Find Tan(-105°) Exact Value Without Calculator

Hey guys! Let's dive into a fun trigonometry problem where we'll find the exact value of tan(-105°) without reaching for that calculator. We'll be using trigonometric identities and some clever angle manipulation to solve this. So, buckle up, and let's get started!

Understanding the Problem

Our main goal here is to find the value of tan(-105°). Now, -105° isn't one of those angles we usually memorize on the unit circle, like 30°, 45°, or 60°. So, we need to get a bit creative. The key idea is to express -105° as a sum or difference of angles whose tangent values we do know. Think about angles like 45°, 60°, and their multiples. These are our friends in this trigonometric adventure.

Before we jump into the solution, let's quickly recap the tangent addition and subtraction formulas:

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

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

These formulas are going to be our bread and butter for this problem. They allow us to break down the tangent of a complex angle into simpler, more manageable parts. We'll also use the fact that the tangent function is negative in the second and fourth quadrants, which will help us determine the sign of our final answer.

Breaking Down -105°

The first step is to think about how we can express -105° as a sum or difference of angles we know well. One way to do this is to notice that -105° = -60° - 45°. Both -60° and -45° are angles whose tangent values we can easily recall.

Now, you might be wondering, “Why did we choose -60° and -45°?” Well, it’s all about making our lives easier. We know that tan(-60°) = -√3 and tan(-45°) = -1. These are standard values that pop up frequently in trigonometry, so it’s super handy to have them memorized. If you don't have them memorized, no worries! You can always derive them using the unit circle or special right triangles (30-60-90 and 45-45-90).

Another way we could have broken down -105° is as -105° = -45° - 60°. This is essentially the same as our previous breakdown, just with the order switched. The important thing is that we've expressed our target angle as a combination of angles with known tangent values. This is the crucial step that allows us to use the tangent addition or subtraction formulas.

Applying the Tangent Subtraction Formula

Now that we've broken down -105° into -60° - 45°, we can use the tangent subtraction formula:

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

In our case, A = -60° and B = 45°. Plugging these values into the formula, we get:

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

Remember those tangent values we talked about? Now's their time to shine! We know that tan(-60°) = -√3 and tan(-45°) = -1. Let's substitute these values into our equation:

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

This looks a bit messy, but don't worry, we'll simplify it step by step. The key is to take it slow and be careful with our signs. A small sign error can throw off the whole calculation.

Simplifying the Expression

Let's simplify the numerator first: -√3 - (-1) = -√3 + 1. Not too bad, right?

Now, let's tackle the denominator: 1 + (-√3)(-1) = 1 + √3. Okay, we're making progress!

So, our expression now looks like this:

tan(-105°) = (-√3 + 1) / (1 + √3)

This is a valid answer, but it's not in the simplest form. Mathematicians like to get rid of radicals in the denominator (it's a matter of convention and aesthetics, really). To do this, we'll use a technique called rationalizing the denominator.

Rationalizing the Denominator

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 1 + √3 is 1 - √3. Remember, multiplying by the conjugate is a clever trick because it eliminates the radical in the denominator.

So, we multiply both the top and bottom of our fraction by (1 - √3):

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

Now, we need to multiply out the numerators and the denominators. Let's start with the numerator:

(-√3 + 1)(1 - √3) = -√3(1) + √3(√3) + 1(1) - 1(√3)

Simplifying this, we get:

-√3 + 3 + 1 - √3 = 4 - 2√3

Great! Now, let's move on to the denominator:

(1 + √3)(1 - √3) = 1(1) - 1(√3) + √3(1) - √3(√3)

Simplifying this, we get:

1 - √3 + √3 - 3 = -2

Notice how the radical terms canceled out in the denominator? That's the magic of using the conjugate!

Now, our expression looks like this:

tan(-105°) = (4 - 2√3) / -2

Final Simplification

We're almost there! We just need to simplify this fraction. We can divide both terms in the numerator by -2:

tan(-105°) = (4 / -2) - (2√3 / -2)

This simplifies to:

tan(-105°) = -2 + √3

And there we have it! We've found the exact value of tan(-105°) without using a calculator. The answer is -2 + √3, which can also be written as √3 - 2.

The Answer and Conclusion

So, the final answer is √3 - 2. We found this by breaking down -105° into -60° - 45°, using the tangent subtraction formula, and then simplifying the resulting expression. We also had to rationalize the denominator to get our answer in the simplest form.

This problem is a great example of how we can use trigonometric identities and algebraic manipulation to find exact values of trigonometric functions. It might seem a bit daunting at first, but by breaking it down into smaller steps and being careful with our calculations, we can solve it. Keep practicing these types of problems, and you'll become a trigonometry whiz in no time!

Keywords: exact value, trigonometry, tan(-105°), tangent subtraction formula, rationalize the denominator, unit circle, trigonometric identities