Trigonometric Exploration Calculating Cos²((α + Β)/2) And Sin²((α - Β)/2)

In the fascinating world of trigonometry, understanding the relationships between angles and their trigonometric functions is crucial. This article delves into a specific scenario involving acute angles ${\alpha}$ and ${\beta}$, where ${\cos \alpha = \frac{3}{5}}$ and ${\cos \beta = \frac{5}{13}}$. We will explore how to leverage these values to determine other trigonometric expressions, specifically focusing on ${\cos^2\left(\frac{\alpha + \beta}{2}\right)}$ and ${\sin^2\left(\frac{\alpha - \beta}{2}\right)}$. This exploration will not only enhance your understanding of trigonometric identities but also demonstrate practical applications of these concepts. We will break down the problem step by step, ensuring a comprehensive and easy-to-follow approach.

Determining Sine Values for Angles ${\alpha}$ and ${\beta}$

To begin, we are given that ${\cos \alpha = \frac{3}{5}}$ and ${\cos \beta = \frac{5}{13}}$, where both ${\alpha}$ and ${\beta}$ are acute angles. This means that both angles lie between 0 and ${\frac{\pi}{2}}$ radians (or 0 and 90 degrees). In this range, sine values are positive. We can use the Pythagorean identity, which states that ${\sin^2 \theta + \cos^2 \theta = 1}$, to find the values of ${\sin \alpha}$ and ${\sin \beta}$. For angle ${\alpha}$, we have:

${\sin^2 \alpha + \cos^2 \alpha = 1}$ ${\sin^2 \alpha + \left(\frac{3}{5}\right)^2 = 1}$ ${\sin^2 \alpha + \frac{9}{25} = 1}$ ${\sin^2 \alpha = 1 - \frac{9}{25}}$ ${\sin^2 \alpha = \frac{16}{25}}$ ${\sin \alpha = \sqrt{\frac{16}{25}} = \frac{4}{5}}$

Since ${\alpha}$ is an acute angle, ${\sin \alpha}$ is positive, so we take the positive square root. Similarly, for angle ${\beta}$, we have:

${\sin^2 \beta + \cos^2 \beta = 1}$ ${\sin^2 \beta + \left(\frac{5}{13}\right)^2 = 1}$ ${\sin^2 \beta + \frac{25}{169} = 1}$ ${\sin^2 \beta = 1 - \frac{25}{169}}$ ${\sin^2 \beta = \frac{144}{169}}$

${\sin \beta = \sqrt{\frac{144}{169}} = \frac{12}{13}}$

Again, since ${\beta}$ is an acute angle, ${\sin \beta}$ is positive. Thus, we have ${\sin \alpha = \frac{4}{5}}$ and ${\sin \beta = \frac{12}{13}}$. These values will be crucial in our subsequent calculations. Understanding how to derive these values from the given cosine values is a fundamental aspect of trigonometric problem-solving. The use of the Pythagorean identity is a cornerstone technique, and mastering it is essential for tackling more complex trigonometric problems. Furthermore, recognizing the sign conventions for trigonometric functions in different quadrants (in this case, the first quadrant where both sine and cosine are positive) is vital for accurate calculations.

Calculating ${\cos(\alpha + \beta)}$ and ${\cos(\alpha - \beta)}$

Now that we have the values of ${\sin \alpha}$, ${\cos \alpha}$, ${\sin \beta}$, and ${\cos \beta}$, we can proceed to calculate ${\cos(\alpha + \beta)}$ and ${\cos(\alpha - \beta)}$. These calculations are essential for finding ${\cos^2\left(\frac{\alpha + \beta}{2}\right)}$ and ${\sin^2\left(\frac{\alpha - \beta}{2}\right)}$ later on. We will use the angle sum and difference identities for cosine, which are:

${\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta}$ ${\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta}$

Substituting the values we found earlier, we get:

${\cos(\alpha + \beta) = \left(\frac{3}{5}\right)\left(\frac{5}{13}\right) - \left(\frac{4}{5}\right)\left(\frac{12}{13}\right)}$ ${\cos(\alpha + \beta) = \frac{15}{65} - \frac{48}{65}}$ ${\cos(\alpha + \beta) = -\frac{33}{65}}$

And:

${\cos(\alpha - \beta) = \left(\frac{3}{5}\right)\left(\frac{5}{13}\right) + \left(\frac{4}{5}\right)\left(\frac{12}{13}\right)}$ ${\cos(\alpha - \beta) = \frac{15}{65} + \frac{48}{65}}$

${\cos(\alpha - \beta) = \frac{63}{65}}$

Therefore, we have found that ${\cos(\alpha + \beta) = -\frac{33}{65}}$ and ${\cos(\alpha - \beta) = \frac{63}{65}}$. These values are crucial intermediate steps in our journey to find the final answers. Understanding and applying the angle sum and difference identities is a fundamental skill in trigonometry. These identities allow us to express trigonometric functions of sums and differences of angles in terms of trigonometric functions of the individual angles. This is a powerful technique that simplifies many trigonometric problems. The correct application of these identities, along with careful arithmetic, is essential for obtaining the correct results.

