Finding Sin Α And Cos Α: A Step-by-Step Guide

by ADMIN 46 views

Hey math enthusiasts! Today, we're diving into a classic trigonometry problem: finding the values of sine (sin α) and cosine (cos α) when given the tangent (tan α). This is a fundamental concept, and understanding it will give you a solid base for more advanced trigonometry. Let's break it down, step by step, so you can ace this type of question. If you're ready to learn, keep reading!

Understanding the Basics: Tangent, Sine, and Cosine

Alright, before we jump into the calculations, let's make sure we're all on the same page with the core concepts. Tangent, sine, and cosine are the three main trigonometric functions. They relate the angles of a right triangle to the lengths of its sides. Think of it like a secret code that helps us understand how angles and sides connect in these special triangles. Here's a quick refresher:

  • Tangent (tan α): This is the ratio of the opposite side to the adjacent side of a right triangle with angle α. Remember the mnemonic SOH CAH TOA? Tan is the TOA. This means tan α = opposite / adjacent.
  • Sine (sin α): This is the ratio of the opposite side to the hypotenuse of the right triangle. In SOH CAH TOA, sine is the SOH. That means sin α = opposite / hypotenuse.
  • Cosine (cos α): This is the ratio of the adjacent side to the hypotenuse. In SOH CAH TOA, cosine is the CAH. That means cos α = adjacent / hypotenuse.

Now, let's apply these definitions. You are given that tan α = 3/9. This tells us the ratio of the opposite side to the adjacent side. But, we need to find the sine and cosine, which involve the hypotenuse. So, we'll need a little help to find that hypotenuse. Remember, we will break down the steps and keep it simple. It's really not that hard, you just need to understand the connection between these ratios. Once you do, it becomes a lot easier!

Step-by-Step Solution: Finding sin α and cos α

Alright, let's get our hands dirty and actually solve this problem. We know that tan α = 3/9. Here's how we'll find sin α and cos α:

  1. Visualize the Right Triangle: Imagine a right triangle where α is one of the acute angles. Since tan α = opposite/adjacent = 3/9, we can say that the opposite side has a length of 3 units, and the adjacent side has a length of 9 units. Note, you could also say opposite = 1 and adjacent = 3, but using 3 and 9 will help us stick to whole numbers.

  2. Find the Hypotenuse: We can use the Pythagorean theorem to find the length of the hypotenuse (let's call it 'h'). The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, we have: h² = opposite² + adjacent².

    • Substitute the values we know: h² = 3² + 9²
    • Calculate: h² = 9 + 81
    • Simplify: h² = 90
    • Find h: h = √90
    • Simplify the square root: h = 3√10

  3. Calculate sin α: Now that we know the length of the hypotenuse, we can find sin α. Remember, sin α = opposite / hypotenuse. We know the opposite side is 3, and we calculated the hypotenuse to be 3√10. So:

    • sin α = 3 / (3√10)
    • Simplify by canceling out the 3s: sin α = 1 / √10
    • Rationalize the denominator (multiply the numerator and denominator by √10): sin α = (1 * √10) / (√10 * √10) = √10 / 10

  4. Calculate cos α: Finally, let's find cos α. We know cos α = adjacent / hypotenuse. We know the adjacent side is 9, and the hypotenuse is 3√10. So:

    • cos α = 9 / (3√10)
    • Simplify: cos α = 3 / √10
    • Rationalize the denominator: cos α = (3 * √10) / (√10 * √10) = 3√10 / 10

And there you have it! We've successfully calculated both sin α and cos α. So, for tan α = 3/9, we find that sin α = √10 / 10, and cos α = 3√10 / 10. Great job, you guys!

Important Considerations and Tips

There are a couple of things to keep in mind when working through these types of problems, and a few tips that can make your life easier.

  • Simplifying Fractions: Always simplify your fractions as much as possible. This makes your answers easier to understand and reduces the chances of errors. In our example, we simplified fractions when possible.
  • Rationalizing the Denominator: Make sure to rationalize the denominator. This means getting rid of any square roots in the denominator. This is a standard practice in mathematics and helps to standardize the format of your answers. We did this for both sin α and cos α.
  • Checking Your Work: It's always a good idea to double-check your work, especially when dealing with square roots and fractions. You can quickly review your calculations and ensure that all steps are correct.
  • Understanding the Unit Circle: If you're feeling adventurous, you can take a look at the unit circle. It’s a great way to visualize trigonometric functions and to understand their relationships better. This will come in handy when you start working with radians and angles beyond 90 degrees. This visualization tool can help you see where these values fit in a broader context.
  • Practice, Practice, Practice: The more you practice, the better you'll become at solving these types of problems. Try working through several examples to reinforce your understanding. There are a ton of online resources with practice problems and solutions.

Conclusion: Mastering the Trigonometric Ratios

So, we've successfully navigated the process of finding sin α and cos α when given tan α! You guys have learned how to use the basic trigonometric functions, apply the Pythagorean theorem, and simplify expressions. Remember that trigonometry is all about understanding the relationships between angles and sides in triangles. You've got this! Keep practicing, and you'll become a pro in no time.

Keep in mind, understanding these basic concepts is key to unlocking more complex trigonometric problems. If you're comfortable with this, you can now move on to more complicated things, such as inverse trigonometric functions, trigonometric identities, and the unit circle. Keep up the excellent work, and always remember to double-check your calculations. Happy calculating, and keep exploring the fascinating world of mathematics!

If you have any questions or want to see more examples, feel free to ask. Cheers!