Finding Trig Functions When Cot(x) = 5/8: A Step-by-Step Guide

Have you ever been given the cotangent of an angle and needed to find the other trigonometric functions? It might seem daunting at first, but with a step-by-step approach and a bit of trigonometric know-how, it's totally achievable! In this article, we'll break down how to find the values of all trigonometric functions when given that cot(x) = 5/8 and 0 < x < π. So, let's dive in and make trigonometry a little less mysterious!

Understanding the Problem: Cotangent and the Unit Circle

When dealing with trigonometric functions, it's essential to understand what they represent. Our main keyword here is cotangent, so let’s break it down. The cotangent (cot) of an angle x is defined as the ratio of the adjacent side to the opposite side in a right triangle, or equivalently, as 1/tan(x) or cos(x)/sin(x). Knowing this definition is our starting point. We are given that cot(x) = 5/8, which means that for an angle x, the ratio of the adjacent side to the opposite side is 5/8. This ratio is crucial in determining the other trigonometric functions. The constraint 0 < x < π tells us that x lies in the first or second quadrant of the unit circle. This information is vital because it helps us determine the signs of the trigonometric functions. In the first quadrant, all trigonometric functions are positive. In the second quadrant, sine (sin) and its reciprocal, cosecant (csc), are positive, while cosine (cos), tangent (tan), secant (sec), and cotangent (cot) are negative. Understanding these fundamental concepts is key to successfully solving the problem. We’ll use the given information to build a right triangle and then use the Pythagorean theorem to find the missing side. This will allow us to determine the values of sine, cosine, tangent, and their reciprocals. By considering the quadrant in which x lies, we can accurately assign the correct signs to each trigonometric function. Let's delve deeper into the steps to solve this problem and gain a clear understanding of how to approach similar trigonometric questions.

Step 1: Visualizing the Triangle and Finding the Sides

To tackle this problem, let's start by visualizing a right triangle. Since we know that cot(x) = 5/8, we can think of this as the adjacent side being 5 and the opposite side being 8. Remember, cot(x) is the ratio of the adjacent side to the opposite side. This is a crucial piece of the puzzle. Now, let's sketch a right triangle where the side adjacent to angle x is 5 units long, and the side opposite to angle x is 8 units long. This visualization helps bring the abstract trigonometric concepts into a more tangible form. With these two sides, we can find the hypotenuse using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, this is represented as a² + b² = c², where a and b are the lengths of the legs (the adjacent and opposite sides), and c is the length of the hypotenuse. In our case, a = 5 and b = 8. Plugging these values into the Pythagorean theorem, we get: 5² + 8² = c² which simplifies to 25 + 64 = c². Therefore, c² = 89. To find the length of the hypotenuse c, we take the square root of both sides: c = √89. So, the hypotenuse of our triangle is √89. Now that we have all three sides of the triangle – the adjacent side (5), the opposite side (8), and the hypotenuse (√89) – we are well-equipped to find the remaining trigonometric functions. The next step involves using these values to calculate sine, cosine, tangent, and their reciprocals. Let's move on to the next phase of our trigonometric journey!

Step 2: Calculating Sine and Cosine

Now that we have the lengths of all three sides of our right triangle, we can move on to calculating the sine and cosine of angle x. These are fundamental trigonometric functions, and understanding how to find them is crucial. Let’s start with sine. By definition, the sine (sin) of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. In our case, the opposite side is 8, and the hypotenuse is √89. Therefore, sin(x) = 8 / √89. However, it's standard practice to rationalize the denominator, which means we want to eliminate the square root from the denominator. To do this, we multiply both the numerator and the denominator by √89: sin(x) = (8 * √89) / (√89 * √89) = (8√89) / 89. So, we have found that sin(x) = (8√89) / 89. Next, let's calculate the cosine. The cosine (cos) of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. In our triangle, the adjacent side is 5, and the hypotenuse is √89. Thus, cos(x) = 5 / √89. Again, we need to rationalize the denominator. Multiplying both the numerator and the denominator by √89, we get: cos(x) = (5 * √89) / (√89 * √89) = (5√89) / 89. Therefore, cos(x) = (5√89) / 89. We've now successfully calculated the sine and cosine of angle x. These two values are pivotal because they allow us to find the other trigonometric functions, such as tangent, cosecant, secant, and, of course, we already know the cotangent. Let's proceed to the next step and find the remaining trigonometric functions using the values we've just calculated.

