Exact Value Of Tan 105 Degrees Using Sum Or Difference Identities

In the realm of trigonometry, finding the exact values of trigonometric functions for specific angles can sometimes seem like a daunting task. However, with the use of sum and difference identities, we can tackle these problems with elegance and precision. Today, we're going to dive into finding the exact value of $\tan 105^{\circ}$ using these powerful identities. So, buckle up, math enthusiasts, and let's get started!

Understanding Sum and Difference Identities

Before we jump into the problem, let's briefly discuss the sum and difference identities for tangent. These identities are the cornerstone of our solution and allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. The specific identity we'll be using is the sum identity for tangent, which is given by:

$$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$

This identity tells us that the tangent of the sum of two angles, A and B, is equal to the sum of the tangents of the individual angles divided by 1 minus the product of their tangents. Similarly, there's a difference identity, but for this particular problem, the sum identity will be our focus. Mastering these trigonometric identities is crucial for solving various problems in trigonometry and calculus. They provide a way to break down complex angles into simpler ones, making calculations easier. When you understand these identities well, you can manipulate trigonometric expressions more effectively and solve a wide range of problems. Whether you're dealing with trigonometric equations, simplifying expressions, or evaluating functions at specific angles, these identities are your reliable tools.

Breaking Down 105 Degrees

The key to solving this problem lies in recognizing that $105^{\circ}$ can be expressed as the sum of two angles whose tangent values we know. Think about common angles like $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$. Can you see a combination that adds up to $105^{\circ}$? You got it! We can write $105^{\circ}$ as the sum of $60^{\circ}$ and $45^{\circ}$:

$$105^{\circ} = 60^{\circ} + 45^{\circ}$$

Now, why is this helpful? Because we know the exact values of the tangent function for both $60^{\circ}$ and $45^{\circ}$. Specifically:

• $\tan 60^{\circ} = \sqrt{3}$
• $\tan 45^{\circ} = 1$

By breaking down $105^{\circ}$ into the sum of two familiar angles, we've paved the way for using the sum identity we discussed earlier. This is a common strategy in trigonometry: transforming a difficult angle into a combination of angles that are easier to work with. Recognizing these relationships is an important skill in trigonometry. It allows you to apply trigonometric identities effectively and simplify complex expressions. This decomposition approach is widely used in various applications, such as solving equations, analyzing waveforms, and even in computer graphics for rotations and transformations.

Applying the Sum Identity

Now comes the fun part: applying the sum identity! We'll substitute $A = 60^{\circ}$ and $B = 45^{\circ}$ into the sum identity for tangent:

$$\tan(105^{\circ}) = \tan(60^{\circ} + 45^{\circ}) = \frac{\tan 60^{\circ} + \tan 45^{\circ}}{1 - \tan 60^{\circ} \tan 45^{\circ}}$$

Next, we'll plug in the values we know for $\tan 60^{\circ}$ and $\tan 45^{\circ}$:

$$\tan(105^{\circ}) = \frac{\sqrt{3} + 1}{1 - \sqrt{3} imes 1} = \frac{\sqrt{3} + 1}{1 - \sqrt{3}}$$

So, we've found an expression for $\tan 105^{\circ}$, but it's not in the simplest form yet. We have a radical in the denominator, which is generally considered unaesthetic in mathematical expressions. To get rid of it, we'll need to rationalize the denominator. Carefully substituting the known values is crucial in this step. A simple mistake here can lead to an incorrect result. This process highlights the importance of accuracy in mathematical calculations. Remember, each step builds upon the previous one, so ensuring precision at every stage is essential for arriving at the correct solution.

Rationalizing the Denominator

To rationalize the denominator, we'll multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of $1 - \sqrt{3}$ is $1 + \sqrt{3}$. This process is a standard technique for eliminating radicals from the denominator of a fraction. Remember, multiplying by the conjugate utilizes the difference of squares pattern, $(a - b)(a + b) = a^2 - b^2$, which helps to eliminate the square root. It's a handy algebraic trick that simplifies expressions and makes them easier to work with.

Multiplying both the numerator and denominator, we get:

$$\tan(105^{\circ}) = \frac{(\sqrt{3} + 1)(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})}$$

Now, let's expand both the numerator and the denominator:

$$\begin{aligned} \tan(105^{\circ}) &= \frac{\sqrt{3} + 3 + 1 + \sqrt{3}}{1 - 3} \\ &= \frac{4 + 2\sqrt{3}}{-2} \end{aligned}$$

We're almost there! We can simplify this fraction further by dividing both the numerator and denominator by -2:

Simplifying the Expression

Dividing both the numerator and the denominator by -2, we get:

$$\tan(105^{\circ}) = \frac{4 + 2\sqrt{3}}{-2} = -2 - \sqrt{3}$$

And there you have it! We've found the exact value of $\tan 105^{\circ}$ using sum identities and a little algebraic manipulation. The final answer is:

$$\tan(105^{\circ}) = -2 - \sqrt{3}$$

Simplifying the expression is the final touch that presents the answer in its most elegant form. In mathematics, we often strive to express solutions in their simplest possible form. This not only makes the answer easier to interpret but also highlights the underlying mathematical relationships more clearly. Simplifying expressions often involves factoring, canceling common factors, and rationalizing denominators, as we've seen in this example. It's a crucial step in problem-solving and demonstrates a deep understanding of the mathematical principles involved.

Conclusion

Finding the exact value of $\tan 105^{\circ}$ might have seemed challenging at first, but by using the sum identity for tangent and rationalizing the denominator, we were able to arrive at the solution. This problem showcases the power of trigonometric identities and algebraic techniques in simplifying complex expressions. So, the next time you encounter a similar problem, remember the steps we've taken here, and you'll be well-equipped to tackle it! Mastering these skills opens doors to more advanced topics in mathematics and its applications. Keep practicing, and you'll find yourself becoming more confident and proficient in solving trigonometric problems. Remember, math is a journey, not a destination, and every problem you solve is a step forward in your understanding.

I hope this explanation helps you to understand how to solve this type of problem! Keep exploring the world of trigonometry, and you'll be amazed at the beauty and power of these mathematical tools.