Solving For Tan Θ Given Sin Θ And Quadrant Information

Hey guys! 👋 Let's dive into a super interesting trigonometry problem today. We're given that $\sin \theta = \frac{7}{25}$ and $\frac{\pi}{2} < \theta < \pi$. Our mission, should we choose to accept it, is to find the value of $\tan \theta$, and we need to estimate our answer to two decimal places. Sounds like a plan? Awesome! Let’s break it down step by step. So, stick around, and let's unravel this trigonometric mystery together! We'll make it super easy and fun, I promise! 😉

Understanding the Givens

First off, let's really dig into what we already know. We're told that $\sin \theta = \frac{7}{25}$. Okay, that’s our starting point. Remember, sine is the ratio of the opposite side to the hypotenuse in a right-angled triangle. So, we can picture a right triangle where the side opposite to angle $\theta$ is 7 units long, and the hypotenuse is 25 units long. But wait, there's more! We also know that $\frac{\pi}{2} < \theta < \pi$. This little tidbit is super important because it tells us which quadrant our angle $\theta$ is chilling in. Think of the unit circle – $\frac{\pi}{2}$ is 90 degrees, and $\pi$ is 180 degrees. So, $\theta$ is somewhere between 90 and 180 degrees, which means it's hanging out in the second quadrant. Why does this matter? Well, in the second quadrant, sine is positive (which we already knew), but cosine and tangent are negative. Keep this in your mental toolbox because it's gonna be crucial in figuring out the sign of our final answer. We've got the sine, we know the quadrant, and we're armed with the knowledge that tangent will be negative. Let’s keep moving!

Calculating the Adjacent Side

Now that we've got our bearings, the next step is to figure out the length of the adjacent side of our right triangle. Remember the Pythagorean theorem? $a^2 + b^2 = c^2$, where a and b are the lengths of the two shorter sides (adjacent and opposite), and c is the length of the hypotenuse. In our case, we know the hypotenuse (c = 25) and the opposite side (b = 7). We need to find the adjacent side (a). Let's plug in the values and do some math magic! So, we have $a^2 + 7^2 = 25^2$. That simplifies to $a^2 + 49 = 625$. Now, subtract 49 from both sides: $a^2 = 625 - 49$, which gives us $a^2 = 576$. To find a, we take the square root of both sides: $a = \sqrt{576}$. And guess what? $\sqrt{576}$ is 24! So, the length of the adjacent side is 24 units. But hold on! Remember we're in the second quadrant where cosine (and therefore the adjacent side in this context) is negative? So, the adjacent side is actually -24. This is a super important detail, so make sure you've got it locked in. We’re making great progress! We've found the adjacent side, and we've remembered the quadrant rule. Let's keep this momentum going!

Finding tan θ

Alright, guys, we're in the home stretch! We've got all the pieces of the puzzle, and now it's time to put them together to find $\tan \theta$. Remember, tangent is the ratio of the opposite side to the adjacent side. In our case, the opposite side is 7, and the adjacent side is -24 (don't forget that negative sign!). So, $\tan \theta = \frac{opposite}{adjacent} = \frac{7}{-24}$. That means $\tan \theta = -\frac{7}{24}$. We're almost there, I can feel it! Now, the final step is to estimate this fraction as a decimal to two decimal places. Grab your calculators (or your mental math skills if you're feeling ambitious!), and let's do this. Divide 7 by 24, and you'll get approximately 0.291666... Since we want two decimal places, we look at the third decimal place. It's a 1, so we don't round up. But remember, we have a negative sign, so our final answer is approximately -0.29. And there you have it! We've successfully found the value of $\tan \theta$. High fives all around!

Final Answer

So, after all that trigonometric sleuthing, we've discovered that if $\sin \theta = \frac{7}{25}$ and $\frac{\pi}{2} < \theta < \pi$, then $\tan \theta \approx -0.29$. We took it step by step, remembering our trig ratios, the Pythagorean theorem, and the importance of quadrant signs. This is the kind of problem that really helps solidify your understanding of trigonometry. You guys did awesome! 🎉 Keep practicing, and these trig problems will become second nature. Remember, the key is to break it down, understand the givens, and apply the right concepts. And most importantly, have fun with it! 😊

• Given: $\sin \theta = \frac{7}{25}$ and $\frac{\pi}{2} < \theta < \pi$
• Quadrant: Second Quadrant ($\tan \theta$ is negative)
• Pythagorean Theorem: Found adjacent side = -24
• $\tan \theta = \frac{opposite}{adjacent} = \frac{7}{-24}$
• Estimate: $\tan \theta \approx -0.29$

