Find Sin 112.5 Degrees Using The Half-Angle Formula

Hey there, math enthusiasts! Today, we're diving into the fascinating world of trigonometry to unravel the mystery of $\sin 112.5^{\circ}$. We're not just going to pull out a calculator; instead, we'll flex our mathematical muscles and use the powerful half-angle formula. This approach not only gives us the answer but also deepens our understanding of trigonometric identities. So, buckle up and let's embark on this mathematical journey together!

Understanding the Half-Angle Formula

Before we jump into the problem, let's first understand the half-angle formula, our trusty tool for this task. The half-angle formulas are a set of trigonometric identities that express the trigonometric functions of an angle that is half of another angle. Specifically, for sine, the half-angle formula is given by:

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

Where the $\pm$ sign indicates that the result can be either positive or negative, depending on the quadrant in which the angle $\frac{x}{2}$ lies. This is a crucial detail we'll need to consider later. Now that we have our formula, let's think about how it applies to our specific problem: finding $\sin 112.5^{\circ}$. The key here is to recognize that $112.5^{\circ}$ is half of $225^{\circ}$, an angle whose cosine we know. By cleverly using the half-angle formula, we can break down a seemingly complex problem into simpler, manageable steps. This is the beauty of trigonometry – it provides us with tools to relate angles and their trigonometric functions, allowing us to solve problems in creative and insightful ways. So, let's keep this formula in mind as we move forward and apply it to find the exact value of $\sin 112.5^{\circ}$.

Applying the Half-Angle Formula to Find $\sin 112.5^{\circ}$

Okay, guys, let's get down to business! Our mission is to find $\sin 112.5^{\circ}$, and we're armed with the half-angle formula. As we discussed, $112.5^{\circ}$ is half of $225^{\circ}$. This is our golden ticket! So, we can rewrite our target as $\sin(\frac{225^{\circ}}{2})$. Now, we can directly apply the half-angle formula:

$$\sin(\frac{225^{\circ}}{2}) = \pm \sqrt{\frac{1 - \cos(225^{\circ})}{2}}$$

Fantastic! We've successfully transformed our problem into something more tangible. But remember that $\pm$ sign? We'll deal with that in a moment. First, we need to figure out what $\cos(225^{\circ})$ is. Think about the unit circle – $225^{\circ}$ lies in the third quadrant, where both cosine and sine are negative. $225^{\circ}$ is $45^{\circ}$ past $180^{\circ}$, so it's a reference angle of $45^{\circ}$. We know that $\cos(45^{\circ}) = \frac{\sqrt{2}}{2}$, so $\cos(225^{\circ}) = -\frac{\sqrt{2}}{2}$. Now we have all the pieces we need. Let's plug this value back into our formula:

$$\sin(\frac{225^{\circ}}{2}) = \pm \sqrt{\frac{1 - (-\frac{\sqrt{2}}{2})}{2}}$$

Simplifying this expression will give us the value of $\sin 112.5^{\circ}$. We are getting closer to the final answer step by step. The thrill of solving a math problem lies in breaking it down into smaller parts and tackling each part strategically. So, let's move on to the next step: simplifying the expression inside the square root.

Simplifying the Expression and Choosing the Correct Sign

Alright, let's simplify the expression under the square root. We've got:

$$\sin(\frac{225^{\circ}}{2}) = \pm \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}$$

To make things easier, let's get rid of the fraction within the fraction by multiplying the numerator and denominator inside the square root by 2:

$$\sin(\frac{225^{\circ}}{2}) = \pm \sqrt{\frac{2 + \sqrt{2}}{4}}$$

Now, we can take the square root of the denominator, which is a perfect square:

$$\sin(\frac{225^{\circ}}{2}) = \pm \frac{\sqrt{2 + \sqrt{2}}}{2}$$

We're almost there! We've got a simplified expression, but we still need to decide whether to use the positive or negative sign. Remember, the sign depends on the quadrant in which $112.5^{\circ}$ lies. $112.5^{\circ}$ is between $90^{\circ}$ and $180^{\circ}$, placing it in the second quadrant. In the second quadrant, sine is positive. Therefore, we choose the positive sign.

$$\sin 112.5^{\circ} = \frac{\sqrt{2 + \sqrt{2}}}{2}$$

Boom! We've done it! We've successfully found the exact value of $\sin 112.5^{\circ}$ using the half-angle formula. This is a fantastic example of how trigonometric identities can be used to solve seemingly complex problems. It's not just about memorizing formulas; it's about understanding how they work and applying them strategically.

Rationalizing the Denominator (If Necessary)

In our case, the denominator is already a rational number (2), so rationalizing the denominator is not necessary. Our final answer is clean and elegant:

$$\sin 112.5^{\circ} = \frac{\sqrt{2 + \sqrt{2}}}{2}$$

This is the exact value of $\sin 112.5^{\circ}$. We found it by applying the half-angle formula, simplifying the expression, and carefully considering the sign based on the quadrant of the angle. This problem highlights the power and beauty of trigonometry. By understanding the relationships between angles and their trigonometric functions, we can solve a wide range of problems. It also demonstrates the importance of breaking down complex problems into smaller, more manageable steps. Each step, from applying the formula to simplifying the expression, brings us closer to the final solution. So next time you encounter a trigonometric challenge, remember the half-angle formula and the strategies we've discussed here, and you'll be well-equipped to tackle it with confidence! Remember, math is not just about finding the answer; it's about the journey of discovery and the understanding we gain along the way.

Final Answer

Therefore,

$$\sin 112.5^{\circ} = \frac{\sqrt{2 + \sqrt{2}}}{2}$$

So the answer is:

$$\sin 112.5^{\circ}=\frac{\sqrt{2+\sqrt{2}}}{2}$$

In the context of the original question, the blanks would be filled as follows:

$$\sin 112.5^{\circ}=\frac{\sqrt{[2]+\sqrt{2}}}{2}$$