Calculating ${\cos^2\left(\frac{\alpha + \beta}{2}\right)}$

Now, we aim to find the value of ${\cos^2\left(\frac{\alpha + \beta}{2}\right)}$. To do this, we will use the half-angle identity for cosine, which is derived from the double-angle formula. The relevant form of the half-angle identity is:

${\cos^2\left(\frac{\theta}{2}\right) = \frac{1 + \cos \theta}{2}}$

In our case, ${\theta = \alpha + \beta}$, so we have:

${\cos^2\left(\frac{\alpha + \beta}{2}\right) = \frac{1 + \cos(\alpha + \beta)}{2}}$

We already calculated ${\cos(\alpha + \beta) = -\frac{33}{65}}$, so we substitute this value into the equation:

${\cos^2\left(\frac{\alpha + \beta}{2}\right) = \frac{1 + \left(-\frac{33}{65}\right)}{2}}$

${\cos^2\left(\frac{\alpha + \beta}{2}\right) = \frac{1 - \frac{33}{65}}{2}}$

${\cos^2\left(\frac{\alpha + \beta}{2}\right) = \frac{\frac{65 - 33}{65}}{2}}$

${\cos^2\left(\frac{\alpha + \beta}{2}\right) = \frac{\frac{32}{65}}{2}}$

${\cos^2\left(\frac{\alpha + \beta}{2}\right) = \frac{32}{65} \cdot \frac{1}{2}}$

${\cos^2\left(\frac{\alpha + \beta}{2}\right) = \frac{16}{65}}$

Thus, we find that ${\cos^2\left(\frac{\alpha + \beta}{2}\right) = \frac{16}{65}}$. This result is one of the key objectives of our exploration. The half-angle identity is a powerful tool in trigonometry, allowing us to find trigonometric functions of half-angles when we know the trigonometric functions of the full angle. This identity is particularly useful in simplifying expressions and solving equations. The ability to correctly apply this identity, along with accurate substitution and simplification, is crucial for arriving at the correct answer.

Calculating ${\sin^2\left(\frac{\alpha - \beta}{2}\right)}$

Next, we will calculate ${\sin^2\left(\frac{\alpha - \beta}{2}\right)}$. Similar to the previous calculation, we will use the half-angle identity for sine. The relevant form of the half-angle identity is:

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

In this case, ${\theta = \alpha - \beta}$, so we have:

${\sin^2\left(\frac{\alpha - \beta}{2}\right) = \frac{1 - \cos(\alpha - \beta)}{2}}$

We previously found that ${\cos(\alpha - \beta) = \frac{63}{65}}$, so we substitute this value into the equation:

${\sin^2\left(\frac{\alpha - \beta}{2}\right) = \frac{1 - \frac{63}{65}}{2}}$

${\sin^2\left(\frac{\alpha - \beta}{2}\right) = \frac{\frac{65 - 63}{65}}{2}}$

${\sin^2\left(\frac{\alpha - \beta}{2}\right) = \frac{\frac{2}{65}}{2}}$

${\sin^2\left(\frac{\alpha - \beta}{2}\right) = \frac{2}{65} \cdot \frac{1}{2}}$

${\sin^2\left(\frac{\alpha - \beta}{2}\right) = \frac{1}{65}}$

Therefore, we find that ${\sin^2\left(\frac{\alpha - \beta}{2}\right) = \frac{1}{65}}$. This is the second key result we aimed to determine. This calculation further demonstrates the power and utility of the half-angle identities in trigonometry. By correctly applying the half-angle identity for sine and using the previously calculated value of ${\cos(\alpha - \beta)}$, we were able to efficiently find the value of ${\sin^2\left(\frac{\alpha - \beta}{2}\right)}$. This reinforces the importance of understanding and being able to apply these fundamental trigonometric identities.

In this comprehensive exploration, we successfully determined the values of ${\cos^2\left(\frac{\alpha + \beta}{2}\right)}$ and ${\sin^2\left(\frac{\alpha - \beta}{2}\right)}$ given that ${\cos \alpha = \frac{3}{5}}$ and ${\cos \beta = \frac{5}{13}}$, where ${\alpha}$ and ${\beta}$ are acute angles. We started by finding the sine values for both angles using the Pythagorean identity. Then, we calculated ${\cos(\alpha + \beta)}$ and ${\cos(\alpha - \beta)}$ using the angle sum and difference identities. Finally, we applied the half-angle identities to find the desired values. This step-by-step approach highlights the interconnectedness of trigonometric concepts and the importance of mastering fundamental identities. Understanding and applying these principles allows for the efficient and accurate solution of a wide range of trigonometric problems. The journey through this problem reinforces the significance of trigonometric identities as powerful tools in mathematical analysis and problem-solving. By breaking down complex problems into manageable steps and utilizing the appropriate identities, we can navigate the intricacies of trigonometry with confidence and precision.