KCSE 2012 Question 10 Solution: Finding Cos(90 - X) Given Tan X

Jul 31, 2025 by ADMIN 64 views

Decoding KCSE 2012 Question 10: A Deep Dive into Trigonometry

Hey guys! Today, we're going to break down a classic trigonometry problem from the KCSE 2012 exam, Question 10 to be exact. This question is a fantastic example of how seemingly simple trigonometric relationships can be used to solve more complex problems. So, grab your calculators (or your mental math muscles!) and let's dive in!

The Question: A Quick Recap

Before we get started, let's quickly remind ourselves of the question. It goes something like this: "Given that tan x = 3/4, find cos (90 - x), giving the answer to 4 significant figures." Sounds a bit intimidating at first, right? But don't worry, we'll tackle it step by step and you'll see it's not as scary as it looks.

Understanding the Basics: Tan x and Right-Angled Triangles

The first key concept we need to understand is what tan x actually represents. Remember your SOH CAH TOA? Tangent (tan) is the ratio of the opposite side to the adjacent side in a right-angled triangle. So, if tan x = 3/4, we can imagine a right-angled triangle where the side opposite to angle x has a length of 3 units, and the side adjacent to angle x has a length of 4 units. This is crucial for visualizing the problem and setting up our solution.

Now, how do we find the hypotenuse of this triangle? That's where our old friend the Pythagorean theorem comes in! The Pythagorean theorem states that in a right-angled 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. In our case, this means: hypotenuse² = 3² + 4² = 9 + 16 = 25. Taking the square root of both sides, we find that the hypotenuse has a length of 5 units. Now we have all three sides of our triangle: opposite = 3, adjacent = 4, and hypotenuse = 5. This is a classic 3-4-5 right-angled triangle, a pattern you'll often see in trigonometry problems.

With this triangle in mind, we can easily determine the values of sine and cosine for angle x. Remember SOH CAH TOA? Sine (sin) is the ratio of the opposite side to the hypotenuse, so sin x = 3/5. Cosine (cos) is the ratio of the adjacent side to the hypotenuse, so cos x = 4/5. Keep these values handy, as they'll be important later in our solution. The understanding of trigonometric ratios within a right-angled triangle is the cornerstone for tackling this question. By visualizing the triangle and applying the Pythagorean theorem, we've laid a solid foundation for the next step. Mastering this basic concept not only helps with this specific problem but also builds a strong base for more advanced trigonometry topics. Remember, practice makes perfect! The more you work with these ratios and triangles, the more comfortable you'll become with them, making problems like this seem much less daunting. This initial step is where many students often get stuck, so taking the time to fully grasp the relationship between trigonometric ratios and right-angled triangles is essential for success in trigonometry. So, let’s move on and see how we can use this knowledge to solve the actual question!

The Core of the Problem: cos (90 - x) and Co-functions

Okay, so we've figured out the sides of our triangle and we know sin x and cos x. But the question asks us to find cos (90 - x). This is where the concept of co-functions comes into play. This is the second crucial piece of the puzzle.

Co-functions are trigonometric functions that are related to each other in a special way. Specifically, the cosine of an angle is equal to the sine of its complement, and vice versa. What's a complement, you ask? Well, two angles are complementary if they add up to 90 degrees. So, the complement of x is (90 - x).

This means that cos (90 - x) = sin x! Isn't that neat? This identity is a game-changer for this problem. Instead of trying to calculate cos (90 - x) directly, we can simply use the value of sin x that we already found. Remember when we calculated sin x from our triangle? We found that sin x = 3/5. So, cos (90 - x) = 3/5 as well! This trigonometric identity, cos (90° - x) = sin x, is incredibly powerful. It allows us to transform one trigonometric expression into another, often simplifying the problem significantly. It’s essential to memorize these co-function relationships, as they appear frequently in trigonometry and calculus. Beyond just memorization, understanding why this identity works is key. It stems from the inherent relationship between the angles and sides in a right-angled triangle. The angle (90° - x) is the other acute angle in our triangle, and the side opposite to it is the same side that is adjacent to angle x. This geometric relationship directly translates into the trigonometric identity. Mastering these co-function relationships is like unlocking a secret weapon in your trigonometry toolkit. It allows you to approach problems from different angles (pun intended!) and find elegant solutions. So, make sure you're comfortable with these identities and how to apply them. Now that we've simplified the problem using this identity, we're just one step away from the final answer. Let's nail it!

