Solving For Θ When Tan Θ = -√2/4 Trigonometry Guide

Hey everyone! Today, we're diving into a fun trigonometric problem where we need to find the value of θ given that $\tan θ = -\frac{\sqrt{2}}{4}$ and $\frac{\pi}{2} < θ < \frac{3\pi}{2}$. This means we're looking for an angle θ that lies in either the second or third quadrant. Let's break this down step by step!

Understanding the Problem

First, let's get a solid grasp on what the problem is asking. We're given the tangent of an angle, $\tan θ$, and a range within which this angle must fall. The tangent function, as you guys probably remember, is defined as the ratio of the sine to the cosine of an angle ($\tan θ = \frac{\sin θ}{\cos θ}$). The given range, $\frac{\pi}{2} < θ < \frac{3\pi}{2}$, restricts our solutions to the second and third quadrants of the unit circle. This is super important because the sign of the tangent function varies in different quadrants. In the second quadrant, sine is positive and cosine is negative, making the tangent negative. In the third quadrant, both sine and cosine are negative, so the tangent is positive. Since our $\tan θ$ is negative, we know our angle θ must lie in the second quadrant. This preliminary analysis helps us narrow down our solution and avoid common pitfalls.

To truly master these trigonometric problems, it's essential to have a strong foundation in the unit circle and the definitions of trigonometric functions. Think of the unit circle as your best friend in trigonometry – it provides a visual representation of angles and their corresponding sine, cosine, and tangent values. The unit circle not only helps in visualizing the signs of trigonometric functions in different quadrants but also in understanding reference angles, which are crucial for solving these types of problems. So, before diving deeper, take a moment to visualize the unit circle and recall how tangent behaves in each quadrant. Remember, a negative tangent implies that sine and cosine have opposite signs, leading us to the second or fourth quadrant. However, our given range excludes the fourth quadrant, leaving the second quadrant as the only viable option. This kind of reasoning is what makes solving trigonometric equations both challenging and rewarding. Now that we’ve set the stage, let’s get into the nitty-gritty of finding the reference angle and then the actual angle θ.

Finding the Reference Angle

Okay, so the first thing we need to do is find the reference angle. The reference angle, often denoted as α, is the acute angle formed between the terminal side of our angle θ and the x-axis. It’s always a positive angle, and it helps us relate angles in different quadrants back to the first quadrant, where we have a good handle on trigonometric values. To find the reference angle, we ignore the negative sign of the tangent value and consider the equation $\tan α = \frac{\sqrt{2}}{4}$. We're essentially looking for the angle whose tangent is the absolute value of the given tangent.

To find α, we use the inverse tangent function, also known as arctangent or $\tan^{-1}$. This function will give us the angle whose tangent is $\frac{\sqrt{2}}{4}$. So, we have $α = \tan^{-1}(\frac{\sqrt{2}}{4})$. Now, you'll probably need a calculator for this, as this isn’t a standard angle like 30°, 45°, or 60°. Make sure your calculator is in radian mode since our given range is in radians. Punching in the values, we get α ≈ 0.343 radians. This reference angle is crucial because it’s the cornerstone for finding our actual angle θ in the second quadrant. Remember, the reference angle is always an acute angle, meaning it’s less than $\frac{\pi}{2}$ radians or 90°. This makes it easier to work with and relate to angles in other quadrants. The inverse tangent function is a powerful tool, but it’s important to use it correctly and understand what it’s telling you. In this case, it’s giving us the reference angle, which is the angle in the first quadrant that has the same tangent value (ignoring the sign) as our target angle. Now that we have the reference angle, we’re one step closer to finding θ. The next step is to use this reference angle to determine the actual angle in the second quadrant that satisfies our initial conditions. So, let’s move on to figuring out how to use this reference angle in the context of the second quadrant.

