What Is The Exact Value Of Tan 105

Unveiling the Exact Value of tan 105°: A Deep Dive into Trigonometric Identities

Finding the exact value of tan 105° might seem daunting at first, especially if you're only familiar with the tangent values of common angles like 0°, 30°, 45°, 60°, and 90°. However, with a strategic application of trigonometric identities and a bit of algebraic manipulation, we can elegantly arrive at the precise answer. This article will guide you through the process, explaining the underlying principles and providing a thorough understanding of how to solve similar problems. We'll explore multiple approaches, demonstrating the versatility of trigonometric tools. By the end, you'll not only know the exact value of tan 105° but also possess a deeper understanding of trigonometric calculations.

Understanding the Challenge: Why We Can't Directly Find tan 105°

The tangent function, denoted as tan(x), represents the ratio of the sine and cosine functions: tan(x) = sin(x) / cos(x). While we readily know the sine and cosine values for standard angles, 105° isn't one of them. Therefore, we need to employ strategies to express 105° in terms of angles whose trigonometric values are known.

Method 1: Using the Sum-to-Product Formula

This approach leverages the sum formula for tangent: tan(A + B) = (tan A + tan B) / (1 - tan A tan B). We can express 105° as the sum of two angles with known tangent values, such as 60° and 45°.

Steps:

1. Express 105° as a sum: 105° = 60° + 45°
2. Apply the sum formula: tan(105°) = tan(60° + 45°) = (tan 60° + tan 45°) / (1 - tan 60° tan 45°)
3. Substitute known values: We know that tan 60° = √3 and tan 45° = 1. Substituting these values, we get: tan(105°) = (√3 + 1) / (1 - √3 * 1) = (√3 + 1) / (1 - √3)
4. Rationalize the denominator: To express the result in a more standard form, we rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator (1 + √3): tan(105°) = [(√3 + 1)(1 + √3)] / [(1 - √3)(1 + √3)] = (1 + 2√3 + 3) / (1 - 3) = (4 + 2√3) / (-2) = -2 - √3

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

Method 2: Utilizing the Difference Formula

Alternatively, we can express 105° as a difference of angles: 105° = 150° - 45°. The difference formula for tangent is: tan(A - B) = (tan A - tan B) / (1 + tan A tan B).

Steps:

1. Express 105° as a difference: 105° = 150° - 45°
2. Apply the difference formula: tan(105°) = tan(150° - 45°) = (tan 150° - tan 45°) / (1 + tan 150° tan 45°)
3. Substitute known values: tan 150° = -tan(180° - 150°) = -tan(30°) = -1/√3 and tan 45° = 1. Substituting these values: tan(105°) = (-1/√3 - 1) / (1 + (-1/√3)(1)) = (-1 - √3) / (√3 - 1)
4. Rationalize the denominator: Multiplying the numerator and denominator by the conjugate (√3 + 1): tan(105°) = [(-1 - √3)(√3 + 1)] / [(√3 - 1)(√3 + 1)] = (-√3 - 1 - 3 - √3) / (3 - 1) = (-4 - 2√3) / 2 = -2 - √3

Again, we arrive at the exact value of tan 105° as -2 - √3.

Method 3: Employing the Half-Angle Formula

The half-angle formula for tangent offers yet another pathway: tan(x/2) = (1 - cos x) / sin x = sin x / (1 + cos x). We can use this to find tan 210°, which is equivalent to tan (105° x 2). This will require the manipulation of the obtained result. This is a less straightforward approach for this specific problem, but demonstrates the versatility of trigonometric identities.

Steps:

1. Double the angle: We aim to find tan(210°), which is double the angle of 105°.
2. Use the half-angle formula (in reverse): This involves working backwards from tan(210°) to find tan(105°). Since tan(210°) = tan(180° + 30°) = tan(30°) = 1/√3, this method is not ideal for this specific problem because it utilizes a different angle and is unnecessarily complex.

Method 4: Using the Angle Addition Formula with Negative Angles

We can represent 105° as the sum of 135° and -30°. This approach demonstrates flexibility in choosing angles for the calculation.

Steps:

1. Express 105° as a sum: 105° = 135° + (-30°)
2. Apply the sum formula: tan(105°) = tan(135° - 30°) = [tan(135°) + tan(-30°)] / [1 - tan(135°)tan(-30°)]
3. Substitute known values: tan(135°) = -1 and tan(-30°) = -tan(30°) = -1/√3.
4. Substitute and Simplify: tan(105°) = [-1 - (1/√3)] / [1 - (-1)(-1/√3)] = (-√3 - 1) / (√3 - 1).
5. Rationalize the denominator: Multiply the numerator and denominator by (√3 + 1): tan(105°) = [(-√3 - 1)(√3 + 1)] / [(√3 - 1)(√3 + 1)] = (-3 - 2√3 -1) / (3 - 1) = (-4 - 2√3) / 2 = -2 - √3

Why Multiple Methods? The Power of Trigonometric Identities

The fact that we've reached the same answer, -2 - √3, using different methods highlights the inherent interconnectedness of trigonometric identities. Each method offers a unique pathway to the solution, demonstrating the power and flexibility of these mathematical tools. Choosing the most efficient method often depends on the specific problem and your familiarity with different identities.

Frequently Asked Questions (FAQ)

• Q: Can I use a calculator to find tan 105°? A: Yes, a calculator will give you an approximate decimal value. However, the methods outlined above provide the exact value, which is often crucial in mathematical proofs and more advanced calculations.

• Q: Why is the answer negative? A: The tangent function is negative in the second quadrant (90° to 180°), and 105° lies within this quadrant.

• Q: Are there other angles whose tangent value can be expressed exactly? A: Yes, many angles, especially those that are multiples or fractions of 30°, 45°, and 60°, have exact tangent values that can be derived using trigonometric identities.

• Q: How can I practice solving similar problems? A: Practice is key! Try finding the exact values of other tangents using the sum, difference, and half-angle formulas. Start with angles that are combinations of known angles and gradually increase the complexity.

Conclusion: Mastering Trigonometric Identities

Finding the exact value of tan 105° is a valuable exercise in applying trigonometric identities. It demonstrates the elegance and power of these mathematical tools, revealing the interconnectedness of seemingly disparate angles. By mastering these techniques, you not only solve specific problems but also gain a deeper understanding of the fundamental principles of trigonometry, enabling you to tackle more complex mathematical challenges with confidence. Remember, the key is to strategically choose the appropriate identity and apply it systematically, step-by-step. The journey of understanding trigonometric identities is rewarding, leading to a more profound appreciation for the beauty and precision of mathematics.