Find Sin Α, Cos Α, Tan Α If Cot Α = -√8, Α ∈ (3π/2, 2π)

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Hey guys! Let's dive into this trigonometric problem where we need to figure out the values of sin α, cos α, and tan α. We're given that cot α = -√8, and α lies in the interval (3π/2, 2π). This means α is in the fourth quadrant, which is super important because it tells us the signs of our trigonometric functions.

Understanding the Basics

Before we jump into the solution, let’s quickly recap some fundamental trigonometric identities and concepts. These will be our bread and butter for solving this problem.

  • Cotangent (cot α): Remember that cot α is the reciprocal of tan α, so cot α = 1/tan α. It’s also equal to cos α / sin α.
  • Tangent (tan α): tan α is sin α / cos α.
  • Pythagorean Identity: The most crucial identity here is sin² α + cos² α = 1. This connects sine and cosine, allowing us to find one if we know the other.
  • Quadrants: In the fourth quadrant (3π/2 < α < 2π), cosine is positive, while sine and tangent are negative. This sign information is critical.

Why is the quadrant important? Well, the quadrant tells us whether sine, cosine, and tangent are positive or negative. In the fourth quadrant, only cosine is positive. Sine and tangent are negative. This is essential for determining the correct signs in our final answers. Make sure you always consider the quadrant when solving trig problems!

Step-by-Step Solution

Now, let's break down how to solve this problem step by step.

1. Find tan α

We know that cot α = -√8. Since tan α is the reciprocal of cot α, we can easily find tan α:

tan α = 1 / cot α = 1 / (-√8) = -1/√8

To rationalize the denominator, we multiply the numerator and denominator by √8:

tan α = (-1/√8) * (√8/√8) = -√8 / 8

So, we have tan α = -√8 / 8. Remember this value, we'll need it later.

2. Use the Identity 1 + tan² α = 1/cos² α

This identity is derived from the Pythagorean identity and is extremely useful when we know tan α and want to find cos α. Let's plug in our value for tan α:

1 + (-√8 / 8)² = 1 / cos² α

1 + (8 / 64) = 1 / cos² α

1 + (1 / 8) = 1 / cos² α

9 / 8 = 1 / cos² α

Now, take the reciprocal of both sides to find cos² α:

cos² α = 8 / 9

3. Find cos α

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

cos α = ±√(8 / 9) = ±(√8 / 3)

Since α is in the fourth quadrant, where cosine is positive, we choose the positive value:

cos α = √8 / 3

We can simplify √8 as 2√2, so:

cos α = 2√2 / 3

Great! We've found cos α = 2√2 / 3. This is another crucial piece of the puzzle.

4. Use the Identity sin² α + cos² α = 1 to find sin α

Now that we have cos α, we can use the Pythagorean identity to find sin α. Plug in the value of cos α:

sin² α + (2√2 / 3)² = 1

sin² α + (8 / 9) = 1

sin² α = 1 - (8 / 9)

sin² α = 1 / 9

5. Find sin α

Take the square root of both sides:

sin α = ±√(1 / 9) = ±(1 / 3)

Since α is in the fourth quadrant, where sine is negative, we choose the negative value:

sin α = -1 / 3

Fantastic! We've found sin α = -1 / 3.

6. Verify tan α using sin α and cos α

As a final check, let's verify that our values for sin α and cos α give us the correct value for tan α. We know that tan α = sin α / cos α.

tan α = (-1 / 3) / (2√2 / 3)

To divide fractions, we multiply by the reciprocal:

tan α = (-1 / 3) * (3 / 2√2) = -1 / 2√2

Rationalize the denominator:

tan α = (-1 / 2√2) * (√2 / √2) = -√2 / 4

Now, let’s simplify our earlier value for tan α, which was -√8 / 8:

-√8 / 8 = -(2√2) / 8 = -√2 / 4

Both values match! This confirms our calculations are correct.

Final Answers

Alright, guys! We've successfully found the values of sin α, cos α, and tan α. Here they are:

  • sin α = -1 / 3
  • cos α = 2√2 / 3
  • tan α = -√8 / 8 = -√2 / 4

sin α (-1 / 3): Understanding sine is crucial in trigonometry. It represents the y-coordinate on the unit circle and helps us understand vertical motion and oscillations. In this case, a negative sine value in the fourth quadrant aligns perfectly with trigonometric principles.

cos α (2√2 / 3): Cosine, on the other hand, represents the x-coordinate on the unit circle. A positive cosine in the fourth quadrant indicates that the angle's terminal side lies to the right of the y-axis, which is consistent with our quadrant specification.

tan α (-√8 / 8): Tangent, the ratio of sine to cosine, combines these two aspects. A negative tangent in this quadrant confirms that the sine and cosine have opposite signs, reinforcing the accuracy of our calculations.

Tips for Trigonometry Problems

Before we wrap up, here are some golden nuggets of wisdom for tackling trigonometry problems:

  1. Know Your Identities: Memorize the basic trigonometric identities. They are your best friends.
  2. Understand Quadrants: Always determine the quadrant to get the correct signs.
  3. Draw Diagrams: Visualizing the problem can make it easier to understand.
  4. Practice, Practice, Practice: The more you practice, the better you'll get.

Conclusion

So, there you have it! We've successfully navigated through this trigonometric problem, finding sin α, cos α, and tan α given cot α and the interval for α. Remember, the key is to understand the definitions, use the identities wisely, and pay attention to the quadrants. Keep practicing, and you'll become a trig superstar in no time!