Solving Trigonometry: Find Sin¹¹⁹x + 20 Cosec X

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Hey there, math enthusiasts! Today, we're diving into a cool trigonometry problem that's perfect for those prepping for their CBSE Board exams. The problem is: If sin x + cosec x = 2, then find the value of sin¹¹⁹x + 20 cosec x. It might look a bit intimidating at first, but trust me, we'll break it down step by step to make it super clear. Ready to get started? Let's go!

Understanding the Problem

First things first, let's make sure we understand what the problem is asking. We're given an equation involving sine and cosecant functions and we need to find the value of a slightly more complex expression. Remember that cosecant is just the reciprocal of sine (cosec x = 1/sin x). This relationship is key to solving this problem. Our goal is to find the value of sin¹¹⁹x + 20 cosec x. Sounds like a fun challenge, right?

To get started, let's first write down what we know. We are given: sin x + cosec x = 2. From this equation, our aim is to figure out the values of sin x and cosec x. The trick here is to recognize the reciprocal relationship between sin x and cosec x. This relationship is our secret weapon in solving this problem. By using this reciprocal identity, we will make some smart substitutions, perform some simple algebraic manipulations, and finally, we will find the values we need.

Before we proceed, it is important to understand a few core trigonometric concepts: Sine, cosecant, and their reciprocal relationship. Grasping these concepts is very important to understand the solution. Remember, understanding the basic principles is like having a strong foundation for a house. This will allow us to navigate the problem step-by-step, making sure we don't get lost in the mathematical maze. We will try to explore the underlying concepts, and then move on to apply these in a logical way to solve the problem and get the right solution. Remember, in math, every step has a reason, and understanding that reason is the key to success!

Solving Step by Step

Alright, let's roll up our sleeves and start solving this problem. The key is to start with the equation sin x + cosec x = 2. Since we know that cosec x = 1/sin x, we can substitute that into our equation. So now it looks like this: sin x + 1/sin x = 2. Now, to make things a bit easier, let's multiply the entire equation by sin x. This will help us get rid of the fraction and make it a bit more manageable. Doing this gives us: sin²x + 1 = 2sin x. Notice how we're slowly but surely simplifying the equation? That's the goal!

Now, let's rearrange the terms to form a quadratic equation. We'll move everything to one side to get: sin²x - 2sin x + 1 = 0. This is a quadratic equation, and it's in a form that we can easily factor. The left side of the equation is actually a perfect square trinomial! It factors into (sin x - 1)² = 0. From this, we can see that sin x - 1 = 0, which means sin x = 1. We have successfully figured out the value of sin x!

Now that we have the value of sin x, let's find cosec x. Since cosec x = 1/sin x, and we know sin x = 1, then cosec x = 1/1 = 1. Awesome, we've found both sin x and cosec x. Now, to the grand finale! We're going to substitute these values into the expression sin¹¹⁹x + 20 cosec x. So, we have 1¹¹⁹ + 20 * 1. And, 1¹¹⁹ is just 1, so we have 1 + 20 * 1 = 1 + 20 = 21. So, the value of sin¹¹⁹x + 20 cosec x is 21. And there you have it, we've solved the problem. Wasn't that fun?

Key Takeaways and Tips

So, what can we learn from this problem? Well, the key is recognizing the reciprocal relationship between sine and cosecant. This is a common trick in trigonometry, so always be on the lookout for it! Simplifying the equation by getting rid of fractions and rearranging it into a recognizable form (like a quadratic equation) is another important skill. And, of course, the ability to factor and solve quadratic equations comes in handy as well.

Here are a few tips to keep in mind when tackling similar problems:

  • Always look for reciprocal relationships. These are often the key to simplifying the problem.
  • Try to rewrite the equation to make it simpler. This often involves getting rid of fractions or rearranging terms.
  • Familiarize yourself with common algebraic techniques, such as factoring and solving quadratic equations.
  • Practice, practice, practice! The more problems you solve, the better you'll get at recognizing patterns and applying the right techniques.

Remember, trigonometry can be super rewarding. With a bit of practice, you will become a trigonometry expert. Keep practicing, and you will become more confident in your skills. Don’t be afraid to ask for help or go back to the basics if you feel stuck. Good luck with your CBSE Board exams and all future math adventures!

Conclusion

So, guys, we've successfully solved the trigonometry problem! We started with sin x + cosec x = 2, and after a few steps, we found the value of sin¹¹⁹x + 20 cosec x to be 21. We used the reciprocal relationship between sine and cosecant, simplified the equation, and applied some basic algebraic techniques. Remember, math is all about breaking down complex problems into smaller, manageable steps. And with practice, you can solve any problem thrown your way. Keep learning, keep practicing, and never give up. This is just one example of the many exciting things you can do with trigonometry. Keep exploring and keep having fun with math! Remember, with the right approach and a little bit of effort, anything is possible. Now go forth and conquer those trigonometry problems! You've got this!