What Is The Exact Value Of Tan 105

What is the Exact Value of tan 105°? A Comprehensive Guide

Determining the exact value of trigonometric functions for angles beyond the standard 0°, 30°, 45°, 60°, and 90° often requires employing trigonometric identities and a bit of clever manipulation. Finding the exact value of tan 105° is a classic example of this process. This article will guide you through several methods to achieve this, explaining the underlying principles and offering a deep dive into the mathematics involved.

Understanding the Problem: Why Not a Simple Calculation?

Unlike simpler angles, 105° isn't a readily available angle on the unit circle. We can't directly look up its tangent value in a standard trigonometric table. Therefore, we need to break down 105° into a combination of angles whose tangent values are known. This process involves utilizing trigonometric identities, specifically the tangent addition formula.

Method 1: Using the Tangent Addition Formula

The tangent addition formula provides a powerful tool for solving this problem. The formula states:

tan(A + B) = (tan A + tan B) / (1 - tan A * tan B)

We can express 105° as the sum of two angles whose tangents we know: 60° and 45°. Thus, A = 60° and B = 45°.

Substituting these values into the formula:

tan(105°) = tan(60° + 45°) = (tan 60° + tan 45°) / (1 - tan 60° * tan 45°)

We know that:

• tan 60° = √3
• tan 45° = 1

Substituting these known values:

tan(105°) = (√3 + 1) / (1 - √3 * 1) = (√3 + 1) / (1 - √3)

This result is an exact value, but it's not in a simplified form. To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator (1 + √3):

tan(105°) = [(√3 + 1) / (1 - √3)] * [(1 + √3) / (1 + √3)]

Expanding this expression:

tan(105°) = (√3 + 3 + 1 + √3) / (1 - 3) = (2√3 + 4) / (-2) = -(√3 + 2)

Therefore, the exact value of tan 105° is -2 - √3.

Method 2: Utilizing the Tangent Subtraction Formula

Alternatively, we could express 105° as the difference between two known angles. For example, 105° = 150° - 45°. The tangent subtraction formula is:

tan(A - B) = (tan A - tan B) / (1 + tan A * tan B)

Here, A = 150° and B = 45°. However, we need to find tan 150°. We can use the property that tan(180° - x) = -tan(x). Therefore, tan 150° = tan(180° - 30°) = -tan 30° = -1/√3.

Substituting into the subtraction formula:

tan(105°) = (tan 150° - tan 45°) / (1 + tan 150° * tan 45°) = (-1/√3 - 1) / (1 + (-1/√3) * 1)

Simplifying:

tan(105°) = (-1 - √3) / (√3 - 1)

Again, we rationalize the denominator by multiplying by the conjugate (√3 + 1):

tan(105°) = [(-1 - √3) / (√3 - 1)] * [(√3 + 1) / (√3 + 1)] = (-√3 - 3 - 1 - √3) / (3 - 1) = (-2√3 - 4) / 2 = -√3 - 2

This confirms our previous result: the exact value of tan 105° is -2 - √3.

Method 3: Employing the Half-Angle Formula

The half-angle formula for tangent is another useful approach. The formula is:

tan(x/2) = (1 - cos x) / sin x = sin x / (1 + cos x)

We can use this formula if we can express 105° as half of a known angle. While not immediately obvious, we can use 210°. We know that:

• sin 210° = -1/2
• cos 210° = -√3/2

Substituting into the half-angle formula:

tan(210°/2) = tan(105°) = (1 - cos 210°) / sin 210° = (1 - (-√3/2)) / (-1/2) = (1 + √3/2) / (-1/2) = -2 - √3

Alternatively, using the other form of the half-angle formula:

tan(105°) = sin 210° / (1 + cos 210°) = (-1/2) / (1 - √3/2) = (-1/2) / [(2 - √3)/2] = -1 / (2 - √3)

Rationalizing the denominator:

tan(105°) = [-1 / (2 - √3)] * [(2 + √3) / (2 + √3)] = -(2 + √3) / (4 - 3) = -2 - √3

This method again confirms that the exact value of tan 105° is -2 - √3.

Verification and Conclusion

Through three different methods, employing various trigonometric identities, we consistently arrive at the same result: the exact value of tan 105° is -2 - √3. This demonstrates the power and versatility of trigonometric identities in solving complex trigonometric problems. Remember that careful manipulation and rationalization of denominators are crucial steps in obtaining the simplified and exact form of the answer. Understanding these methods not only allows you to calculate the exact value of tan 105° but provides a solid foundation for tackling other similar trigonometric problems. This comprehensive approach ensures you not only get the correct answer but also understand the underlying mathematical reasoning. The use of multiple methods for verification reinforces the accuracy of the solution and provides a deeper understanding of the concept.