Solving \(\sin^2 75^\circ - \cos^2 75^\circ\): A Math Guide

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Hey guys! Today, we're diving into a super interesting trigonometric problem: finding the value of the expression sin⁡275∘−cos⁡275∘{\sin^2 75^\circ - \cos^2 75^\circ}. This might look intimidating at first glance, but don't worry! We're going to break it down step by step, making sure everyone can follow along. Whether you're a student tackling homework or just a math enthusiast, this guide is for you. We'll explore the fundamental trigonometric identities, apply some clever algebraic manipulations, and ultimately reveal the answer. So, grab your thinking caps, and let's get started on this mathematical adventure!

Understanding the Trigonometric Identity

Before we jump into solving sin⁡275∘−cos⁡275∘{\sin^2 75^\circ - \cos^2 75^\circ}, it’s crucial to understand and remember a fundamental trigonometric identity. This identity will serve as the cornerstone of our solution. The identity we're talking about is the double-angle formula for cosine, which states:

cos⁡(2x)=cos⁡2(x)−sin⁡2(x).{\cos(2x) = \cos^2(x) - \sin^2(x).}

Now, you might be wondering, “Why this identity?” Well, take a close look at the expression we need to evaluate: sin⁡275∘−cos⁡275∘{\sin^2 75^\circ - \cos^2 75^\circ}. Notice that it’s almost the same as the right side of our identity, but the terms are switched around. This is a key observation. To make our expression match the identity, we can simply factor out a -1:

sin⁡275∘−cos⁡275∘=−(cos⁡275∘−sin⁡275∘).{\sin^2 75^\circ - \cos^2 75^\circ = -(\cos^2 75^\circ - \sin^2 75^\circ).}

Now, we have something that perfectly aligns with the double-angle formula. By recognizing this connection, we’ve taken the first major step towards simplifying the problem. Remember, in mathematics, spotting these kinds of relationships is often half the battle. With this identity in our toolkit, we're well-equipped to move forward and find the value of the given expression. So, let's keep this identity in mind as we proceed to the next steps!

Applying the Double-Angle Formula

Okay, now that we've identified the relevant trigonometric identity, it's time to put it to work. We've established that:

sin⁡275∘−cos⁡275∘=−(cos⁡275∘−sin⁡275∘).{\sin^2 75^\circ - \cos^2 75^\circ = -(\cos^2 75^\circ - \sin^2 75^\circ).}

And we know from the double-angle formula for cosine that:

cos⁡(2x)=cos⁡2(x)−sin⁡2(x).{\cos(2x) = \cos^2(x) - \sin^2(x).}

The magic happens when we substitute x=75∘{x = 75^\circ} into the double-angle formula:

cos⁡(2×75∘)=cos⁡2(75∘)−sin⁡2(75∘).{\cos(2 \times 75^\circ) = \cos^2(75^\circ) - \sin^2(75^\circ).}

This simplifies to:

cos⁡(150∘)=cos⁡2(75∘)−sin⁡2(75∘).{\cos(150^\circ) = \cos^2(75^\circ) - \sin^2(75^\circ).}

Do you see where we're going with this? We've successfully transformed the expression inside the parentheses into something we can actually calculate! Remember that extra negative sign we factored out? Now, we need to bring it back into the equation:

sin⁡275∘−cos⁡275∘=−cos⁡(150∘).{\sin^2 75^\circ - \cos^2 75^\circ = -\cos(150^\circ).}

We’ve reduced the original expression to the negative of the cosine of 150 degrees. This is a huge simplification! Now, all that's left is to figure out the value of cos⁡(150∘){\cos(150^\circ)}. We’re on the home stretch now, guys! Stay with me as we move on to the final calculation.

Calculating cos⁡(150∘){\cos(150^\circ)}

To find the value of cos⁡(150∘){\cos(150^\circ)}, we'll use our knowledge of the unit circle and reference angles. Think of the unit circle as our trusty map for navigating trigonometric values. Remember, angles are measured counter-clockwise from the positive x-axis.

An angle of 150∘{150^\circ} lies in the second quadrant. The reference angle, which is the acute angle formed between the terminal side of our angle and the x-axis, is:

180∘−150∘=30∘.{180^\circ - 150^\circ = 30^\circ.}

So, 150∘{150^\circ} has a reference angle of 30∘{30^\circ}. This is great because we know the cosine of 30∘{30^\circ}. We remember from our special right triangles (the 30-60-90 triangle, specifically) that:

cos⁡(30∘)=32.{\cos(30^\circ) = \frac{\sqrt{3}}{2}.}

But wait! There's one more crucial detail. We need to consider the sign of cosine in the second quadrant. In the second quadrant, cosine is negative. This is because the x-coordinate (which corresponds to cosine) is negative in the second quadrant. Therefore:

cos⁡(150∘)=−32.{\cos(150^\circ) = -\frac{\sqrt{3}}{2}.}

We’ve cracked it! We now know the value of cos⁡(150∘){\cos(150^\circ)}. Remember that the original expression we were trying to solve was:

sin⁡275∘−cos⁡275∘=−cos⁡(150∘).{\sin^2 75^\circ - \cos^2 75^\circ = -\cos(150^\circ).}

Now we can simply substitute the value we just found for cos⁡(150∘){\cos(150^\circ)} into this equation.

The Final Solution

Alright, guys, it's time to bring it all together and reveal the final answer! We've journeyed through trigonometric identities, applied the double-angle formula, and navigated the unit circle. Now, let's wrap it up.

We know that:

sin⁡275∘−cos⁡275∘=−cos⁡(150∘).{\sin^2 75^\circ - \cos^2 75^\circ = -\cos(150^\circ).}

And we've calculated that:

cos⁡(150∘)=−32.{\cos(150^\circ) = -\frac{\sqrt{3}}{2}.}

So, substituting this value into our equation, we get:

sin⁡275∘−cos⁡275∘=−(−32).{\sin^2 75^\circ - \cos^2 75^\circ = -\left(-\frac{\sqrt{3}}{2}\right).}

Simplifying, we arrive at the final solution:

sin⁡275∘−cos⁡275∘=32.{\sin^2 75^\circ - \cos^2 75^\circ = \frac{\sqrt{3}}{2}.}

That's it! We've successfully found the value of the expression sin⁡275∘−cos⁡275∘{\sin^2 75^\circ - \cos^2 75^\circ}. The answer is 32{\frac{\sqrt{3}}{2}}. Great job, team! You tackled a potentially tricky problem with confidence and skill. Remember, the key to success in trigonometry (and math in general) is understanding the fundamental concepts and knowing how to apply them. I hope this breakdown was helpful and clear. Keep practicing, and you'll become a math whiz in no time!