Trigonometry Challenge: Solving For Tan(A+B) And Cot(A+B)

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Hey math enthusiasts! Today, we're diving into a cool trigonometry problem. We'll be working with angles A and B, figuring out their tangents and cotangents. Specifically, we're given some interesting info:

  • sin⁡A=213\sin A = \frac{2}{\sqrt{13}}, where angle A is in the second quadrant (QII).
  • tan⁡B=−158\tan B = -\frac{15}{8}, and angle B is also in the second quadrant (QII).

Our mission, should we choose to accept it, is to find:\

a. tan⁡(A+B)\tan (A + B) \

b. cot⁡(A+B)\cot (A + B)

So, grab your calculators (or your brains, if you're feeling extra sharp) and let's get started! This problem is a fantastic way to flex your trigonometry muscles, putting your knowledge of trigonometric identities and quadrant rules to the test. Ready? Let's go!

Unveiling tan(A + B): A Step-by-Step Guide

Alright, folks, let's tackle the first part of our challenge: finding tan⁡(A+B)\tan(A + B). To do this, we'll need to use the tangent addition formula. Do you remember it? Here it is:

tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡A⋅tan⁡B\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}

See? Not so scary, right? Now, let's break down how to use this formula with the information we have. We already know tan⁡B\tan B, but we need to figure out tan⁡A\tan A. Here's how we'll do it:

  1. Finding cos A: Since we know sin⁡A\sin A, we can use the Pythagorean identity: sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1. Let's plug in our value for sin⁡A\sin A:

    (213)2+cos⁡2A=1\left(\frac{2}{\sqrt{13}}\right)^2 + \cos^2 A = 1
    413+cos⁡2A=1\frac{4}{13} + \cos^2 A = 1
    cos⁡2A=913\cos^2 A = \frac{9}{13}

    Now, take the square root of both sides:

    cos⁡A=±313\cos A = \pm \frac{3}{\sqrt{13}}

    But wait! Angle A is in the second quadrant (QII). In QII, cosine is negative. So, we choose the negative value:

    cos⁡A=−313\cos A = -\frac{3}{\sqrt{13}}

  2. Finding tan A: Now that we know sin⁡A\sin A and cos⁡A\cos A, we can find tan⁡A\tan A using the formula: tan⁡A=sin⁡Acos⁡A\tan A = \frac{\sin A}{\cos A}.

    tan⁡A=213−313\tan A = \frac{\frac{2}{\sqrt{13}}}{-\frac{3}{\sqrt{13}}}
    tan⁡A=−23\tan A = -\frac{2}{3}

    Great! We've found tan⁡A\tan A.

  3. Applying the Tangent Addition Formula: Now, let's plug our values for tan⁡A\tan A and tan⁡B\tan B into the tangent addition formula:

    tan⁡(A+B)=−23+(−158)1−(−23)⋅(−158)\tan(A + B) = \frac{-\frac{2}{3} + (-\frac{15}{8})}{1 - (-\frac{2}{3}) \cdot (-\frac{15}{8})}
    tan⁡(A+B)=−23−1581−3024\tan(A + B) = \frac{-\frac{2}{3} - \frac{15}{8}}{1 - \frac{30}{24}}
    tan⁡(A+B)=−1624−45241−54\tan(A + B) = \frac{-\frac{16}{24} - \frac{45}{24}}{1 - \frac{5}{4}}
    tan⁡(A+B)=−6124−14\tan(A + B) = \frac{-\frac{61}{24}}{-\frac{1}{4}}
    tan⁡(A+B)=616\tan(A + B) = \frac{61}{6}

    And there you have it, folks! The value of tan⁡(A+B)\tan(A + B) is 616\frac{61}{6}. Awesome, right? We've successfully navigated the first part of our mission. This journey highlights the importance of understanding the fundamental trigonometric identities and how to apply them, along with paying close attention to the quadrant of an angle to determine the correct signs of trigonometric functions. It's all about piecing together the information like a puzzle, and voila – the solution appears!