Alright, let's talk about some sneaky pitfalls that often trip up students when tackling problems like this. Knowing these common mistakes can help you steer clear and ace your trigonometry! 🏆 One of the biggest slip-ups is forgetting about the quadrant. We can't stress this enough – the quadrant is crucial. It tells you the signs of sine, cosine, and tangent, and ignoring it can lead to a completely wrong answer. In our problem, knowing we were in the second quadrant clued us in that tangent was negative. Always, always, always check the quadrant! Another frequent fumble is misapplying the Pythagorean theorem. It's tempting to just plug in numbers without thinking, but make sure you're putting the values in the right places. Remember, $a^2 + b^2 = c^2$, where c is the hypotenuse. Getting those mixed up can throw off your entire calculation. And speaking of calculations, double-check your arithmetic! It's so easy to make a small mistake, like a sign error or a simple addition or subtraction goof, and that can derail your whole solution. Take a moment to review your steps and make sure everything adds up (literally!). Also, don't forget the negative sign when calculating the adjacent side in the second or third quadrant. Since cosine is negative in these quadrants, the adjacent side will also be negative. This is a super common error, so keep it top of mind! Finally, be careful with rounding. If the problem asks for a specific number of decimal places, make sure you round correctly. Look at the digit after the desired decimal place to determine whether to round up or down. And if you're using a calculator, be mindful of how it rounds – sometimes it can be a little sneaky. By being aware of these common mistakes, you can avoid them and boost your confidence in tackling trigonometry problems. You've got this! 💪

Okay, guys, now that we've conquered that problem and talked about common pitfalls, it's time to put your newfound skills to the test! Practice makes perfect, right? So, let's dive into some practice problems that are similar to what we just worked through. This will help solidify your understanding and build your confidence. Let's get started! 🚀

1. Problem 1: If $\cos \theta = -\frac{5}{13}$ and $\pi < \theta < \frac{3\pi}{2}$, find $\sin \theta$ and $\tan \theta$. Estimate your answers to two decimal places.
2. Problem 2: Given that $\tan \theta = \frac{8}{15}$ and $\pi < \theta < \frac{3\pi}{2}$, determine the values of $\sin \theta$ and $\cos \theta$. Round your answers to two decimal places.
3. Problem 3: If $\sin \theta = -\frac{24}{25}$ and $\frac{3\pi}{2} < \theta < 2\pi$, calculate $\cos \theta$ and $\tan \theta$. Express your answers as decimals rounded to two places.

Remember the steps we used earlier:

• Identify the given information and the quadrant.
• Use the Pythagorean theorem to find the missing side.
• Determine the signs of the trigonometric functions based on the quadrant.
• Calculate the required trigonometric ratios.
• Estimate the final answers to two decimal places.

These problems are designed to give you practice with different scenarios and quadrants. Don't be afraid to take your time, draw diagrams if it helps, and think through each step. And if you get stuck, don't worry! Review the steps we discussed and try again. You've got this! 😉

Alright, let's take a step back from the math for a second and think about why all this trigonometry stuff actually matters in the real world. It's easy to get caught up in the equations and formulas, but it's super cool to see how these concepts pop up in everyday life. So, where does trigonometry actually get used? Well, one big area is navigation. Think about GPS systems, ships at sea, and airplanes in the sky. Trigonometry is essential for calculating angles, distances, and directions. Pilots and sailors use trigonometric principles to chart courses, determine their position, and avoid obstacles. It's literally life-saving stuff! Another fascinating application is in architecture and engineering. When designing buildings, bridges, and other structures, engineers need to calculate angles and forces to ensure stability and safety. Trigonometry helps them determine the correct angles for beams, supports, and other structural elements. Without it, our buildings might not stand up straight! 🏢 Trigonometry also plays a crucial role in physics. From analyzing projectile motion to understanding waves and oscillations, trigonometric functions are used to model a wide range of physical phenomena. For example, when studying the motion of a pendulum or the behavior of light waves, trigonometry is indispensable. And let's not forget about surveying. Surveyors use trigonometry to measure land, create maps, and determine property boundaries. They rely on trigonometric techniques to calculate distances and elevations, which are essential for accurate land management and construction projects. But wait, there's more! Trigonometry even has applications in computer graphics and game development. When creating 3D models and animations, trigonometric functions are used to rotate, scale, and position objects in space. This is what makes video games and animated movies look so realistic. So, as you can see, trigonometry isn't just some abstract math concept – it's a powerful tool that's used in countless ways to make our world work. Next time you see a tall building, use your GPS, or play a video game, remember that trigonometry is working behind the scenes. Pretty cool, right? 😎