Finding Sin(θ/2) Given Tan(θ) A Trigonometry Guide

Hey there, math enthusiasts! Today, we're diving into a trigonometric problem that might seem a bit daunting at first, but trust me, we'll break it down step by step. We're going to tackle the challenge of finding $\sin{\frac{\theta}{2}}$ given that $\tan{\theta} = \frac{8}{3}$ and $0 < \theta < \frac{\pi}{2}$. This problem is a fantastic exercise in applying trigonometric identities and understanding the relationships between different trigonometric functions. So, grab your calculators (or your thinking caps!) and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we fully grasp what we're dealing with. The problem gives us the value of the tangent of an angle $\theta$ and tells us that this angle lies in the first quadrant (between 0 and $\frac{\pi}{2}$ radians). Our mission, should we choose to accept it, is to find the sine of half this angle. This involves a bit of trigonometric maneuvering, but don't worry, we'll navigate it together.

The Importance of the Quadrant

The fact that $\theta$ lies in the first quadrant is crucial. Why? Because in the first quadrant, all trigonometric functions (sine, cosine, tangent, etc.) are positive. This helps us avoid any ambiguity when we're calculating square roots or using inverse trigonometric functions. Knowing the quadrant helps us determine the sign of the trigonometric functions, which is essential for arriving at the correct answer. So, always pay close attention to the given range of the angle!

The Half-Angle Formula

The key to solving this problem lies in the half-angle formula for sine. This formula allows us to express the sine of half an angle in terms of the cosine of the original angle. Specifically, the formula states:

$\sin{\frac{\theta}{2}} = \pm\sqrt{\frac{1 - \cos{\theta}}{2}}$

Notice the $\pm$ sign in front of the square root. This is where the quadrant information comes in handy. Since $0 < \theta < \frac{\pi}{2}$, it follows that $0 < \frac{\theta}{2} < \frac{\pi}{4}$. This means that $\frac{\theta}{2}$ also lies in the first quadrant, where sine is positive. Therefore, we can confidently take the positive square root.

So, our working formula becomes:

$\sin{\frac{\theta}{2}} = \sqrt{\frac{1 - \cos{\theta}}{2}}$

Now, we have a clear path forward: we need to find $\cos{\theta}$, plug it into this formula, and simplify.

Finding Cosine from Tangent

We're given $\tan{\theta} = \frac{8}{3}$, but we need $\cos{\theta}$. How do we bridge this gap? The answer lies in the fundamental trigonometric identities. One identity that's particularly useful here is:

$1 + \tan^2{\theta} = \sec^2{\theta}$

where $\sec{\theta}$ is the secant function, which is the reciprocal of cosine: $\sec{\theta} = \frac{1}{\cos{\theta}}$.

Let's plug in our value for $\tan{\theta}$:

$1 + \left(\frac{8}{3}\right)^2 = \sec^2{\theta}$

$1 + \frac{64}{9} = \sec^2{\theta}$

$\frac{9}{9} + \frac{64}{9} = \sec^2{\theta}$

$\frac{73}{9} = \sec^2{\theta}$

Now, we take the square root of both sides:

$\sec{\theta} = \pm\sqrt{\frac{73}{9}} = \pm\frac{\sqrt{73}}{3}$

Again, we need to consider the quadrant. Since $\theta$ is in the first quadrant, cosine is positive, and therefore secant is also positive. So, we take the positive root:

$\sec{\theta} = \frac{\sqrt{73}}{3}$

Now, we can easily find $\cos{\theta}$ by taking the reciprocal:

$\cos{\theta} = \frac{1}{\sec{\theta}} = \frac{3}{\sqrt{73}}$

To rationalize the denominator (which is good mathematical practice), we multiply the numerator and denominator by $\sqrt{73}$:

$\cos{\theta} = \frac{3\sqrt{73}}{73}$

Great! We've found $\cos{\theta}$. Now, we're ready to plug it into the half-angle formula.

Applying the Half-Angle Formula

Remember our formula?

$\sin{\frac{\theta}{2}} = \sqrt{\frac{1 - \cos{\theta}}{2}}$

Let's substitute the value of $\cos{\theta}$ we just found:

$\sin{\frac{\theta}{2}} = \sqrt{\frac{1 - \frac{3\sqrt{73}}{73}}{2}}$

Now, we need to simplify this expression. The first step is to get a common denominator in the numerator:

$\sin{\frac{\theta}{2}} = \sqrt{\frac{\frac{73}{73} - \frac{3\sqrt{73}}{73}}{2}}$

$\sin{\frac{\theta}{2}} = \sqrt{\frac{\frac{73 - 3\sqrt{73}}{73}}{2}}$

Next, we divide by 2, which is the same as multiplying by $\frac{1}{2}$:

$\sin{\frac{\theta}{2}} = \sqrt{\frac{73 - 3\sqrt{73}}{146}}$

And that's our answer! We've successfully found $\sin{\frac{\theta}{2}}$.

The Final Answer

$\sin{\frac{\theta}{2}} = \sqrt{\frac{73 - 3\sqrt{73}}{146}}$

Key Takeaways

Let's recap what we've learned in this problem:

1. Understanding the problem: We carefully read the problem and identified what was given and what we needed to find.
2. The importance of the quadrant: We recognized that the quadrant of the angle helped us determine the signs of the trigonometric functions.
3. The half-angle formula: We recalled and applied the half-angle formula for sine.
4. Finding cosine from tangent: We used the identity $1 + \tan^2{\theta} = \sec^2{\theta}$ to find $\cos{\theta}$ from $\tan{\theta}$.
5. Simplifying the expression: We carefully simplified the expression, paying attention to order of operations and rationalizing the denominator.

Trigonometry can seem tricky at times, but with practice and a solid understanding of the fundamental identities, you can conquer even the most challenging problems. Keep practicing, and you'll become a trig whiz in no time!

Practice Problems

Want to test your understanding? Try these practice problems:

1. If $\cos{\theta} = \frac{1}{4}$ and $0 < \theta < \frac{\pi}{2}$, find $\sin{\frac{\theta}{2}}$.
2. If $\sin{\theta} = \frac{2}{5}$ and $\frac{\pi}{2} < \theta < \pi$, find $\cos{\frac{\theta}{2}}$.

Good luck, and happy problem-solving!

Conclusion

So, there you have it! We've successfully navigated the world of trigonometric identities and half-angle formulas to find the value of $\sin{\frac{\theta}{2}}$. Remember, the key to mastering trigonometry is understanding the relationships between the functions and practicing applying the formulas. Don't be afraid to break down complex problems into smaller, more manageable steps. And most importantly, have fun with it! Math can be a beautiful and rewarding journey, and we're here to guide you every step of the way. Keep exploring, keep learning, and keep those trigonometric skills sharp!