Finding the Value of Tan -585°: A Comprehensive Guide

In trigonometry, understanding the periodic properties of trigonometric functions is essential for simplifying and solving complex problems. This article focuses on finding the value of tan-585°, using the periodic nature of the tangent function.

Understanding the Periodic Property of the Tangent Function

The tangent function is periodic with a period of 180°. This means that for any angle θ, the value of tan(θ k ? 180°) is the same as the value of tan(θ), where k is an integer. The formula is expressed as:

tan(θ) tan(θ k ? 180°)

Reduction to an Equivalent Angle

To find the value of tan-585°, we first reduce -585° to an equivalent angle between 0° and 360° by adding 360° repeatedly until the result is positive. Here's the step-by-step process:

The equivalent angle for -585° in the range of 0° to 360° is 135°.

Calculating the Tangent Value

Now, we can calculate tan135° using the properties of the tangent function. Note that 135° is in the second quadrant, where the tangent function is negative. Furthermore, we can use the reference angle to simplify the calculation:

tan135° tan(180° - 45°) -tan45° -1

Thus, the value of tan-585° is:

boxed{-1}

Alternative Methods for Finding the Value

There are other methods to find the value of tan-585°. Here are two additional approaches:

Method 1: Using the Property tan(-x) -tan(x)

Given that tan-585° -tan585°, and applying the periodic property of the tangent function:

-tan585° -tan(6 × 90° 45°) -(-tan45°) -1

Method 2: Using the Periodicity Directly

By reducing -585° to 135° and directly using the periodic property:

tan-585° tan135° -tan45° -1

In this method, we recognize that 135° is in the second quadrant, where the tangent function is negative, and the reference angle is 45°. Therefore, the value is -1.

Conclusion

Understanding the periodic properties of trigonometric functions, such as the tangent function, is crucial for solving complex trigonometric problems. By using these properties, we can simplify angles to their equivalent values within the 0° to 360° range and then use the known values and sign rules to find the final answer.

In this article, we have demonstrated how to find the value of tan-585° using the periodic property and the periodicity of the tangent function. The value of tan-585° is -1, as confirmed by the methods described.