $\sin x = -\frac{1}{2}$ are given by:
$x = \arcsin\left(-\frac{1}{2}\right) + 2k\pi \quad \text{and} \quad x = \pi - \arcsin\left(-\frac{1}{2}\right) + 2k\pi$
where $k$ is any integer.

STEP 5

Identify specific solutions within the interval $[0, 2\pi]$ :
$\arcsin\left(-\frac{1}{2}\right) = -\frac{\pi}{6}$ , so:
x = -\frac{\pi}{6} + 2k\pi \quad \Rightarrow \quad x = \frac{11\pi}{6} \quad \text{(within [0, 2\pi])}
x = \pi - \left(-\frac{\pi}{6}\right) + 2k\pi \quad \Rightarrow \quad x = \frac{7\pi}{6} \quad \text{(within [0, 2\pi])}

STEP 6

Graph the function $y = 2\sin x + 1$ and indicate the solutions:
1. Plot the graph of $y = 2\sin x + 1$ .
2. Identify where the graph intersects the line $y = 0$ .
3. Mark the solutions $x = \frac{7\pi}{6}$ and $x = \frac{11\pi}{6}$ on the graph.
The solutions to the equation $2 \sin x + 1 = 0$ within the interval $[0, 2\pi]$ are:
$x = \frac{7\pi}{6}, \frac{11\pi}{6}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms