The smalest positive number for which 3sin(2x-6)=1 is x=

To find the smallest positive value of \( x \) for which \( 3\sin(2x - 6) = 1 \), follow these steps:

### Step 1: Isolate the sine function
First, divide both sides of the equation by 3:
\[
\sin(2x - 6) = \frac{1}{3}
\]

### Step 2: Solve for the angle inside the sine function
Now, take the inverse sine (arcsin) of both sides to find the general solution:
\[
2x - 6 = \arcsin\left(\frac{1}{3}\right)
\]
Let's denote \( \arcsin\left(\frac{1}{3}\right) \) as \( \theta \).

Thus:
\[
2x - 6 = \theta
\]

### Step 3: Solve for \( x \)
Add 6 to both sides:
\[
2x = \theta + 6
\]
Divide by 2:
\[
x = \frac{\theta + 6}{2}
\]

### Step 4: Consider the periodic nature of the sine function
The sine function is periodic with a period of \( 2\pi \). Therefore, the general solution for \( x \) is:
\[
x = \frac{\theta + 6}{2} + \frac{k\pi}{2}
\]
where \( k \) is an integer.

For the smallest positive \( x \), set \( k = 0 \):
\[
x = \frac{\arcsin\left(\frac{1}{3}\right) + 6}{2}
\]

### Step 5: Calculate the value
Using a calculator, find \( \arcsin\left(\frac{1}{3}\right) \approx 0.3398 \) radians.

Now:
\[
x = \frac{0.3398 + 6}{2} = \frac{6.3398}{2} = 3.1699 \text{

Learn how to find the smallest positive number for which the equation 3sin(2x - 6) = 1 holds true using trigonometric principles and inverse sine function. This problem involves solving trigonometric equations and is suitable for students in Grades 11-12.

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