Understanding the Exact Value of tan -810°
In trigonometry, it is often necessary to determine the exact values of various trigonometric functions at angles that may be negative or not within the standard range of 0° to 360°. This article will guide you through the process of finding the exact value of tan -810° and provide a deeper understanding of negative angles and their trigonometric implications.
Introduction to Trigonometric Functions
The tangent function, or tan, is one of the primary trigonometric functions and is defined as the ratio of the sine function to the cosine function: tan(theta) frac{sin(theta)}{cos(theta)}. It is important to understand that the tangent function is periodic with a period of 180°, meaning its behavior repeats every 180° and is undefined at certain points such as 90° and -90°.
Reducing the Angle to the Principal Value
The first step in finding the exact value of tan -810° is to reduce it to an equivalent angle within the standard range of 0° to 360°.
Since a full circle is 360°, we can add or subtract multiples of 360° to obtain an equivalent positive angle:
-810° 360° -450°
-450° 360° -90°
Therefore, the equivalent angle in the standard range is -90°. It's important to note that this angle is measured from the positive x-axis in a clockwise direction.
Behavior of Tangent at Special Angles
When dealing with angles such as -90°, we need to understand the behavior of the tangent function at these points. The tangent of 90° (or its negative counterparts) is undefined due to the fact that the cosine of 90° is zero, leading to a division by zero.
Mathematically, we have:
tan 90° frac{sin 90°}{cos 90°} frac{1}{0} (undefined)
Since the tangent function is an odd function (i.e., tan(-theta) -tan(theta)), we can use this property to evaluate tan -90°:
tan -90° -tan 90° -infty
Thus, the exact value of tan -810° is:
tan -810° tan -90° -infty
Summary and Further Exploration
In conclusion, the exact value of tan -810° is -infty. This result is based on the periodic nature of the tangent function and the behavior of the tangent at special angles such as 90°. Negative angles like -90° change the direction of measurement but do not change the fundamental properties of the trigonometric functions.
Understanding these concepts can be incredibly useful for solving more complex trigonometric problems and can provide a deeper insight into the nature of periodic functions. For further exploration, you can join the logicxonomy Telegram group for free study materials and additional resources.
Sanjay Chakradeo