Solving Trigonometric Equations: A Step-by-Step Guide

Hey there, math enthusiasts! Ever stumbled upon a trigonometric equation and felt a bit lost? Don't sweat it! We're going to break down how to solve the equation ${7 \sin 3\theta + 6 = 0}$ for ${\theta}$ within the interval ${\[0^{\circ}, 360^{\circ}}$). It might seem tricky at first, but with a little patience and these easy steps, you'll be acing these problems in no time. So, let's dive in and make trigonometry a whole lot less intimidating, okay?

Isolating the Trigonometric Function

Okay, guys, the first thing we need to do is get that sine function all by itself. Think of it like this: we're trying to uncover the secret of ${\theta}$, and to do that, we have to peel away all the extra layers. We start by subtracting 6 from both sides of the equation. This gives us:

${7 \sin 3\theta = -6}$

Now, to isolate the sine function completely, we'll divide both sides by 7:

${\sin 3\theta = -\frac{6}{7}}$

See? We've simplified the equation, getting closer to finding our angle ${\theta}$. This step is crucial because it sets the stage for using our knowledge of the unit circle or inverse trigonometric functions. Always remember that isolating the trig function is the cornerstone of solving these equations. In this case, we've made the equation much more manageable, and we're ready for the next steps. Keep going; you're doing great!

Now, the expression is ready to determine the reference angle. To do this, we use the inverse sine function. Make sure your calculator is in degree mode. This is super important, because radians and degrees are different units for measuring angles, and mixing them up is a common mistake that can lead to wrong answers. So, double-check your calculator settings before you proceed! The inverse sine of ${-\frac{6}{7}}$ is approximately -59.03 degrees. However, the reference angle is always the positive acute angle formed by the terminal side of the angle and the x-axis. So we'll use 59.03 degrees as our reference angle.

Finding the Reference Angle

Alright, now that we have isolated the trigonometric function, it's time to find the reference angle. This is the acute angle formed between the terminal side of the angle and the x-axis. To find this reference angle, we'll use the inverse sine function. Specifically, we'll take the inverse sine of the absolute value of ${-\frac{6}{7}}$. Remember to keep your calculator in degree mode! So, here's the deal:

${\sin^{-1}\left(\frac{6}{7}\right) \approx 59.03^{\circ}}$

This tells us the reference angle, which we'll call ${\alpha}$, is approximately 59.03 degrees. This reference angle is super important because it helps us figure out the other angles that satisfy the original equation. We're not quite done yet, but we're getting closer. We'll use this reference angle to find all possible values of ${3\theta}$ in the next step. Keep going, you're doing great!

Determining the Angles in the Specified Interval

Okay, so here's where things get a bit more interesting. Remember how we found that reference angle of 59.03 degrees? Now we need to figure out which quadrants our angle ${3\theta}$ lies in. Since ${\sin 3\theta}$ is negative ${-\frac{6}{7}}$, we know that ${3\theta}$ must be in either the third or fourth quadrants. That's because the sine function is negative in those quadrants. Here's how we find the angles:

• In the third quadrant: ${3\theta = 180^{\circ} + \alpha}$
• In the fourth quadrant: ${3\theta = 360^{\circ} - \alpha}$

So, let's plug in our reference angle:

• ${3\theta = 180^{\circ} + 59.03^{\circ} = 239.03^{\circ}}$
• ${3\theta = 360^{\circ} - 59.03^{\circ} = 300.97^{\circ}}$

But wait, there's more! Because the sine function has a period of 360 degrees, we need to consider all angles coterminal with these two angles. That means we need to add multiples of 360 degrees to each of these angles until we exceed the interval. In other words, we need to solve the general solutions for ${3\theta}$, which are:

${3\theta = 239.03^{\circ} + 360^{\circ}n}$

${3\theta = 300.97^{\circ} + 360^{\circ}n}$

Here, ${n}$ is any integer. Let's determine a range of values that could apply for ${n}$, and find the angles. Keep in mind, the range is [0, 360), so we need to pick an ${n}$ such that the resulting value for ${3\theta}$ is inside the range, but we must then divide each result by 3 to obtain the angle ${\theta}$, the thing we're after!

Solving for θ

Alright, now we are in the home stretch! We've found the values of ${3\theta}$, and now it is time to get the values of ${\theta}$. To do this, we simply divide each solution by 3:

• ${\theta = \frac{239.03^{\circ}}{3} \approx 79.68^{\circ}}$
• ${\theta = \frac{300.97^{\circ}}{3} \approx 100.32^{\circ}}$

But don't forget! The sine function has a period of ${360^{\circ}}$, which means that we must determine whether there are any other valid results within the specified range. Given that our original problem specified ${3\theta}$, the total period is 120 degrees. Therefore, we need to add 120 degrees to the angles we already found, and check if the answers are still within the range:

• ${79.68^{\circ} + 120^{\circ} = 199.68^{\circ}}$
• ${100.32^{\circ} + 120^{\circ} = 220.32^{\circ}}$

Because we have a ${3\theta}$ in the original equation, there are a total of six solutions, given that we had two basic solutions from the quadrants and three full periods (360 degrees). So, we need to add another 120 degrees to our results:

• ${199.68^{\circ} + 120^{\circ} = 319.68^{\circ}}$
• ${220.32^{\circ} + 120^{\circ} = 340.32^{\circ}}$

Therefore, the solutions for ${\theta}$ in the interval ${\[0^{\circ}, 360^{\circ}}$) are approximately:

${\theta \approx 79.68^{\circ}, 100.32^{\circ}, 199.68^{\circ}, 220.32^{\circ}, 319.68^{\circ}, 340.32^{\circ}}$

Conclusion

And there you have it! We've successfully solved the trigonometric equation ${7 \sin 3\theta + 6 = 0}$ within the interval ${\[0^{\circ}, 360^{\circ}}$). Remember, solving these equations involves isolating the trigonometric function, finding the reference angle, determining the angles in the correct quadrants, and then solving for ${\theta}$. It's all about breaking down the problem step by step. With practice, you'll become a pro at this in no time. Keep practicing and exploring, and don't be afraid to ask for help. You've got this!

Key Takeaways

• Isolate the trig function: Always start by isolating the trigonometric function. This simplifies the equation and makes it easier to solve.
• Find the reference angle: Use the inverse trig function to find the reference angle.
• Determine the quadrants: Knowing the sign of the trig function helps you determine which quadrants the angle can be in.
• Solve for the angle: Use the reference angle and the quadrant information to find all possible solutions within the interval.
• Consider the period: Account for the periodicity of the trigonometric function to find all solutions.

Final Thoughts

Solving trigonometric equations might seem daunting at first, but with practice and a systematic approach, it becomes much more manageable. Remember to always double-check your work and pay close attention to the intervals specified in the problem. By mastering these concepts, you'll not only excel in your math classes but also gain a deeper appreciation for the beauty and power of trigonometry. Keep practicing, and don't hesitate to seek help if you need it. You're on your way to becoming a trigonometry master! Happy solving, guys!