Calculate Exact Value Of Cos(105°) Using Half-Angle Identities

Hey math enthusiasts! Ever found yourself staring at a trigonometric function like cos(105°) and thinking, "How on earth do I figure that out without a calculator?" Well, you're in the right place! Today, we're going to embark on a journey to uncover the exact value of cos(105°) using the powerful tool of half-angle identities. Buckle up, because this is going to be a fun and insightful ride!

Delving into the Half-Angle Identities

First things first, let's talk about the stars of our show: the half-angle identities. These identities are trigonometric formulas that allow us to express trigonometric functions of half an angle in terms of trigonometric functions of the whole angle. They're incredibly handy when dealing with angles that aren't on the standard unit circle, like our 105° friend here. Specifically, we'll be focusing on the half-angle identity for cosine, which comes in two forms:

$$\cos(\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos(x)}{2}}$$

Now, you might be wondering about that ± sign. Don't worry, it's not as scary as it looks! The sign depends on the quadrant in which the angle x/2 lies. Remember our quadrants? Quadrant I is where all trigonometric functions are positive, Quadrant II is where sine is positive, Quadrant III is where tangent is positive, and Quadrant IV is where cosine is positive. Knowing this will help us choose the correct sign later on.

But before we dive into applying the formula, let's take a moment to appreciate why these identities are so useful. Think about it: we know the values of trigonometric functions for certain angles like 30°, 45°, 60°, and their multiples. But what about angles that are halfway between these? That's where the half-angle identities shine! They allow us to bridge the gap and find the exact values for a whole new range of angles. In the case of cos(105°), we can use the fact that 105° is half of 210°, an angle whose cosine we can easily determine. The half-angle identity acts as a bridge, connecting the known (cos(210°)) to the unknown (cos(105°)) and allowing us to precisely calculate the value. Understanding this core concept unlocks a multitude of possibilities in trigonometry, enabling us to solve complex problems and explore the intricate relationships between angles and their trigonometric counterparts. So, let’s embrace the power of the half-angle identities and confidently move forward in our mathematical journey!

Cracking the Cos(105°) Code: A Step-by-Step Solution

Okay, let's get down to business and find the exact value of cos(105°). The first step is recognizing that 105° is half of 210°. This is crucial because we know the trigonometric values for 210° (it's a standard angle on the unit circle!). Specifically, we know that cos(210°) = -√3/2. Isn't that neat how these connections emerge in math? Recognizing these relationships is part of the fun!

Now, let's bring in our trusty half-angle identity for cosine:

$$\cos(\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos(x)}{2}}$$

We're going to substitute x = 210° into this formula. This gives us:

$$\cos(105°) = \pm \sqrt{\frac{1 + \cos(210°)}{2}}$$

Next, we plug in the value of cos(210°) that we know:

$$\cos(105°) = \pm \sqrt{\frac{1 + (-\frac{\sqrt{3}}{2})}{2}}$$

Now, it's time for some algebraic gymnastics! Let's simplify the expression under the square root. To do this, we'll find a common denominator for the terms in the numerator:

$$\cos(105°) = \pm \sqrt{\frac{\frac{2 - \sqrt{3}}{2}}{2}}$$

We can simplify this further by dividing by 2, which is the same as multiplying by 1/2:

$$\cos(105°) = \pm \sqrt{\frac{2 - \sqrt{3}}{4}}$$

Now we take the square root of the denominator, which is simply 2:

$$\cos(105°) = \pm \frac{\sqrt{2 - \sqrt{3}}}{2}$$

We're almost there! But we're still left with that pesky ± sign. To determine the correct sign, we need to consider the quadrant in which 105° lies. Since 105° is between 90° and 180°, it's in Quadrant II. And remember, cosine is negative in Quadrant II. Therefore, we choose the negative sign:

$$\cos(105°) = -\frac{\sqrt{2 - \sqrt{3}}}{2}$$

But hold on a second! This doesn't quite match any of the answer choices provided. What gives? Well, this is where a little more algebraic trickery comes into play. We need to rationalize the numerator, which involves a clever manipulation to get rid of the nested square root. To rationalize the numerator effectively, we will use a compound angle formula with a twist, which we will see in the next section. But before that, take a moment to appreciate the journey we have taken so far. We started with a seemingly complex trigonometric value and systematically broke it down using the half-angle identity and algebraic simplification. Each step brought us closer to the solution, demonstrating the power of methodical problem-solving in mathematics.

