Identifying Points On The Graph Of Y = Tan(x)
The tangent function, denoted as y = tan(x), is a fundamental trigonometric function with a unique graph and properties. This article delves into understanding the tangent function, its behavior, and how to identify key points on its graph. Specifically, we will address the question of which point the graph of the parent function y = tan(x) passes through, analyzing the provided options and explaining the underlying concepts.
To determine which point the graph of y = tan(x) passes through, we must first understand the tangent function itself. The tangent function is defined as the ratio of the sine function to the cosine function: tan(x) = sin(x) / cos(x). This definition is crucial because it links the tangent function to the unit circle and the behavior of sine and cosine functions.
The unit circle is a circle with a radius of 1 centered at the origin in the Cartesian plane. Angles are measured counterclockwise from the positive x-axis. For any angle x, the coordinates of the point where the terminal side of the angle intersects the unit circle are (cos(x), sin(x)). The tangent function then gives the ratio of the y-coordinate (sin(x)) to the x-coordinate (cos(x)).
The graph of y = tan(x) exhibits several key characteristics: it has vertical asymptotes at points where cos(x) = 0, as division by zero is undefined. These asymptotes occur at x = (2n + 1)π/2, where n is an integer. The function has a period of π, meaning its pattern repeats every π units along the x-axis. The function is also odd, meaning tan(-x) = -tan(x), which implies symmetry about the origin.
The tangent function increases from negative infinity to positive infinity between each pair of consecutive vertical asymptotes. At x = 0, tan(0) = 0. As x approaches π/2 from the left, tan(x) approaches positive infinity, and as x approaches -π/2 from the right, tan(x) approaches negative infinity.
Understanding these properties is essential for identifying points that lie on the graph of y = tan(x). We need to evaluate the tangent function at specific x-values and check if the resulting y-value matches the coordinates of the given points. This involves recalling the values of sine and cosine for common angles, such as π/6, π/4, π/3, and π/2, and then computing the ratio to find the tangent.
Now, let's analyze the provided options to determine which point the graph of y = tan(x) passes through. The options are:
- (√3/3, π/3)
- (π/3, √3/3)
- (π/3, √3)
- (√3, π/3)
To determine the correct point, we need to evaluate tan(x) for the x-coordinate of each point and see if the result matches the y-coordinate. Let's examine each option:
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(√3/3, π/3): Here, x = √3/3 and y = π/3. We need to check if tan(√3/3) = π/3. This is not a straightforward calculation, and it's unlikely to be correct since the tangent of √3/3 (approximately 0.577) is not equal to π/3 (approximately 1.047). This option seems incorrect.
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(π/3, √3/3): In this case, x = π/3 and y = √3/3. We need to evaluate tan(π/3) and see if it equals √3/3. Recall that π/3 radians is 60 degrees. We know that sin(π/3) = √3/2 and cos(π/3) = 1/2. Therefore, tan(π/3) = sin(π/3) / cos(π/3) = (√3/2) / (1/2) = √3. Since √3 ≠√3/3, this option is incorrect.
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(π/3, √3): Here, x = π/3 and y = √3. As we calculated in the previous step, tan(π/3) = √3. Therefore, the point (π/3, √3) satisfies the equation y = tan(x). This option appears to be correct.
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(√3, π/3): For this point, x = √3 and y = π/3. We need to check if tan(√3) = π/3. The value of √3 is approximately 1.732 radians, which is about 99.2 degrees. The tangent of √3 radians is not a standard value, but it's definitely not equal to π/3 (approximately 1.047). Thus, this option is incorrect.
Based on our analysis, the graph of the parent function y = tan(x) passes through the point (π/3, √3). This is because when x = π/3, tan(x) = tan(π/3) = √3, which matches the y-coordinate of the point.
Identifying key points on the graph of the tangent function is crucial for understanding its behavior and for solving various trigonometric problems. The point (π/3, √3) is one such key point because it corresponds to a standard angle (π/3 radians or 60 degrees) where the tangent function has a well-defined value (√3).
Other important points on the graph include (0, 0), where the tangent function is zero, and points near the vertical asymptotes, such as x = π/2 and x = -π/2. The tangent function approaches positive or negative infinity as x approaches these asymptotes, which is a key characteristic of the function.
Understanding these key points and the overall behavior of the tangent function allows us to sketch the graph accurately and to solve equations involving tangents. It also helps in understanding more complex trigonometric functions and their applications in various fields such as physics, engineering, and computer graphics.
In summary, the graph of the parent function y = tan(x) passes through the point (π/3, √3). This conclusion was reached by evaluating the tangent function at the x-coordinates of the given options and verifying which point satisfies the equation. The analysis also highlighted the importance of understanding the definition and properties of the tangent function, including its relationship to the unit circle, its periodicity, and the location of its vertical asymptotes. Grasping these concepts is essential for working with trigonometric functions and their applications in various mathematical and scientific contexts. By correctly identifying key points and understanding the function's behavior, we can solve trigonometric problems and gain a deeper appreciation for the nature of this important mathematical function.