Evaluating Trigonometric Expressions A Step By Step Guide

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Hey guys! Today, we're diving into a fun math problem that involves evaluating a trigonometric expression. Don't worry, it's not as scary as it sounds! We'll break it down step by step, so you can follow along easily. Let's get started!

Understanding the Problem

Before we jump into the solution, let's take a closer look at the expression we need to evaluate:

tan⁑260∘+4sin⁑245∘+3sec⁑230∘+5cos⁑290∘cosec⁑30∘+sec⁑60βˆ˜βˆ’cot⁑230∘\frac{\tan ^2 60^{\circ}+4 \sin ^2 45^{\circ}+3 \sec ^2 30^{\circ}+5 \cos ^2 90^{\circ}}{\operatorname{cosec} 30^{\circ}+\sec 60^{\circ}-\cot ^2 30^{\circ}}

This might seem intimidating at first glance, but it's just a combination of different trigonometric functions at specific angles. Our goal is to find the values of these functions and then plug them into the expression to get our final answer. Remember those special trigonometric angles and their values? They're going to be our best friends here! We'll use the known values of trigonometric functions for standard angles like 30Β°, 45Β°, 60Β°, and 90Β° to simplify this expression. This involves recalling the values of sine, cosine, tangent, cosecant, secant, and cotangent for these angles. For instance, we know that sin⁑45∘=12\sin 45^{\circ} = \frac{1}{\sqrt{2}}, tan⁑60∘=3\tan 60^{\circ} = \sqrt{3}, and so on. By substituting these known values, we transform the expression into a numerical one, which can then be simplified using basic arithmetic operations. This methodical approach helps in accurately evaluating complex trigonometric expressions. So, let’s equip ourselves with the necessary trigonometric values and embark on this mathematical journey!

Key Trigonometric Values

To solve this, we need to know the values of trigonometric functions for some common angles. Let's quickly recap them:

  • sin⁑45∘=12\sin 45^{\circ} = \frac{1}{\sqrt{2}}
  • cos⁑45∘=12\cos 45^{\circ} = \frac{1}{\sqrt{2}}
  • tan⁑45∘=1\tan 45^{\circ} = 1
  • sin⁑30∘=12\sin 30^{\circ} = \frac{1}{2}
  • cos⁑30∘=32\cos 30^{\circ} = \frac{\sqrt{3}}{2}
  • tan⁑30∘=13\tan 30^{\circ} = \frac{1}{\sqrt{3}}
  • sin⁑60∘=32\sin 60^{\circ} = \frac{\sqrt{3}}{2}
  • cos⁑60∘=12\cos 60^{\circ} = \frac{1}{2}
  • tan⁑60∘=3\tan 60^{\circ} = \sqrt{3}
  • sin⁑90∘=1\sin 90^{\circ} = 1
  • cos⁑90∘=0\cos 90^{\circ} = 0
  • tan⁑90∘=undefined\tan 90^{\circ} = \text{undefined}

And remember the reciprocal trigonometric functions:

  • cosec⁑θ=1sin⁑θ\operatorname{cosec} \theta = \frac{1}{\sin \theta}
  • sec⁑θ=1cos⁑θ\sec \theta = \frac{1}{\cos \theta}
  • cot⁑θ=1tan⁑θ\cot \theta = \frac{1}{\tan \theta}

These key trigonometric values are the building blocks for solving our problem. We need to have these values at our fingertips, or at least know how to quickly derive them using the unit circle or special triangles (30-60-90 and 45-45-90). Mastering these values will make simplifying the expression much easier. Remember, practice makes perfect! Try memorizing these, or create a small table you can refer to. The more familiar you are with these values, the faster and more accurately you'll be able to solve trigonometric problems. It's like knowing your multiplication tables – it just makes everything else flow more smoothly. So, let's make sure we've got these values down pat before we move on to the next step. Think of them as your secret weapon for conquering trigonometric expressions! Ready to put them to use? Let's do it!

Step-by-Step Solution

Now that we have our trigonometric values ready, let's solve the expression step by step.

  1. Evaluate the numerator:

    • tan⁑260∘=(3)2=3\tan ^2 60^{\circ} = (\sqrt{3})^2 = 3
    • sin⁑245∘=(12)2=12\sin ^2 45^{\circ} = (\frac{1}{\sqrt{2}})^2 = \frac{1}{2}
    • sec⁑230∘=(1cos⁑30∘)2=(132)2=(23)2=43\sec ^2 30^{\circ} = (\frac{1}{\cos 30^{\circ}})^2 = (\frac{1}{\frac{\sqrt{3}}{2}})^2 = (\frac{2}{\sqrt{3}})^2 = \frac{4}{3}
    • cos⁑290∘=02=0\cos ^2 90^{\circ} = 0^2 = 0

    So, the numerator becomes:

    3+4(12)+3(43)+5(0)=3+2+4+0=93 + 4(\frac{1}{2}) + 3(\frac{4}{3}) + 5(0) = 3 + 2 + 4 + 0 = 9

