Why Tan(5π/6) ≠ Tan(5π/3)? Explained Simply!

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Why tan(5π/6) ≠ tan(5π/3)? Explained Simply!

Hey guys! Let's dive into a trigonometric head-scratcher: Why is the tangent of 5π/6 not equal to the tangent of 5π/3? This might seem confusing at first, but once we break it down, it's actually pretty straightforward. We're going to explore the unit circle, quadrants, and the behavior of the tangent function to understand why these two values are different. So, buckle up and let's get started!

Understanding the Unit Circle and Angles

First off, let's quickly refresh our understanding of the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian plane. Angles are measured counterclockwise from the positive x-axis. Key points on the unit circle correspond to specific angles, and these points give us the cosine and sine values of those angles. Remember, in the unit circle, the coordinates of a point are (cos θ, sin θ), where θ is the angle.

Now, let's pinpoint our angles: 5π/6 and 5π/3. To visualize these, it helps to think in terms of fractions of π. The angle 5π/6 is slightly less than π (which is 6π/6), placing it in the second quadrant. On the other hand, 5π/3 is more than 3π/2 (which is 4.5π/3) but less than 2π (which is 6π/3), so it lands in the fourth quadrant. These quadrant placements are super crucial in understanding the signs of trigonometric functions.

To really grasp the essence, let's consider reference angles. A reference angle is the acute angle formed between the terminal side of our angle and the x-axis. For 5π/6, the reference angle is π/6 (since π - 5π/6 = π/6). For 5π/3, the reference angle is π/3 (since 2π - 5π/3 = π/3). Notice that although the angles themselves are different, their reference angles are related but not identical, which plays a significant role in their tangent values.

Tangent: Sine Over Cosine

Okay, now let's talk about tangent. Remember that the tangent function is defined as the ratio of sine to cosine: tan θ = sin θ / cos θ. This is a fundamental identity in trigonometry, and it's the key to understanding why tangent behaves the way it does in different quadrants. The sign of the tangent will depend on the signs of sine and cosine in each quadrant.

In the first quadrant, both sine and cosine are positive, so tangent is also positive. In the second quadrant, sine is positive, but cosine is negative, making tangent negative. In the third quadrant, both sine and cosine are negative, so tangent is positive again (a negative divided by a negative is a positive). Finally, in the fourth quadrant, sine is negative, and cosine is positive, so tangent is negative. This quadrantal behavior is a cornerstone concept.

Let's circle back to our angles. For 5π/6, which is in the second quadrant, we know that tangent will be negative. For 5π/3, which is in the fourth quadrant, tangent will also be negative. But here's the catch: while both are negative, they won't necessarily be the same value. The magnitude of the tangent depends on the specific angle and the ratio of sine to cosine.

Evaluating tan(5π/6) and tan(5π/3)

Alright, let's put some numbers to this! To find tan(5π/6), we need to find sin(5π/6) and cos(5π/6). We know that the reference angle for 5π/6 is π/6. The sine of π/6 is 1/2, and the cosine of π/6 is √3/2. Since 5π/6 is in the second quadrant, sine is positive, and cosine is negative. Therefore:

sin(5π/6) = 1/2 cos(5π/6) = -√3/2

So, tan(5π/6) = sin(5π/6) / cos(5π/6) = (1/2) / (-√3/2) = -1/√3, which simplifies to -√3/3.

Now, let's tackle tan(5π/3). The reference angle for 5π/3 is π/3. The sine of π/3 is √3/2, and the cosine of π/3 is 1/2. Since 5π/3 is in the fourth quadrant, sine is negative, and cosine is positive. Thus:

sin(5π/3) = -√3/2 cos(5π/3) = 1/2

So, tan(5π/3) = sin(5π/3) / cos(5π/3) = (-√3/2) / (1/2) = -√3.

Notice the difference? tan(5π/6) is -√3/3, while tan(5π/3) is -√3. They're both negative, but they have different magnitudes. This difference arises from the specific ratios of sine and cosine for each angle.

Why the Difference? Key Reasons

Let's bring it all together and nail down the key reasons why tan(5π/6) ≠ tan(5π/3):

  1. Different Quadrants: The most crucial reason is that 5π/6 is in the second quadrant, and 5π/3 is in the fourth quadrant. While tangent is negative in both quadrants, the specific ratio of sine to cosine varies. This is our primary determinant.
  2. Reference Angles: Although both angles have related reference angles (π/6 and π/3), they are not the same. The tangent function's value depends on the reference angle, but the sign depends on the quadrant. The reference angle difference contributes to the varying magnitudes.
  3. Sine and Cosine Ratios: The exact values of sine and cosine for 5π/6 and 5π/3 are different. This leads to different ratios when you divide sine by cosine to find the tangent. The precise sine and cosine values are critical.

In Conclusion

So, there you have it! The reason why tan(5π/6) is not equal to tan(5π/3) boils down to their positions in different quadrants, their distinct reference angles, and the specific ratios of their sine and cosine values. Understanding the unit circle and how trigonometric functions behave in each quadrant is fundamental to mastering these concepts. Keep practicing, and you'll be a trig whiz in no time!