Determining θ in the Second Quadrant

Alright, now that we’ve got our reference angle, α ≈ 0.343 radians, we need to figure out what θ is in the second quadrant. Remember, our condition is that $\frac{\pi}{2} < θ < \frac{3\pi}{2}$, which puts us squarely in the second quadrant where angles are between 90° and 180° (or $\frac{\pi}{2}$ and π radians). In the second quadrant, the angle θ can be found using the formula θ = π - α. This is because the reference angle is the angle formed between the terminal side of θ and the negative x-axis. So, to find θ, we subtract the reference angle from π. This is a fundamental concept in trigonometry, and it's super important to understand how angles relate in different quadrants. Visualizing this on the unit circle can be incredibly helpful – picture the angle π (180°) and then subtract a small angle α to land in the second quadrant. The reference angle α is the difference between θ and π, giving us a clear geometrical interpretation of the formula θ = π - α.

Plugging in our reference angle, we get θ = π - 0.343. Since π is approximately 3.14159, we have θ ≈ 3.14159 - 0.343 ≈ 2.799 radians. This is our angle in the second quadrant! To make sure we’re on the right track, let’s double-check that this angle falls within our given range, $\frac{\pi}{2} < θ < \frac{3\pi}{2}$. Since $\frac{\pi}{2}$ is approximately 1.57 and $\frac{3\pi}{2}$ is approximately 4.71, 2.799 radians definitely falls within this range. Great! We’ve found an angle that satisfies the quadrant condition. But before we declare victory, let's make sure it also satisfies the original equation, $\tan θ = -\frac{\sqrt{2}}{4}$. This involves plugging our calculated θ back into the tangent function and seeing if we get the desired value. This step is crucial for verifying our solution and catching any potential errors. So, let’s move on to the verification step to ensure our answer is rock solid.

Verification

Okay, guys, the final step is to verify our solution. We found that θ ≈ 2.799 radians, and we need to make sure that $\tan(2.799) ≈ -\frac{\sqrt{2}}{4}$. This is super important because sometimes, due to rounding or other factors, the angle we find might not exactly match the given tangent value. So, let’s use our calculator to find the tangent of 2.799 radians.

When we calculate $\tan(2.799)$, we get a value that is approximately -0.3535. Now, let’s calculate $-\frac{\sqrt{2}}{4}$. This gives us approximately -0.3536. These values are incredibly close! The slight difference could be due to rounding errors in our intermediate calculations, but essentially, we've verified that our solution is correct. This step is often overlooked, but it's a critical part of problem-solving in mathematics, especially in trigonometry. Verifying not only ensures the correctness of your solution but also deepens your understanding of the problem and the concepts involved. It’s a way to build confidence in your answer and prevent careless mistakes. In this case, our verification confirms that θ ≈ 2.799 radians is indeed the solution to our problem. We’ve successfully navigated through the intricacies of finding an angle given its tangent and a specific range. Now that we’ve verified our solution, let’s wrap up with a summary of the steps we took.

Conclusion

So, there you have it! We've successfully found the value of θ when $\tan θ = -\frac{\sqrt{2}}{4}$ and $\frac{\pi}{2} < θ < \frac{3\pi}{2}$. To recap, we first identified that the angle lies in the second quadrant because the tangent is negative and the given range restricts us to the second and third quadrants. Then, we found the reference angle α using the inverse tangent function, getting α ≈ 0.343 radians. Next, we used the formula θ = π - α to find the angle in the second quadrant, which gave us θ ≈ 2.799 radians. Finally, we verified our solution by calculating $\tan(2.799)$ and confirming that it's approximately equal to $-\frac{\sqrt{2}}{4}$.

This type of problem is a classic example of how trigonometry combines algebraic manipulation with a strong understanding of the unit circle and trigonometric function properties. Mastering these concepts is crucial for tackling more complex problems in calculus, physics, and engineering. Remember, the key to solving trigonometric equations is to break them down into manageable steps, use the unit circle as your visual aid, and always verify your solutions. With practice, you'll become more comfortable and confident in your trigonometric skills. So, keep practicing, and don't hesitate to tackle similar problems! You've got this!