Solve Sin(3θ) = -0.3728: Find 2 Smallest Solutions

Hey guys! Let's dive into solving a trig problem that involves finding the smallest two solutions for the equation $\sin(3\theta) = -0.3728$ within the interval $[0, 2\pi)$. This might sound a bit intimidating at first, but don't worry, we'll break it down step by step. We'll go through the concepts, the calculations, and the reasoning so that you'll be able to tackle similar problems with confidence. Ready? Let's get started!

Understanding the Problem

Before we jump into the calculations, it's crucial to understand what the problem is asking. We're given a trigonometric equation, $\sin(3\theta) = -0.3728$, and we need to find the two smallest values of $\theta$ that satisfy this equation within the interval $[0, 2\pi)$. Remember that $2\pi$ represents a full circle in radians, so we're essentially looking for solutions within one full rotation.

Now, a key thing to recognize here is the 3\theta inside the sine function. This means that the sine function's argument is three times the angle we're trying to find. This will affect how we find our solutions, as we'll need to consider the periodic nature of the sine function and how it's compressed or stretched by this factor of 3. The sine function oscillates between -1 and 1, and it repeats its cycle every $2\pi$ radians. The negative value of -0.3728 tells us that we're looking for angles where the sine function is negative, which occurs in the third and fourth quadrants.

Graphically, you can visualize the sine wave oscillating. When you have $\sin(3\theta)$, you're essentially compressing the sine wave horizontally, meaning it oscillates three times as fast. This is why we'll get more solutions within the $[0, 2\pi)$ interval compared to a regular $\sin(\theta)$ equation. We will find three times as many solutions since the period of $\sin(3\theta)$ is $\frac{2\pi}{3}$. This is an important concept to understand because it directly impacts the number of solutions we expect to find and where they'll be located. The periodic nature of trigonometric functions is fundamental to solving these kinds of equations, and understanding how the coefficient of $\theta$ affects the period is crucial.

Finding the Reference Angle

Okay, so the first step in solving this equation is to find what we call the reference angle. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. Essentially, it’s the angle we'd have if we ignored the sign of the trigonometric value. To find this, we'll use the inverse sine function (also known as arcsin or $\sin^{-1}$) on the absolute value of our given value, which is 0.3728. So, we're calculating $\sin^{-1}(0.3728)$.

Using a calculator, make sure it's in radian mode (very important, guys!), we find that the reference angle, which we'll call $\alpha$, is approximately 0.3826 radians. That is, $\alpha = \sin^{-1}(0.3728) \approx 0.3826$. This angle $\alpha$ is in the first quadrant, but remember, we're looking for solutions where sine is negative. Sine is negative in the third and fourth quadrants. This is where understanding the unit circle and the CAST rule (or ASTC rule) comes in handy. CAST helps us remember which trigonometric functions are positive in each quadrant: C (Cosine) in the fourth, A (All) in the first, S (Sine) in the second, and T (Tangent) in the third. Since we want sine to be negative, we're interested in the third and fourth quadrants.

Now that we have the reference angle, we can find the angles in the third and fourth quadrants that have the same sine value (but negative). In the third quadrant, the angle is $\pi + \alpha$, and in the fourth quadrant, it’s $2\pi - \alpha$. These angles give us the solutions for $3\theta$, not $\theta$ directly, so we're one step closer to our final answer. Don't forget this key difference; we'll deal with it in the next step when we solve for $\theta$ itself.

Solving for 3θ

Now that we have our reference angle ($\alpha \approx 0.3826$ radians), we can find the angles in the third and fourth quadrants where the sine function will give us -0.3728. Remember, we are solving for $3\theta$ first, not $\theta$ directly. In the third quadrant, the angle is given by $\pi + \alpha$. So,

$$3\theta_1 = \pi + 0.3826 \approx 3.5242$$

In the fourth quadrant, the angle is given by $2\pi - \alpha$. So,

$$3\theta_2 = 2\pi - 0.3826 \approx 5.9006$$

These are the two initial solutions for $3\theta$ within the first cycle (0 to $2\pi$). However, because we have $3\theta$, we need to consider that the sine function will complete three full cycles within the interval $[0, 2\pi)$ for $\theta$. This means we need to find additional solutions by adding multiples of $2\pi$ to our initial solutions until we exceed the range for $3\theta$, which is $[0, 6\pi)$.

So, let's add $2\pi$ to our first solution:

$$3\theta_3 = 3.5242 + 2\pi \approx 9.8074$$

And again:

$$3\theta_4 = 3.5242 + 4\pi \approx 16.0906$$

Now, let's do the same for our second solution:

$$3\theta_5 = 5.9006 + 2\pi \approx 12.1842$$

And again:

$$3\theta_6 = 5.9006 + 4\pi \approx 18.4674$$

We stop here because adding another $2\pi$ would make the values greater than $6\pi$ (which is approximately 18.85), the upper bound for $3\theta$ since $\theta$ is in the range $[0, 2\pi)$. So, we have six potential solutions for $3\theta$: approximately 3.5242, 5.9006, 9.8074, 12.1842, 16.0906, and 18.4674. These solutions are crucial intermediates that will lead us to the values of $\theta$ we are seeking.

Solving for θ

Okay, guys, we've made some great progress! We've found six solutions for $3\theta$. Now, the final step is to find the solutions for $\theta$ itself. This is actually quite straightforward: we simply divide each of our $3\theta$ solutions by 3. Remember, we found the following solutions for $3\theta$:

• $3\theta_1 \approx 3.5242$
• $3\theta_2 \approx 5.9006$
• $3\theta_3 \approx 9.8074$
• $3\theta_4 \approx 12.1842$
• $3\theta_5 \approx 16.0906$
• $3\theta_6 \approx 18.4674$

Now, let's divide each of these by 3 to get the corresponding values for $\theta$:

• $\theta_1 \approx 3.5242 / 3 \approx 1.1747$
• $\theta_2 \approx 5.9006 / 3 \approx 1.9669$
• $\theta_3 \approx 9.8074 / 3 \approx 3.2691$
• $\theta_4 \approx 12.1842 / 3 \approx 4.0614$
• $\theta_5 \approx 16.0906 / 3 \approx 5.3635$
• $\theta_6 \approx 18.4674 / 3 \approx 6.1558$

So, we have six potential solutions for $\theta$ within the interval $[0, 2\pi)$. However, the question asks for the smallest two solutions. Looking at our list, the two smallest values are approximately 1.1747 and 1.9669. These are the two angles within the specified interval that satisfy the original equation $\sin(3\theta) = -0.3728$.

It’s essential to verify that these solutions indeed fall within the given interval of $[0, 2\pi)$, which they do since $2\pi$ is approximately 6.2832. Also, remember that when you divide by the coefficient of $\theta$, you are essentially