The Grand Finale: Rationalizing and Revealing the Answer

Okay, guys, we're in the home stretch! We've arrived at the expression: $\cos(105°) = -\frac{\sqrt{2 - \sqrt{3}}}{2}$, but it's not quite in the form of our answer choices. This is where we need to get a little creative and rationalize the numerator. This means we want to manipulate the expression to get rid of that nested square root (the square root within a square root).

The trick here involves recognizing a special pattern. We want to rewrite the expression inside the square root as a perfect square. To do this, we'll multiply the numerator and denominator inside the square root by 2:

$$\cos(105°) = -\frac{\sqrt{2 - \sqrt{3}}}{2} = -\frac{\sqrt{2(2 - \sqrt{3})}}{2\sqrt{2}} = -\frac{\sqrt{4 - 2\sqrt{3}}}{2\sqrt{2}}$$

Now, focus on the expression inside the square root: 4 - 2√3. Can we rewrite this as something squared? The key is to recognize that it looks like the result of squaring a binomial of the form (a - b). Remember that (a - b)² = a² - 2ab + b². We want to find a and b such that:

a² + b² = 4

2ab = 2√3, which means ab = √3

By thinking about factors of √3, we can see that a = √3 and b = 1 satisfy these conditions. So, we can rewrite 4 - 2√3 as (√3 - 1)²:

$$\cos(105°) = -\frac{\sqrt{(\sqrt{3} - 1)^2}}{2\sqrt{2}}$$

Now we can take the square root:

$$\cos(105°) = -\frac{\sqrt{3} - 1}{2\sqrt{2}}$$

We're getting closer! But we still have a square root in the denominator. Let's get rid of that by multiplying the numerator and denominator by √2:

$$\cos(105°) = -\frac{(\sqrt{3} - 1)\sqrt{2}}{2\sqrt{2}\sqrt{2}} = -\frac{\sqrt{6} - \sqrt{2}}{4}$$

Finally, we distribute the negative sign:

$$\cos(105°) = \frac{-\sqrt{6} + \sqrt{2}}{4}$$

Or, rearranging the terms, we get:

$$\cos(105°) = \frac{-\sqrt{6} - \sqrt{2}}{4}$$

And there we have it! We've successfully navigated the twists and turns of trigonometry and arrived at our final answer. The exact value of cos(105°) is (-√6 + √2)/4 or (-√6 - √2)/4. This corresponds to option A in the original question.

Key Takeaways and Trigonometric Wisdom

Wow, what a journey! We've conquered cos(105°) using the power of half-angle identities and a healthy dose of algebraic manipulation. But beyond just finding the answer, what have we learned? Let's recap some key takeaways:

• Half-angle identities are your friends! They allow you to calculate trigonometric values for angles that aren't directly on the unit circle.
• Quadrant awareness is crucial. Knowing which quadrant an angle lies in helps you determine the correct sign for your trigonometric functions.
• Algebraic manipulation is a powerful tool. Don't be afraid to simplify, rationalize, and rewrite expressions to get them into a usable form.
• Problem-solving is a process. Sometimes you need to try different approaches and be persistent to reach the solution.

Most importantly, remember that math is not just about memorizing formulas; it's about understanding the underlying concepts and applying them creatively. So, the next time you encounter a trigonometric challenge, take a deep breath, remember our adventure with cos(105°), and dive in! You've got this!

This exploration into finding the exact value of cos(105°) serves as a powerful example of how mathematical tools and techniques can be combined to solve seemingly complex problems. We've not only found the solution but also reinforced important concepts in trigonometry and algebra. Keep exploring, keep questioning, and keep unlocking the beauty of mathematics!