Final Answer: Giving it in 4 Significant Figures

We're almost there! We know that cos (90 - x) = sin x = 3/5. Now, the question asks us to give the answer to 4 significant figures. This simply means we need to express 3/5 as a decimal with 4 digits that carry meaning. This is the final step in our journey.

Converting 3/5 to a decimal is pretty straightforward: 3 ÷ 5 = 0.6. But we need 4 significant figures. So, we simply add zeros until we have 4 digits after the first non-zero digit. In this case, 0.6 becomes 0.6000. This is our answer to 4 significant figures!

So, the final answer to the question is cos (90 - x) = 0.6000. Woohoo! We did it! We successfully navigated through the trigonometric ratios, the co-function identity, and the significant figures requirement. This seemingly complex problem boiled down to a few key concepts and a little bit of careful calculation. Expressing the answer to a specific number of significant figures is a crucial skill in mathematics and science. It reflects the precision of our measurements and calculations. Understanding significant figures ensures that we present our answers in a way that accurately represents the level of certainty we have. In this case, simply writing 0.6 wouldn't fully answer the question's requirement. The instruction to provide the answer to 4 significant figures implies a certain level of precision, and adding the zeros ensures we meet that requirement. The concept of significant figures is not just a mathematical formality; it's a way of communicating the accuracy and reliability of our results. In practical applications, such as engineering and physics, using the correct number of significant figures is essential for avoiding errors and ensuring the integrity of calculations. So, make sure you're comfortable with this concept, as it's a fundamental part of scientific and mathematical communication. And that’s a wrap! Let’s celebrate our victory over this KCSE question. You've not only learned how to solve this specific problem but also gained a deeper understanding of trigonometric relationships, co-functions, and significant figures. This knowledge will undoubtedly serve you well in future math challenges.

Key Takeaways and Practice Tips

Let's recap the key takeaways from this problem:

Trigonometric Ratios: Remember SOH CAH TOA! Understanding the relationships between sine, cosine, tangent, and the sides of a right-angled triangle is fundamental.
Pythagorean Theorem: A lifesaver for finding missing sides in right-angled triangles.
Co-functions: cos (90 - x) = sin x (and vice versa!). This identity is a powerful tool for simplifying trigonometric expressions.
Significant Figures: Pay attention to the required level of precision in your answers.

To really master these concepts, practice is key. Try solving similar problems from past KCSE papers or your textbook. The more you work with these concepts, the more comfortable and confident you'll become. Remember, mathematics is like learning a language; the more you use it, the better you'll get! Also, don't be afraid to draw diagrams. Visualizing the problem, like drawing the right-angled triangle in this case, can make it much easier to understand and solve. And if you get stuck, don't give up! Go back to the basics, review the definitions and relationships, and try a different approach. There's always a way to solve it. Most importantly, embrace the challenge and enjoy the process of learning! Mathematics can be a fascinating and rewarding subject, and each problem you solve is a step forward in your journey. So, keep practicing, keep exploring, and keep those math muscles flexing! We’ve covered a lot in this deep dive, so make sure you review the concepts and practice similar questions. The more you work with trigonometry, the easier it becomes. You’ve got this!

Final Thoughts

So, there you have it! We've successfully dissected and conquered KCSE 2012 Question 10. Hopefully, this step-by-step explanation has not only helped you understand the solution but also given you a deeper appreciation for the beauty and power of trigonometry. Remember, math isn't just about memorizing formulas; it's about understanding the underlying concepts and applying them creatively to solve problems. Keep practicing, keep exploring, and you'll be amazed at what you can achieve! You guys are awesome, and I know you'll nail your exams. Keep up the great work, and I'll see you in the next problem-solving session! Remember to keep practicing and never give up on your mathematical journey. Every problem you solve makes you stronger and more confident. So, keep challenging yourself, keep learning, and keep shining! Until next time, happy problem-solving!