Unlocking Trig Secrets: Solving For Cotangent And Sine

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Hey math enthusiasts! Today, we're diving into the fascinating world of trigonometry. We're going to tackle a problem where we're given some key information about an angle, θ\theta, and then we'll use that info to find the exact values of some other trig functions. Specifically, we know that cos⁡θ=23\cos \theta = \frac{2}{3} and tan⁡θ>0\tan \theta > 0. Our mission, should we choose to accept it, is to find the exact values of cot⁡θ\cot \theta and sin⁡θ\sin \theta. Sounds like fun, right? Let's break it down step by step, making sure we understand everything clearly. This is a classic example of how to use the relationships between trigonometric functions and how to determine the quadrant of an angle.

Understanding the Given Information: A Solid Foundation

Alright, guys, before we jump into calculations, let's make sure we're on the same page about what we already know. We've got two main pieces of intel: the cosine of the angle and the sign of the tangent. Remember, in trigonometry, these pieces of information are super valuable, like the secret ingredients to a delicious mathematical recipe. Knowing cos⁡θ=23\cos \theta = \frac{2}{3} tells us the ratio of the adjacent side to the hypotenuse in a right triangle. However, it doesn't give us the complete picture of the angle θ\theta. Cosine being positive means the angle is either in the first or fourth quadrant, which is crucial information. Also, tan⁡θ>0\tan \theta > 0 tells us that the tangent of the angle is positive. The tangent function is positive in the first and third quadrants. To find the exact angle, we need to consider both pieces of information together. Since the cosine is positive and the tangent is positive, we know that the angle θ\theta must lie in the first quadrant, where both cosine and tangent are positive. This is the first, and arguably the most important, step to solving this problem. Identifying the correct quadrant helps to determine the signs of the other trigonometric functions we need to calculate.

Let's not forget the basics. The cosine function is defined as the ratio of the adjacent side to the hypotenuse in a right triangle. If we imagine a right triangle where the hypotenuse has a length of 3 and the adjacent side has a length of 2 (in relation to the angle θ\theta), we can use the Pythagorean theorem to find the length of the opposite side. This is essential for finding the sine and cotangent values because they are dependent on the lengths of all three sides of the right triangle. Remember that in a right triangle, the Pythagorean theorem states that a2+b2=c2a^2 + b^2 = c^2, where aa and bb are the lengths of the legs, and cc is the length of the hypotenuse. We'll use this theorem to discover the missing side of our triangle, which will unlock the solutions we're seeking. Knowing the quadrant and the relationships between sides in a right triangle is the cornerstone for solving this kind of trig problem. So, let’s get into the specifics of how to find the values of cotangent and sine using the information we have!

Finding sin⁡θ\sin \theta: The Opposite Side of the Story

Alright, time to find sin⁡θ\sin \theta. We know that cos⁡θ=23\cos \theta = \frac{2}{3} and that the angle is in the first quadrant. Since cos⁡θ\cos \theta is adjacent over hypotenuse, let's picture a right triangle with an adjacent side of 2 and a hypotenuse of 3. We can use the Pythagorean theorem to find the length of the opposite side. Let's call the opposite side oo. So, we have:

22+o2=322^2 + o^2 = 3^2 4+o2=94 + o^2 = 9 o2=5o^2 = 5 o=5o = \sqrt{5}

Since we're in the first quadrant, the opposite side is positive. Now we can find sin⁡θ\sin \theta. Remember that sin⁡θ\sin \theta is opposite over hypotenuse. Therefore:

sin⁡θ=oppositehypotenuse=53\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{5}}{3}

And there you have it, folks! We've successfully calculated the exact value of sin⁡θ\sin \theta. This value is positive because, as we identified earlier, the angle lies in the first quadrant. This positive sign is essential, reflecting the correct direction and orientation of the sine value within the coordinate plane. Remember, paying close attention to the quadrant helps to avoid any errors in determining the signs of the trigonometric functions. This part of the problem shows how the Pythagorean theorem and understanding the unit circle work together to solve trigonometric problems. Also, remember that the sine of an angle gives you the ratio of the opposite side to the hypotenuse, and it is a fundamental function in the field of trigonometry.

Calculating cot⁡θ\cot \theta: The Reciprocal Journey

Now, let's find cot⁡θ\cot \theta. The cotangent function is the reciprocal of the tangent function. That is, cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}. Also, we know that tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}. We already know sin⁡θ=53\sin \theta = \frac{\sqrt{5}}{3} and cos⁡θ=23\cos \theta = \frac{2}{3}. Therefore:

tan⁡θ=sin⁡θcos⁡θ=5323=52\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{2}

Now, since cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}, we have:

cot⁡θ=152=25\cot \theta = \frac{1}{\frac{\sqrt{5}}{2}} = \frac{2}{\sqrt{5}}

To rationalize the denominator, we multiply the numerator and the denominator by 5\sqrt{5}:

cot⁡θ=25⋅55=255\cot \theta = \frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{2\sqrt{5}}{5}

There you go, guys! We've found the exact value of cot⁡θ\cot \theta, using the relationship between cotangent and tangent and the values we already calculated for sine and cosine. This process highlights how all the trigonometric functions are interconnected. Cotangent, like tangent, is positive in the first quadrant, which aligns with our previous analysis of the given information. Knowing this helps to confirm that our solution is correct. Also, rationalizing the denominator is a standard practice in mathematics to simplify the expression and eliminate radicals from the denominator, making it easier to work with. Remember that the cotangent is the ratio of the adjacent side to the opposite side in a right triangle, which allows you to understand the relationship between different sides of the triangle.

Summarizing the Solutions: The Grand Finale

Okay, let's recap everything we've discovered today. Given that cos⁡θ=23\cos \theta = \frac{2}{3} and tan⁡θ>0\tan \theta > 0, we determined the exact values:

  • sin⁡θ=53\sin \theta = \frac{\sqrt{5}}{3}
  • cot⁡θ=255\cot \theta = \frac{2\sqrt{5}}{5}

Congratulations! We've successfully solved this trigonometric problem. We began by understanding the given information, then we used the Pythagorean theorem and relationships between trigonometric functions to find our answers. Understanding how to use the information given, like the signs of the trig functions and the quadrant of the angle, is the key to successfully solving trig problems. Remember that trigonometry is all about the relationships between angles and sides in triangles. Every function has a relationship with the other and with the triangle's sides. Keep practicing, and you'll become a trigonometry whiz in no time. Always remember to consider the signs of the trig functions in different quadrants to ensure your answers are accurate. Also, it is good practice to revisit all the relationships between trigonometric functions to solve problems with ease and confidence. Don’t be afraid to draw diagrams to visualize the problem; it helps a lot. Keep exploring the world of trigonometry, and happy calculating, everyone!

This entire journey, from understanding the given information to calculating the final answers, highlights the interconnectedness of trigonometric concepts. The ability to move seamlessly between different functions, like cosine, sine, tangent, and cotangent, is a crucial skill. The Pythagorean theorem serves as a backbone, allowing you to bridge the gap between known values and unknown ones. Remember, understanding the signs of trigonometric functions in different quadrants is just as important as knowing the formulas. Finally, the ability to rationalize the denominator is often necessary to present your answers in the simplest, most elegant form. Keep practicing, and you'll find that trigonometry becomes more intuitive with each problem you solve. Happy calculating, and keep exploring the fascinating world of mathematics! It is important to remember the fundamentals and keep reviewing the core concepts of trigonometry. With consistent practice and understanding, you can excel in trigonometry, and it will be a piece of cake for you.