Calculating cot(A + B): The Reciprocal Route

Now, let's move on to the second part of our challenge: finding cot⁡(A+B)\cot(A + B). Luckily for us, this is a breeze! Remember that the cotangent is simply the reciprocal of the tangent. In other words:

cot⁡(A+B)=1tan⁡(A+B)\cot(A + B) = \frac{1}{\tan(A + B)}

We already know tan⁡(A+B)\tan(A + B) from our previous calculation, which is 616\frac{61}{6}. So, let's find the reciprocal:

cot⁡(A+B)=1616\cot(A + B) = \frac{1}{\frac{61}{6}}
cot⁡(A+B)=661\cot(A + B) = \frac{6}{61}

And there you have it! The value of cot⁡(A+B)\cot(A + B) is 661\frac{6}{61}. See? That was super easy, wasn't it? This part of the problem reinforces the fundamental relationship between tangent and cotangent, reminding us that with a little understanding of trigonometric functions, complex-looking problems can often be broken down into simpler, manageable steps. Congratulations on solving this challenging trigonometry problem! You've shown that with a clear understanding of trigonometric principles, you can conquer even the most daunting equations. Keep up the excellent work! Now, go forth and spread your newfound trigonometric wisdom!

Delving Deeper: Key Trigonometric Concepts

Let's take a moment to recap some of the essential concepts we used in this problem. Understanding these concepts is key to acing trigonometry:

  • Trigonometric Identities: These are equations that are true for all values of the variables. We used the Pythagorean identity (sin⁡2A+cos⁡2A=1\sin^2 A + \cos^2 A = 1) to find cos⁡A\cos A when we knew sin⁡A\sin A. Familiarizing yourself with these identities is crucial.
  • Quadrant Rules: The unit circle is divided into four quadrants, and each quadrant has specific rules for the signs of sine, cosine, and tangent. For example, in QII, sine is positive, cosine is negative, and tangent is negative. Remembering these rules is vital for determining the correct signs of your answers.
  • Tangent Addition Formula: This formula allows you to find the tangent of the sum of two angles. This is a fundamental formula in trigonometry, and you must know it.
  • Reciprocal Identities: Knowing the reciprocal relationships (cotangent is the reciprocal of tangent, secant is the reciprocal of cosine, and cosecant is the reciprocal of sine) makes solving problems much easier.

Mastering these concepts will provide a solid foundation for tackling more complex trigonometry problems. Always remember to break down the problems into smaller, manageable steps, and don't be afraid to use the formulas and identities you've learned. The more you practice, the more comfortable and confident you'll become!

Practice Makes Perfect: Additional Exercises

Want to sharpen your trigonometry skills even further? Here are a few practice problems to test your knowledge:

  1. Given cos⁡A=12\cos A = \frac{1}{2} with A in QI and sin⁡B=12\sin B = \frac{1}{\sqrt{2}} with B in QI, find tan⁡(A−B)\tan(A - B).
  2. If tan⁡A=2\tan A = 2 and tan⁡B=3\tan B = 3, find tan⁡(A+B)\tan(A + B) and cot⁡(A+B)\cot(A + B).
  3. Given sin⁡A=35\sin A = \frac{3}{5} and cos⁡B=−513\cos B = -\frac{5}{13}, with A in QII and B in QIII, find sin⁡(A+B)\sin(A + B).

These exercises will help you reinforce the concepts we've discussed and boost your confidence in solving trigonometry problems. Remember to work through the problems step-by-step, using the formulas and identities we covered. Good luck, and happy solving!

Conclusion: Mastering the Trigonometry Challenge

Alright, folks, we've reached the finish line! We've successfully calculated tan⁡(A+B)\tan(A + B) and cot⁡(A+B)\cot(A + B) using our knowledge of trigonometric identities, quadrant rules, and the tangent addition formula. We also explored the key concepts and provided some additional exercises to further your understanding.

Remember, trigonometry is all about understanding the relationships between angles and sides of triangles. By mastering the fundamental concepts and practicing regularly, you can confidently tackle any trigonometry challenge that comes your way. So, keep exploring, keep practicing, and keep having fun with math! And always remember, the more you practice, the better you'll become. So, go out there, apply these strategies, and continue to shine in the world of mathematics! Until next time, happy calculating, and keep those math muscles strong!