Finding The Exact Value Of Cos(u+v) A Trigonometric Solution

Hey guys! Today, we're diving deep into the world of trigonometry to solve a fascinating problem. We're going to find the exact value of $\cos (u+v)$, but there's a twist! We're given some information about $\sin u$ and $\sin v$, and we need to use our trigonometric knowledge to piece everything together. Buckle up, because this is going to be an exciting journey!

Breaking Down the Problem

Before we jump into calculations, let's understand what we're dealing with. We know that $\sin u = \frac{-4}{5}$ and $\sin v = \frac{-5}{13}$. The crucial part here is the quadrants: $u$ is in quadrant IV, and $v$ is in quadrant III. These quadrants are our secret keys, telling us the signs of cosine and sine in those regions. Remember, in quadrant IV, cosine is positive, and sine is negative, while in quadrant III, both cosine and sine are negative. This information is vital for determining the correct signs of our trigonometric values later on.

Our mission is to find $\cos (u+v)$. This looks like a job for the cosine addition formula! Do you guys remember it? It states that $\cos (u+v) = \cos u \cos v - \sin u \sin v$. See? We already know $\sin u$ and $\sin v$, but we're missing $\cos u$ and $\cos v$. That's our next challenge!

Finding cos u and cos v

To find the missing cosine values, we'll use the Pythagorean identity: $\sin^2 θ + \cos^2 θ = 1$. This identity is a cornerstone of trigonometry, and it's going to be our best friend here.

Let's start with $\cos u$. We know $\sin u = \frac-4}{5}$, so $\sin^2 u = \left( \frac{-4}{5} \right)^2 = \frac{16}{25}$. Plugging this into the Pythagorean identity, we get $\frac{16}{25} + \cos^2 u = 1$. Solving for $\cos^2 u$, we have $\cos^2 u = 1 - \frac{16}{25} = \frac{9}{25}$. Now, taking the square root of both sides, we get $\cos u = \pm \frac{3}{5}$. Remember, $u$ is in quadrant IV where cosine is positive, so we choose the positive value $\cos u = \frac{3{5}$.

Now, let's tackle $\cos v$. We know $\sin v = \frac-5}{13}$, so $\sin^2 v = \left( \frac{-5}{13} \right)^2 = \frac{25}{169}$. Using the Pythagorean identity again, we have $\frac{25}{169} + \cos^2 v = 1$. Solving for $\cos^2 v$, we get $\cos^2 v = 1 - \frac{25}{169} = \frac{144}{169}$. Taking the square root, we find $\cos v = \pm \frac{12}{13}$. Since $v$ is in quadrant III where cosine is negative, we choose the negative value $\cos v = \frac{-12{13}$.

Great job, guys! We've successfully found both $\cos u$ and $\cos v$. We're now one step closer to solving the original problem.

Plugging into the Cosine Addition Formula

We've gathered all the pieces of the puzzle. We know:

• $$\\sin u = \\frac{-4}{5}$$

• $$\\cos u = \\frac{3}{5}$$

• $$\\sin v = \\frac{-5}{13}$$

• $$\\cos v = \\frac{-12}{13}$$

And we know the cosine addition formula: $\cos (u+v) = \cos u \cos v - \sin u \sin v$.

Let's plug in the values! We get:

$$\\\\\cos (u+v) = \\left( \\frac{3}{5} \\right) \\left( \\frac{-12}{13} \\right) - \\left( \\frac{-4}{5} \\right) \\left( \\frac{-5}{13} \\right)$$

Now, let's simplify. Multiplying the fractions, we have:

$$\\\\\cos (u+v) = \\frac{-36}{65} - \\frac{20}{65}$$

Combining the fractions, we get:

$$\\\\\cos (u+v) = \\frac{-56}{65}$$

And there you have it! We've found the exact value of $\cos (u+v)$. It's $\frac{-56}{65}$.

Mastering Trigonometric Identities and Problem-Solving Strategies

Isn't it amazing how we can use trigonometric identities and a bit of logical thinking to solve these problems? This example beautifully illustrates the power of the cosine addition formula and the Pythagorean identity. But more importantly, it highlights the importance of understanding the quadrants and their implications on the signs of trigonometric functions. This is a crucial concept in trigonometry, guys, so make sure you've got it down!

This problem wasn't just about plugging numbers into a formula; it was about understanding the relationships between trigonometric functions and using that understanding to navigate the problem. We had to first identify the missing pieces (\cos u and \cos v), then use the Pythagorean identity and our knowledge of quadrants to find those pieces. Finally, we plugged everything into the cosine addition formula to arrive at the solution.

Practice Makes Perfect

The best way to master these concepts is through practice. Try solving similar problems with different values and quadrants. Challenge yourselves to think critically and apply the trigonometric identities in creative ways. You'll be amazed at how quickly you improve! Maybe try changing the values of $\sin u$ and $\sin v$, or placing $u$ and $v$ in different quadrants. See how these changes affect the final answer. This kind of exploration will deepen your understanding and make you a trigonometric wizard in no time!

Also, don't hesitate to revisit the fundamental trigonometric identities. Make sure you understand where they come from and how they relate to each other. The more comfortable you are with these identities, the easier it will be to solve complex problems.

Remember, guys, trigonometry is like a puzzle. Each piece of information is a clue, and the identities are the tools you need to put it all together. With practice and a bit of perseverance, you'll be solving trigonometric puzzles like a pro!

Conclusion: You've Conquered cos(u+v)!

Wow, we did it! We successfully navigated the trigonometric landscape and found the exact value of $\cos (u+v)$. We started with some seemingly limited information, but by using our knowledge of quadrants, the Pythagorean identity, and the cosine addition formula, we conquered the problem. Give yourselves a pat on the back, guys! You've earned it.

This journey wasn't just about finding an answer; it was about learning the process of problem-solving in trigonometry. We learned how to break down a complex problem into smaller, manageable steps. We learned the importance of understanding the underlying concepts, like quadrants and trigonometric identities. And we learned the value of perseverance and practice.

So, the next time you encounter a trigonometric challenge, remember the lessons we learned today. Remember to break down the problem, identify the key information, and use the appropriate tools and identities. And most importantly, remember to have fun! Trigonometry can be challenging, but it's also incredibly rewarding. Keep practicing, keep exploring, and keep unlocking the secrets of the trigonometric world!

I hope this comprehensive guide has been helpful, guys! Keep up the amazing work, and I'll see you in the next trigonometric adventure!