Solving Trigonometric Equations Using Half-Angle Identities A Step-by-Step Guide

Hey guys! Today, we're diving into the fascinating world of trigonometry, specifically focusing on half-angle identities. These identities are super useful for finding the sine, cosine, and tangent of an angle that's half the size of a known angle. We're going to break down a problem step-by-step, so you can see exactly how these identities work in action. Trust me, once you get the hang of it, you'll be solving these problems like a pro! Understanding trigonometric identities, especially half-angle formulas, is crucial for solving a variety of problems in mathematics, physics, and engineering. These identities allow us to express trigonometric functions of half angles in terms of trigonometric functions of the full angle. This capability is particularly useful when dealing with angles that are not standard or easily found on the unit circle. In this comprehensive guide, we will explore how to utilize half-angle identities to find the exact values of trigonometric functions. We will specifically focus on sine, cosine, and tangent, providing a clear, step-by-step approach to solving a problem involving these identities. By mastering these techniques, you’ll enhance your problem-solving skills and gain a deeper understanding of the relationships between different trigonometric functions.

The beauty of trigonometric identities lies in their ability to simplify complex expressions and provide alternative ways to represent trigonometric functions. Half-angle identities are a prime example of this, allowing us to bridge the gap between angles and their halves. Whether you're a student grappling with trigonometry for the first time or a seasoned mathematician looking for a refresher, this guide aims to provide a clear and accessible explanation of how to use these identities effectively. So, let's embark on this trigonometric journey together and unlock the power of half-angle identities!

Let's tackle a classic problem. Suppose we know that $\cos \alpha = \frac{3}{5}$ and that $0^{\circ} < \alpha < 90^{\circ}$. Our mission, should we choose to accept it, is to determine the exact values of $\sin \frac{\alpha}{2}$, $\cos \frac{\alpha}{2}$, and $ an \frac{\alpha}{2}$. This problem is a perfect example of how half-angle identities can be applied in a practical context. The given information about the cosine of ${\alpha}$ and the range in which ${\alpha}$ lies is crucial for determining the correct signs and values of the half-angle trigonometric functions. Without this information, we wouldn't be able to narrow down the possibilities and arrive at the precise solutions we need. Therefore, paying close attention to the initial conditions is always the first step in solving trigonometric problems.

We are presented with a scenario where the cosine of an angle ${\alpha}$ is known, and we are asked to find the trigonometric functions of half that angle. This is a common type of problem that highlights the utility of half-angle identities. The fact that ${\alpha}$ lies between 0° and 90° tells us that it is in the first quadrant, which is a key piece of information. In the first quadrant, all trigonometric functions (sine, cosine, and tangent) are positive. However, when we consider half of this angle, the signs of the resulting trigonometric functions will depend on the quadrant in which ${\frac{\alpha}{2}}$ falls. Since ${\alpha}$ is between 0° and 90°, ${\frac{\alpha}{2}}$ will be between 0° and 45°, which is also in the first quadrant. Therefore, we can expect all our answers to be positive, which will help us avoid any confusion with signs. This careful consideration of the quadrant is vital in trigonometric calculations to ensure accurate results.

Before we jump into solving the problem, let's quickly refresh our memory on the half-angle identities. These are the formulas we'll be using to crack this trigonometric nut. We have three main identities to work with:

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

• $$\cos \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos \alpha}{2}}$$

• $$\tan \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} = \frac{\sin \alpha}{1 + \cos \alpha} = \frac{1 - \cos \alpha}{\sin \alpha}$$

Notice the $\pm$ signs in front of the square roots. This is super important! It means we need to figure out whether the result should be positive or negative, based on the quadrant in which $rac{\alpha}{2}$ lies. The half-angle identities are derived from the double-angle formulas, which are also fundamental in trigonometry. Understanding the relationship between these identities can help solidify your understanding of trigonometric functions. These identities provide a direct link between the trigonometric functions of an angle and its half, making them invaluable tools for problem-solving. The tangent half-angle identity, in particular, has multiple forms, each of which can be useful in different situations. Choosing the most appropriate form can often simplify the calculations and lead to a more efficient solution. The multiple forms of the tangent half-angle identity arise from different algebraic manipulations and can be advantageous depending on the given information. For instance, if we already know the sine of ${\alpha}$, the forms involving sine might be more convenient to use. Mastering these identities is not just about memorization; it's about understanding how they are derived and when to apply them effectively.

When working with half-angle identities, the quadrant in which the angle ${\frac{\alpha}{2}}$ lies plays a critical role in determining the sign of the trigonometric function. This is because the trigonometric functions have different signs in different quadrants. In the first quadrant (0° to 90°), all trigonometric functions are positive. In the second quadrant (90° to 180°), sine is positive, while cosine and tangent are negative. In the third quadrant (180° to 270°), tangent is positive, while sine and cosine are negative. Finally, in the fourth quadrant (270° to 360°), cosine is positive, while sine and tangent are negative. Therefore, before applying the half-angle identities, it is essential to determine the quadrant of ${\frac{\alpha}{2}}$ to ensure the correct sign is chosen for the result. This careful consideration of the sign is a key step in using half-angle identities accurately and effectively.

Okay, let's put these identities to work! We know $\cos \alpha = \frac{3}{5}$ and $0^{\circ} < \alpha < 90^{\circ}$. This means $rac{\alpha}{2}$ is between $0^{\circ}$ and $45^{\circ}$, which is in the first quadrant. In the first quadrant, sine, cosine, and tangent are all positive. This is a crucial piece of information because it tells us we'll be using the positive square roots in our half-angle identities. This preliminary analysis helps us avoid common errors and ensures that our final answers are correct. The fact that ${\frac{\alpha}{2}}$ lies in the first quadrant greatly simplifies our task. We can confidently use the positive square root in the half-angle formulas without worrying about the negative sign. This simplification is a direct consequence of understanding the behavior of trigonometric functions in different quadrants.