    Let's break this down even further, guys. First, we tackled tan⁑260∘\tan ^2 60^{\circ}. We know tan⁑60∘\tan 60^{\circ} is 3\sqrt{3}, so squaring it gives us 3. Easy peasy! Next, we looked at sin⁑245∘\sin ^2 45^{\circ}. sin⁑45∘\sin 45^{\circ} is 12\frac{1}{\sqrt{2}}, and squaring that gives us 12\frac{1}{2}. Then, we had 4sin⁑245∘4 \sin ^2 45^{\circ}, which is 4 times 12\frac{1}{2}, resulting in 2. Moving on to sec⁑230∘\sec ^2 30^{\circ}, remember that secant is the reciprocal of cosine. cos⁑30∘\cos 30^{\circ} is 32\frac{\sqrt{3}}{2}, so sec⁑30∘\sec 30^{\circ} is 23\frac{2}{\sqrt{3}}. Squaring that gives us 43\frac{4}{3}. We then multiplied this by 3, giving us 4. Finally, cos⁑290∘\cos ^2 90^{\circ} is simply 020^2, which is 0. Adding all these values together (3 + 2 + 4 + 0), we get a grand total of 9 for the numerator. See? When we break it down piece by piece, it's much less daunting! This methodical approach ensures we don't make any silly mistakes and keeps everything clear and organized. Now, let's conquer the denominator!

  2. Evaluate the denominator:

    • cosec⁑30∘=1sin⁑30∘=112=2\operatorname{cosec} 30^{\circ} = \frac{1}{\sin 30^{\circ}} = \frac{1}{\frac{1}{2}} = 2
    • sec⁑60∘=1cos⁑60∘=112=2\sec 60^{\circ} = \frac{1}{\cos 60^{\circ}} = \frac{1}{\frac{1}{2}} = 2
    • cot⁑230∘=(1tan⁑30∘)2=(113)2=(3)2=3\cot ^2 30^{\circ} = (\frac{1}{\tan 30^{\circ}})^2 = (\frac{1}{\frac{1}{\sqrt{3}}})^2 = (\sqrt{3})^2 = 3

    So, the denominator becomes:

    2+2βˆ’3=12 + 2 - 3 = 1

    Alright, let's decode the denominator now! We have cosec⁑30∘\operatorname{cosec} 30^{\circ}, which is the reciprocal of sin⁑30∘\sin 30^{\circ}. Since sin⁑30∘\sin 30^{\circ} is 12\frac{1}{2}, cosec⁑30∘\operatorname{cosec} 30^{\circ} becomes 112\frac{1}{\frac{1}{2}}, which simplifies to 2. Next up is sec⁑60∘\sec 60^{\circ}, which is the reciprocal of cos⁑60∘\cos 60^{\circ}. cos⁑60∘\cos 60^{\circ} is 12\frac{1}{2}, so sec⁑60∘\sec 60^{\circ} is also 2. Lastly, we have cot⁑230∘\cot ^2 30^{\circ}. Cotangent is the reciprocal of tangent, and tan⁑30∘\tan 30^{\circ} is 13\frac{1}{\sqrt{3}}. Therefore, cot⁑30∘\cot 30^{\circ} is 3\sqrt{3}, and squaring it gives us 3. Putting it all together, we have 2 + 2 - 3, which equals 1. We've successfully navigated the denominator! Remember, the key here is understanding the reciprocal relationships between trigonometric functions. Cosecant is the flip of sine, secant is the flip of cosine, and cotangent is the flip of tangent. Knowing these relationships makes finding these values a breeze. Now that we've conquered both the numerator and the denominator, we're in the home stretch! Let's put it all together and get our final answer.

  3. Final Evaluation:

    Now, we divide the numerator by the denominator:

    91=9\frac{9}{1} = 9

    So, the final answer is 9. Fantastic job, guys! We've successfully evaluated the trigonometric expression. Remember, the secret to tackling these problems is to break them down into smaller, manageable steps. First, identify the trigonometric functions and their corresponding angles. Then, recall or derive the values of these functions for the given angles. Next, substitute these values into the expression and simplify using basic arithmetic. Finally, double-check your work to ensure accuracy. This step-by-step approach will help you avoid errors and build confidence in your trigonometric skills. And remember, practice makes perfect! The more you work through these types of problems, the more comfortable and proficient you'll become. So, keep practicing, keep exploring, and keep having fun with math! You've got this!

Practice Makes Perfect

To solidify your understanding, try solving similar problems. You can change the angles or the trigonometric functions to create new challenges. Remember to always break down the problem into smaller steps and use the key trigonometric values we discussed. Also, don't be afraid to use online resources or textbooks for additional practice problems and explanations. There are tons of great resources out there to help you master trigonometry. The more you practice, the more confident you'll become in your abilities. And remember, even if you get stuck, don't give up! Take a break, review the concepts, and try again. Learning math is a journey, and every problem you solve is a step forward. So, keep practicing, keep learning, and most importantly, keep having fun! Trigonometry can be a fascinating and rewarding subject, and with a little effort, you can master it. So, go out there and conquer those trigonometric expressions!

Conclusion

Evaluating trigonometric expressions might seem tough initially, but with a systematic approach and a good grasp of trigonometric values, it becomes much easier. Remember to break down the problem, substitute the values carefully, and simplify. Keep practicing, and you'll become a pro in no time! We've successfully navigated this problem together, and hopefully, you now feel more confident in your ability to tackle similar challenges. Remember, the key is to stay organized, break down the problem into manageable steps, and utilize the fundamental trigonometric values. Don't be afraid to make mistakes – they're a natural part of the learning process. Just learn from them and keep moving forward. And most importantly, remember to celebrate your successes! Every problem you solve is a victory, and it brings you one step closer to mastering trigonometry. So, keep up the great work, keep practicing, and keep exploring the wonderful world of mathematics! You've got this!