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Understanding trigonometric functions can sometimes feel like navigating a maze. A particularly intriguing question often arises: At Which Of The Following Angles Is the Tangent Function Undefined? This article will demystify the concept, explaining why the tangent function behaves the way it does at specific angles.
The Tangent Function and Its Relationship to Sine and Cosine
The key to understanding where the tangent function is undefined lies in its definition. The tangent of an angle, often written as tan(θ), is defined as the ratio of the sine of the angle to the cosine of the angle: tan(θ) = sin(θ) / cos(θ). This seemingly simple relationship is where the “undefined” mystery unfolds. Think of it like dividing any number by zero; it is mathematically impossible to get an answer! Therefore, when the cosine of an angle is zero, the tangent of that angle becomes undefined.
To pinpoint these angles, we need to recall the unit circle. The cosine of an angle corresponds to the x-coordinate of the point where the angle intersects the unit circle. Therefore, we need to find the angles where the x-coordinate is zero. This happens at 90 degrees (π/2 radians) and 270 degrees (3π/2 radians), and all angles coterminal with them. In essence, we are looking for the points on the unit circle directly above and below the origin. The angles at which the tangent function is undefined can be expressed generally as:
- θ = (π/2) + nπ, where n is any integer.
Let’s illustrate this with a small table:
| Angle (Degrees) | Angle (Radians) | Cosine Value | Tangent Value |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| 90 | π/2 | 0 | Undefined |
| 180 | π | -1 | 0 |
| 270 | 3π/2 | 0 | Undefined |
To further explore the concept of angles where the tangent function is undefined and deepen your understanding, it is recommended to consult reliable mathematical resources. Doing so will provide you with even more clarity on this fascinating aspect of trigonometry.