1. Finding $\sin \frac{\alpha}{2}$

Let's start with sine. We'll use the half-angle identity: $\sin \frac{\alpha}{2} = \sqrt{\frac{1 - \cos \alpha}{2}}$. Plug in $\cos \alpha = \frac{3}{5}$:

$$\sin \frac{\alpha}{2} = \sqrt{\frac{1 - \frac{3}{5}}{2}} = \sqrt{\frac{\frac{2}{5}}{2}} = \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$$

So, $\sin \frac{\alpha}{2} = \frac{\sqrt{5}}{5}$. We used the positive square root because, as we discussed, $rac{\alpha}{2}$ is in the first quadrant where sine is positive. Simplifying the expression under the square root involved finding a common denominator and then dividing fractions. The final step of rationalizing the denominator by multiplying both the numerator and denominator by ${\sqrt{5}}$ gives us a cleaner and more standard form of the answer. This process demonstrates the importance of algebraic manipulation in trigonometric problem-solving.

2. Finding $\cos \frac{\alpha}{2}$

Next up, cosine! We'll use the half-angle identity: $\cos \frac{\alpha}{2} = \sqrt{\frac{1 + \cos \alpha}{2}}$. Again, plug in $\cos \alpha = \frac{3}{5}$:

$$\cos \frac{\alpha}{2} = \sqrt{\frac{1 + \frac{3}{5}}{2}} = \sqrt{\frac{\frac{8}{5}}{2}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$$

Thus, $\cos \frac{\alpha}{2} = \frac{2\sqrt{5}}{5}$. Just like with sine, we used the positive square root because cosine is also positive in the first quadrant. The steps involved in simplifying the cosine expression are similar to those used for sine. We start by finding a common denominator, then dividing fractions, and finally rationalizing the denominator. This consistency in the algebraic techniques makes it easier to apply the half-angle identities effectively. The ability to perform these algebraic manipulations confidently is crucial for success in trigonometry.

3. Finding $\tan \frac{\alpha}{2}$

Last but not least, tangent! We have a couple of options here, but let's use the identity $\tan \frac{\alpha}{2} = \frac{\sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2}}$. We already found $\sin \frac{\alpha}{2} = \frac{\sqrt{5}}{5}$ and $\cos \frac{\alpha}{2} = \frac{2\sqrt{5}}{5}$, so:

$$\tan \frac{\alpha}{2} = \frac{\frac{\sqrt{5}}{5}}{\frac{2\sqrt{5}}{5}} = \frac{\sqrt{5}}{5} \cdot \frac{5}{2\sqrt{5}} = \frac{1}{2}$$

Therefore, $\tan \frac\alpha}{2} = \frac{1}{2}$. Alternatively, we could have used the other half-angle identity for tangent $\tan \frac{\alpha{2} = \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}}$, which would give us the same result. Using the quotient identity for tangent, ${\tan \frac{\alpha}{2} = \frac{\sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2}}}$ simplifies the calculation by directly using the values we already computed for sine and cosine. This approach highlights the interconnectedness of trigonometric functions and identities. The alternative half-angle identity for tangent, ${\tan \frac{\alpha}{2} = \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}}}$, would also lead to the correct answer, but it might involve more algebraic manipulation. Choosing the most efficient method is a skill that develops with practice.

Alright, we've done it! We successfully found the exact values of $\sin \frac{\alpha}{2}$, $\cos \frac{\alpha}{2}$, and $\tan \frac{\alpha}{2}$. Here's a quick recap:

• $$\sin \frac{\alpha}{2} = \frac{\sqrt{5}}{5}$$

• $$\cos \frac{\alpha}{2} = \frac{2\sqrt{5}}{5}$$

• $$\tan \frac{\alpha}{2} = \frac{1}{2}$$

These are the exact values we were looking for. We used the half-angle identities, the given information about $\cos \alpha$ and the range of $\alpha$, and a little bit of algebraic magic to arrive at these answers. This problem provides a solid example of how half-angle identities are used in practice. By breaking down the solution into clear steps, we can see how each identity is applied and how the given information is used to determine the correct signs and values. The process of solving this problem reinforces the importance of understanding the relationships between trigonometric functions and the role of quadrants in determining the signs of these functions. This comprehensive approach ensures that the solutions are not only correct but also well-understood. Mastering these types of problems is essential for building a strong foundation in trigonometry.

So there you have it! We've successfully navigated the world of half-angle identities and solved a trigonometric problem. Remember, the key is to understand the identities, pay attention to the given information (especially the quadrant), and take it one step at a time. With a little practice, you'll be a half-angle identity whiz in no time! Half-angle identities are powerful tools in trigonometry, allowing us to find the trigonometric functions of half angles when we know the trigonometric functions of the full angle. The ability to apply these identities effectively is a valuable skill in mathematics and related fields. By understanding the derivation of these identities and practicing their application, you can develop a deeper appreciation for the elegance and interconnectedness of trigonometric functions. The journey through trigonometry can be challenging, but with consistent effort and a clear understanding of the fundamental concepts, you can master these powerful tools and confidently tackle a wide range of problems. So, keep practicing, keep exploring, and keep enjoying the beauty of mathematics!