Calculating Sine And Cosine From Tangent In A Right Triangle A Trigonometry Guide

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Hey guys! Let's dive into a super important concept in trigonometry: trigonometric ratios in right triangles. Today, we're going to tackle a problem where we know the tangent of an acute angle (alpha) in a right triangle, and our mission is to find the sine and cosine of that angle. Plus, we'll even figure out their approximate values. This is fundamental stuff that will help you in so many areas of math and even in real-world applications, so let's get started!

Problem Overview: Tangent to Sine and Cosine

So, the core of our problem is this: We're given a right triangle with an acute angle called alpha (α). We know that the tangent of alpha (tan α) is equal to 12/5. Our goal is to calculate the sine (sin α) and cosine (cos α) of alpha. Additionally, we want to determine the approximate values for both sine and cosine. This is a classic trigonometry problem that requires us to use the relationships between trigonometric ratios and the sides of a right triangle. Understanding how to solve this will build a strong foundation for more complex trigonometric problems.

Breaking Down the Given Information

First, let's really understand what it means that tan α = 12/5. Remember the SOH-CAH-TOA mnemonic? It's super helpful! It tells us:

  • Sine = Opposite / Hypotenuse
  • Cosine = Adjacent / Hypotenuse
  • Tangent = Opposite / Adjacent

So, tan α = 12/5 means that in our right triangle, the side opposite to angle alpha is 12 units long, and the side adjacent to angle alpha is 5 units long. But what about the hypotenuse? We need that to calculate sine and cosine! This is where the Pythagorean Theorem comes to our rescue.

The Pythagorean Theorem to the Rescue

Finding the Hypotenuse

The Pythagorean Theorem is a cornerstone of geometry, and it's crucial here. It 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, it's expressed as:

  • a² + b² = c²

Where:

  • a and b are the lengths of the two shorter sides (legs) of the right triangle.
  • c is the length of the hypotenuse.

In our case:

  • a = 5 (adjacent side)
  • b = 12 (opposite side)
  • c = ? (hypotenuse, what we need to find)

Let's plug those values into the Pythagorean Theorem:

  • 5² + 12² = c²
  • 25 + 144 = c²
  • 169 = c²

To find c, we take the square root of both sides:

  • c = √169
  • c = 13

Awesome! So, the hypotenuse of our right triangle is 13 units long. Now we have all three sides, and we're ready to calculate sine and cosine.

Calculating Sine and Cosine

Now that we know the lengths of all three sides of the triangle, let's calculate the sine and cosine of angle alpha (α). Remember SOH-CAH-TOA?

Calculating Sine (sin α)

Sine is the ratio of the opposite side to the hypotenuse:

  • sin α = Opposite / Hypotenuse
  • sin α = 12 / 13

So, sin α is 12/13. That's one down!

Calculating Cosine (cos α)

Cosine is the ratio of the adjacent side to the hypotenuse:

  • cos α = Adjacent / Hypotenuse
  • cos α = 5 / 13

And there you have it! cos α is 5/13.

Approximating the Values of Sine and Cosine

Okay, we've found the exact values of sin α and cos α, but sometimes it's helpful to have decimal approximations. This makes it easier to visualize the values and compare them to other trigonometric values. Let's convert our fractions to decimals.

Approximating sin α

We know sin α = 12/13. To get the decimal approximation, we simply divide 12 by 13:

  • 12 ÷ 13 ≈ 0.923

So, sin α is approximately 0.923.

Approximating cos α

Similarly, we know cos α = 5/13. Let's divide 5 by 13:

  • 5 ÷ 13 ≈ 0.385

Therefore, cos α is approximately 0.385.

Putting It All Together and Checking Our Answer

Let's recap what we've done. We started with the information that tan α = 12/5 in a right triangle. We used this to determine the lengths of the opposite and adjacent sides. Then, we used the Pythagorean Theorem to find the length of the hypotenuse. With all three sides known, we calculated the sine and cosine of angle alpha:

  • sin α = 12/13 ≈ 0.923
  • cos α = 5/13 ≈ 0.385

Checking for Reasonableness

It's always a good idea to check if our answers make sense. Remember that in a right triangle, the sine and cosine of an acute angle are always between 0 and 1. Our values of approximately 0.923 and 0.385 fall within this range, which is a good sign! Also, notice that since sin α is larger than cos α, this suggests that angle alpha is closer to 90 degrees than it is to 0 degrees. This aligns with our initial understanding that tan α = 12/5 (a relatively large value) implies a steeper angle.

Conclusion: Mastering Trigonometric Ratios

Great job, guys! We've successfully solved a classic trigonometry problem. We started with the tangent of an angle, used the Pythagorean Theorem, and calculated the sine and cosine of that angle. We even approximated the values to get a better sense of their magnitude. This process demonstrates the interconnectedness of trigonometric ratios and the importance of understanding the relationships between the sides of a right triangle. Keep practicing these types of problems, and you'll become a trigonometry whiz in no time!

Remember, trigonometry is not just about memorizing formulas; it's about understanding the underlying concepts and how they relate to each other. By truly grasping these fundamentals, you'll be able to tackle a wide range of problems and apply trigonometry to real-world situations. So keep exploring, keep questioning, and keep learning! You've got this!

Answer to the specific question

Based on our calculations, none of the provided options (like option a) Sen alfa ≈ 0.8 and Cos alfa ≈ 0.6) accurately reflect the values we computed. The correct approximate values are:

  • Sen alfa ≈ 0.923
  • Cos alfa ≈ 0.385