Find The Exact Value Of Tan(195°): A Step-by-Step Guide

Have you ever wondered how to calculate the exact value of trigonometric functions for angles that aren't the standard 30°, 45°, or 60°? In this article, we'll dive deep into finding the exact value of tan(195°). This seemingly complex problem can be broken down using trigonometric identities and a bit of clever manipulation. So, let's embark on this mathematical journey together!

Understanding the Problem: Why tan(195°) Isn't So Straightforward

When we first encounter tan(195°), it's natural to wonder why we can't simply plug it into a calculator and get a straightforward answer. While calculators can provide decimal approximations, they don't give us the exact value in terms of radicals and integers. This is because 195° isn't one of the special angles (like 30°, 45°, 60°, 90°) for which we have readily memorized trigonometric values. These special angles correspond to common geometric shapes like equilateral triangles and squares, allowing us to derive their trigonometric ratios easily.

So, how do we tackle tan(195°)? The key lies in breaking down 195° into a combination of angles that are special. We need to express 195° as the sum or difference of angles whose tangent values we know. This is where our knowledge of trigonometric identities comes into play. By leveraging identities like the tangent addition formula, we can transform a complex problem into a series of simpler calculations. Think of it as a puzzle – we're taking a single, challenging piece and dissecting it into smaller, more manageable parts. This approach not only allows us to find the exact value but also deepens our understanding of the relationships between different angles and their trigonometric functions.

The Tangent Addition Formula: Our Key Tool

The tangent addition formula is the cornerstone of our solution. It states:

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

This formula allows us to express the tangent of the sum of two angles in terms of the individual tangents. It's a powerful tool for simplifying trigonometric expressions and, in our case, for finding the exact value of tan(195°). Before we can apply the formula, we need to identify two angles that add up to 195° and whose tangent values we know. There are several possibilities, but a particularly useful choice is 150° and 45°. Why these angles? Because both 150° and 45° are related to special angles, and we can easily determine their tangent values.

Let's quickly review the tangent values of these special angles. We know that tan(45°) = 1. For 150°, we can think of it as being in the second quadrant, where the tangent is negative. 150° is 30° away from the 180° mark, and we know that tan(30°) = 1/√3. Therefore, tan(150°) = -1/√3. Now that we have the tangent values for our chosen angles, we're ready to plug them into the tangent addition formula and start simplifying. It's like having all the ingredients for a recipe – now we just need to follow the instructions to bake our final result.

Breaking Down 195°: Choosing the Right Angles

The crucial step in solving this problem is to decompose 195° into the sum of two angles whose tangent values are known. As mentioned earlier, several combinations are possible, but a strategic choice will simplify our calculations. Let's consider a few options:

• 195° = 180° + 15°: While we know tan(180°) = 0, finding the exact value of tan(15°) directly can be a bit challenging. We could use the half-angle formula for tangent, but that adds an extra layer of complexity.
• 195° = 225° - 30°: We know tan(225°) = 1 and tan(30°) = 1/√3, but we'd need to use the tangent subtraction formula, which is similar to the addition formula but with a slight sign change. This is a viable option, but let's explore if we can find an even simpler approach.
• 195° = 150° + 45°: This is the combination we'll focus on. We already discussed why 150° and 45° are convenient choices: they both relate to special angles, and we can easily determine their tangent values. This decomposition allows us to directly apply the tangent addition formula without the need for further angle manipulations.

By carefully selecting 150° and 45°, we've set ourselves up for a more streamlined solution. It's a testament to the power of strategic problem-solving – sometimes, the key to a complex problem lies in finding the right perspective or approach. Now that we have our angles, let's move on to applying the tangent addition formula and simplifying the expression.

Applying the Tangent Addition Formula: Step-by-Step

Now that we've chosen to express 195° as 150° + 45°, we can directly apply the tangent addition formula:

tan(195°) = tan(150° + 45°) = (tan(150°) + tan(45°)) / (1 - tan(150°)tan(45°))

We already know that tan(150°) = -1/√3 and tan(45°) = 1. Let's substitute these values into the formula:

tan(195°) = (-1/√3 + 1) / (1 - (-1/√3)(1))

This expression looks a bit messy, but we can simplify it step by step. First, let's focus on the numerator and denominator separately. The numerator is (-1/√3 + 1). To combine these terms, we need a common denominator. We can rewrite 1 as √3/√3, so the numerator becomes:

(-1/√3 + √3/√3) = (√3 - 1) / √3

Now let's simplify the denominator: (1 - (-1/√3)(1)) = (1 + 1/√3). Again, we need a common denominator to combine the terms. We can rewrite 1 as √3/√3, so the denominator becomes:

(√3/√3 + 1/√3) = (√3 + 1) / √3

Now we have:

tan(195°) = ((√3 - 1) / √3) / ((√3 + 1) / √3)

Dividing by a fraction is the same as multiplying by its reciprocal. So, we can rewrite the expression as:

tan(195°) = ((√3 - 1) / √3) * (√3 / (√3 + 1))

The √3 terms cancel out, leaving us with:

tan(195°) = (√3 - 1) / (√3 + 1)

We've made significant progress, but our answer isn't in the simplest form yet. We need to rationalize the denominator to get rid of the radical in the bottom. This is our next step.

Rationalizing the Denominator: The Final Touch

Our current expression for tan(195°) is (√3 - 1) / (√3 + 1). To rationalize the denominator, we need to eliminate the radical from the denominator. We can do this by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of (√3 + 1) is (√3 - 1).

So, we multiply both the numerator and denominator by (√3 - 1):

tan(195°) = ((√3 - 1) / (√3 + 1)) * ((√3 - 1) / (√3 - 1))

Now, let's multiply out the numerator and denominator separately. The numerator is (√3 - 1)(√3 - 1). We can expand this using the FOIL method (First, Outer, Inner, Last):

• First: √3 * √3 = 3
• Outer: √3 * -1 = -√3
• Inner: -1 * √3 = -√3
• Last: -1 * -1 = 1

Combining these terms, we get 3 - √3 - √3 + 1 = 4 - 2√3. So, the numerator is 4 - 2√3.

Now let's multiply out the denominator: (√3 + 1)(√3 - 1). This is a difference of squares, so it simplifies to (√3)² - (1)² = 3 - 1 = 2. So, the denominator is 2.

Now we have:

tan(195°) = (4 - 2√3) / 2

We can simplify this fraction by dividing both terms in the numerator by 2:

tan(195°) = 2 - √3

And there we have it! The exact value of tan(195°) is 2 - √3. We've successfully navigated the problem by breaking it down into smaller steps, applying the tangent addition formula, and rationalizing the denominator. This result matches option D in the original problem statement.

Conclusion: Mastering Trigonometric Identities

Finding the exact value of tan(195°) might have seemed daunting at first, but we've demonstrated that by using trigonometric identities and a systematic approach, even complex problems can be solved. The key takeaways from this exercise are:

• Understanding the tangent addition formula is crucial for manipulating trigonometric expressions.
• Breaking down angles into sums or differences of special angles simplifies calculations.
• Rationalizing the denominator is often necessary to express answers in their simplest form.

By mastering these techniques, you'll be well-equipped to tackle a wide range of trigonometric problems. Remember, mathematics is often about finding the right tools and applying them strategically. Practice makes perfect, so keep exploring and challenging yourself with new problems!

For further exploration of trigonometric identities and their applications, you can visit Khan Academy's Trigonometry section.