Finding Sin(112.5°) Using The Half-Angle Formula A Step-by-Step Guide

Hey guys! Let's dive into a cool trigonometric problem today. We're going to find the exact value of sin(112.5°) using the half-angle formula. Trigonometry can seem daunting, but trust me, breaking it down step by step makes it super manageable. We'll go through the half-angle formula, apply it to our specific angle, and rationalize the denominator to get our final answer. So, grab your thinking caps, and let’s get started!

Understanding the Half-Angle Formula

Before we jump into solving for sin(112.5°), it's crucial to understand the half-angle formula. This formula allows us to find the trigonometric functions of an angle that is half of another angle whose trigonometric functions we already know. In our case, we want to find sin(112.5°), and we can express 112.5° as half of 225°. The half-angle formula for sine is given by:

$$\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1-\cos(\theta)}{2}}$$

The "±" sign is important because it indicates that the sign of sin(θ/2) depends on the quadrant in which θ/2 lies. This is where our understanding of angles and their quadrants comes into play. We'll need to determine the quadrant of 112.5° to decide whether we should use the positive or negative square root.

The half-angle formulas are derived from the power-reduction formulas and are incredibly handy tools in trigonometry. They bridge the relationship between the trigonometric functions of an angle and its half, allowing us to calculate exact values for angles that aren't standard angles on the unit circle (like 30°, 45°, 60°, etc.). Without these formulas, finding exact values for angles like 112.5° would be much more challenging. In order to successfully tackle this type of problem, it is very important to understand how the half-angle formula works and when to use the positive or negative root. So, keeping this in mind, let's move ahead.

Applying the Half-Angle Formula to sin(112.5°)

Now, let’s apply the half-angle formula to find sin(112.5°). As we discussed, we can express 112.5° as half of 225°. So, θ/2 = 112.5° implies that θ = 225°. The first step is to determine the cosine of 225°. Recall that 225° lies in the third quadrant, where both cosine and sine are negative. The reference angle for 225° is 225° - 180° = 45°. We know that cos(45°) = √2/2, so cos(225°) = -√2/2.

Now, let's plug cos(225°) into the half-angle formula:

$$\sin(112.5^{\circ}) = \sin\left(\frac{225^{\circ}}{2}\right) = \pm \sqrt{\frac{1-\cos(225^{\circ})}{2}}$$

$$\sin(112.5^{\circ}) = \pm \sqrt{\frac{1-\left(-\frac{\sqrt{2}}{2}\right)}{2}}$$

We simplify the expression inside the square root:

$$\sin(112.5^{\circ}) = \pm \sqrt{\frac{1+\frac{\sqrt{2}}{2}}{2}}$$

To get rid of the fraction within the fraction, we multiply the numerator and denominator by 2:

$$\sin(112.5^{\circ}) = \pm \sqrt{\frac{2+\sqrt{2}}{4}}$$

Next, we simplify the square root:

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

Now, we need to determine whether to use the positive or negative sign. Since 112.5° lies in the second quadrant, where sine is positive, we choose the positive sign. Therefore:

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

Rationalizing the Denominator (Not Required Here)

In this particular case, the denominator is already a rational number (2), so we don't need to rationalize it. Rationalizing the denominator is a process we typically perform when there is a square root (or other irrational number) in the denominator of a fraction. It involves multiplying both the numerator and denominator by a suitable expression that eliminates the radical from the denominator. Common techniques include multiplying by the conjugate or by the radical itself.

However, it's always a good practice to check if rationalizing is necessary. In this problem, the final expression we obtained is:

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

The denominator is 2, which is already a rational number. Thus, no further steps are needed for rationalization. Understanding when and how to rationalize denominators is a crucial skill in simplifying expressions and presenting final answers in a standard form. So, if you ever encounter a radical in the denominator, remember the tricks and techniques to get rid of it!

Final Answer

So, after applying the half-angle formula and simplifying, we’ve found that:

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

This is the exact value of sin(112.5°). Isn't it cool how we can use trigonometric formulas to find the exact values of angles that aren't as straightforward as the common ones? Remember, trigonometry is all about understanding relationships and applying the right tools. The half-angle formula is just one of the many tools in your trigonometric toolkit. Keep practicing, and you’ll become a trig whiz in no time! If you have any questions or want to explore other trigonometric problems, feel free to ask. Keep up the great work, everyone!

Practice Problems

To solidify your understanding of the half-angle formula, here are a few practice problems you can try:

1. Find the exact value of cos(15°) using the half-angle formula.
2. Find the exact value of tan(67.5°) using the half-angle formula.
3. Given that cos(θ) = -3/5 and θ is in the second quadrant, find sin(θ/2) and cos(θ/2).

Working through these problems will help you become more comfortable with applying the half-angle formulas and understanding the nuances of choosing the correct sign. Remember, practice makes perfect! So, grab a pen and paper, and give these problems a shot. You've got this!

Conclusion

In conclusion, we've successfully found the exact value of sin(112.5°) using the half-angle formula. We started by understanding the formula itself, then applied it to our specific angle, and finally, we simplified the expression to arrive at our answer. We also discussed the importance of rationalizing the denominator, although it wasn't required in this case.

Trigonometry is a fascinating branch of mathematics with a wide range of applications. Mastering the formulas and techniques, like the half-angle formula, opens doors to solving complex problems and understanding the relationships between angles and their trigonometric functions. Keep exploring, keep practicing, and most importantly, keep enjoying the journey of learning mathematics! You're doing great, and I'm excited to see what you'll conquer next. Until next time, happy problem-solving!