Step 3: Determining Tangent and its Reciprocal Functions

With sine and cosine calculated, we're in a great position to find the remaining trigonometric functions. Let's start with the tangent (tan). The tangent of an angle is defined as the ratio of the sine to the cosine, or equivalently, as the ratio of the opposite side to the adjacent side. We already know that cot(x) = 5/8, and since tangent is the reciprocal of cotangent, we can easily find tan(x). The tangent (tan) of an angle x is defined as 1 / cot(x). Therefore, tan(x) = 1 / (5/8) = 8/5. Alternatively, we can calculate tangent using the sine and cosine values we found earlier. tan(x) = sin(x) / cos(x) = [(8√89) / 89] / [(5√89) / 89]. Simplifying this expression, we get: tan(x) = (8√89) / 89 * 89 / (5√89) = 8/5, which confirms our previous calculation. Now, let's move on to the reciprocal functions: cosecant (csc) and secant (sec). The cosecant (csc) is the reciprocal of sine. Thus, csc(x) = 1 / sin(x). We found that sin(x) = (8√89) / 89, so csc(x) = 1 / [(8√89) / 89] = 89 / (8√89). Rationalizing the denominator, we get: csc(x) = (89 * √89) / (8√89 * √89) = (89√89) / (8 * 89) = √89 / 8. So, csc(x) = √89 / 8. Next, the secant (sec) is the reciprocal of cosine. Hence, sec(x) = 1 / cos(x). We calculated cos(x) = (5√89) / 89, so sec(x) = 1 / [(5√89) / 89] = 89 / (5√89). Rationalizing the denominator, we have: sec(x) = (89 * √89) / (5√89 * √89) = (89√89) / (5 * 89) = √89 / 5. Therefore, sec(x) = √89 / 5. We have now found all six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. To summarize, we have sin(x) = (8√89) / 89, cos(x) = (5√89) / 89, tan(x) = 8/5, csc(x) = √89 / 8, sec(x) = √89 / 5, and cot(x) = 5/8. This comprehensive calculation completes our problem-solving journey. Let's recap our steps and understand the big picture.

Step 4: Considering the Quadrant and Finalizing the Signs

Before we declare victory, there's one crucial step we need to consider: the quadrant in which angle x lies. We were given that 0 < x < π, which means that x could be in the first quadrant (0 < x < π/2) or the second quadrant (π/2 < x < π). This is a critical piece of information because it affects the signs of the trigonometric functions. Let's break it down. In the first quadrant, all trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) are positive. This is because both the x and y coordinates are positive in the first quadrant. In the second quadrant, only sine (and its reciprocal, cosecant) are positive. Cosine, tangent, and their reciprocals (secant and cotangent) are negative in the second quadrant. This is because the x-coordinate is negative, while the y-coordinate is positive in the second quadrant. Now, let's look at our given information: cot(x) = 5/8. Since cotangent is positive, this tells us that x must be in the first quadrant. If x were in the second quadrant, cotangent would be negative. Therefore, all the trigonometric functions we calculated should be positive. Our calculated values are: sin(x) = (8√89) / 89, cos(x) = (5√89) / 89, tan(x) = 8/5, csc(x) = √89 / 8, sec(x) = √89 / 5, and cot(x) = 5/8. Since all these values are positive, and we've confirmed that x is in the first quadrant, our calculations are correct. If we had found any negative values, we would need to adjust them based on the quadrant. For example, if the angle were in the second quadrant, we would need to make cosine, tangent, and secant negative while keeping sine and cosecant positive. This final check ensures that our solution is accurate and complete. Understanding how the quadrant affects the signs of trigonometric functions is a fundamental aspect of trigonometry. It’s a crucial step in solving these types of problems. Let's wrap up with a summary of our findings and some key takeaways.

Conclusion: Summing Up and Key Takeaways

We've successfully navigated through the problem of finding the trigonometric functions given cot(x) = 5/8 and 0 < x < π. Let's recap the journey we took. First, we understood the definitions of trigonometric functions, particularly cotangent, and visualized the problem using a right triangle. We used the given cotangent value to determine the adjacent and opposite sides of the triangle. Then, we employed the Pythagorean theorem to find the length of the hypotenuse. With all three sides known, we calculated the sine and cosine of the angle. From sine and cosine, we easily found tangent and its reciprocal functions: cosecant and secant. Finally, and crucially, we considered the quadrant in which the angle x lies, ensuring that our trigonometric functions had the correct signs. We concluded that all functions were positive since x was in the first quadrant.

The final values we found are:

• sin(x) = (8√89) / 89
• cos(x) = (5√89) / 89
• tan(x) = 8/5
• csc(x) = √89 / 8
• sec(x) = √89 / 5
• cot(x) = 5/8

This step-by-step approach can be applied to similar problems involving other trigonometric functions and different angle ranges. The key takeaway here is that understanding the definitions of trigonometric functions, visualizing triangles, using the Pythagorean theorem, and considering the quadrant are essential tools in solving these problems. Trigonometry can seem complex, but by breaking it down into manageable steps, it becomes much more approachable. Practice is key to mastering these concepts, so try tackling similar problems to solidify your understanding. Remember, each trigonometric function is interconnected, and knowing the value of one can lead you to find the others. Keep exploring and practicing, and you'll become more confident in your trigonometric abilities.

For further learning and practice, you might find resources like Khan Academy's Trigonometry section helpful.