Trigonometric Functions Evaluating Sine And Cosine Values For 1/2 And -1/2

Trigonometric functions play a crucial role in various fields, including mathematics, physics, and engineering. Understanding their values at specific angles is essential for solving many problems. In this article, we will explore the values of sine and cosine functions at different angles and determine which of the given expressions are equal to ${\frac{1}{2}}$ or ${-\frac{1}{2}}$. We will analyze each trigonometric function individually, providing a detailed explanation for each evaluation. Trigonometric functions, such as sine and cosine, form the bedrock of numerous mathematical and scientific applications. Accurately determining their values at specific angles is paramount for problem-solving in fields ranging from physics to engineering. This article embarks on an exploration of these values, focusing particularly on sine and cosine functions, and aims to identify which of the provided expressions yield either ${\frac{1}{2}}$ or ${-\frac{1}{2}}$. We will conduct a thorough, step-by-step analysis of each function, ensuring a clear and comprehensive understanding of the evaluations. Mastering these concepts not only enhances one's grasp of trigonometry but also builds a strong foundation for more advanced mathematical studies and real-world applications, where precision in calculations involving angles and trigonometric ratios is often crucial. This detailed approach will empower readers to confidently tackle similar problems and appreciate the significance of trigonometric functions in various disciplines.

Evaluating ${\sin\left(\frac{7 \pi}{6}\right)}$

To evaluate ${\sin\left(\frac{7 \pi}{6}\right)}$, we need to understand the unit circle and the properties of the sine function. The sine function represents the y-coordinate of a point on the unit circle corresponding to the given angle. The angle ${\frac{7 \pi}{6}}$ is in the third quadrant, where both x and y coordinates are negative. We can express ${\frac{7 \pi}{6}}$ as ${\pi + \frac{\pi}{6}}$. The reference angle in the first quadrant is ${\frac{\pi}{6}}$, and we know that ${\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}}$. Since the sine function is negative in the third quadrant, we have:

${ sin\left(\frac{7 \pi}{6}\right) = -\sin\left(\frac{\pi}{6}\right) = -\frac{1}{2} }$

Therefore, ${\sin\left(\frac{7 \pi}{6}\right)}$ is equal to ${-\frac{1}{2}}$. The process of evaluating trigonometric functions like ${\sin\left(\frac{7 \pi}{6}\right)}$ involves several key steps, each crucial for arriving at the correct result. First and foremost, understanding the unit circle is fundamental. The unit circle provides a visual and conceptual framework for grasping the values of trigonometric functions at various angles. It helps us relate angles to coordinates on a circle with a radius of 1, centered at the origin. The sine function, in particular, corresponds to the y-coordinate of a point on this circle. Knowing this relationship allows us to determine the sine value for any given angle. For the angle ${\frac{7 \pi}{6}}$, we recognize that it lies in the third quadrant. This is a critical observation because the signs of trigonometric functions vary across quadrants. In the third quadrant, both the x and y coordinates are negative, which means that sine (y-coordinate) is negative in this region. Next, we express the angle ${\frac{7 \pi}{6}}$ in terms of its reference angle. The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. Expressing ${\frac{7 \pi}{6}}$ as ${\pi + \frac{\pi}{6}}$ helps us identify that ${\frac{\pi}{6}}$ is the reference angle. Knowing the reference angle is crucial because the trigonometric function values for an angle and its reference angle are the same in magnitude, differing only in sign depending on the quadrant. We recall the sine value of the reference angle, ${\frac{\pi}{6}}$, which is ${\frac{1}{2}}$. This is a standard trigonometric value that is often memorized or can be easily derived from a 30-60-90 triangle. Finally, considering the sign of the sine function in the third quadrant, we conclude that ${\sin\left(\frac{7 \pi}{6}\right)}$ is negative. Therefore, the value is ${-\frac{1}{2}}$. This step-by-step process not only provides the correct answer but also reinforces the underlying principles of trigonometry, such as the unit circle, reference angles, and the signs of trigonometric functions in different quadrants. By mastering these concepts, one can confidently evaluate trigonometric functions for various angles and apply this knowledge to more complex problems in mathematics and other fields.

Analyzing ${\sin\left(\frac{4 \pi}{3}\right)}$

For ${\sin\left(\frac{4 \pi}{3}\right)}$, the angle ${\frac{4 \pi}{3}}$ is also in the third quadrant. We can express ${\frac{4 \pi}{3}}$ as ${\pi + \frac{\pi}{3}}$. The reference angle is ${\frac{\pi}{3}}$, and ${\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}}$. Since sine is negative in the third quadrant:

${ sin\left(\frac{4 \pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2} }$

Thus, ${\sin\left(\frac{4 \pi}{3}\right)}$ is not equal to ${\frac{1}{2}}$ or ${-\frac{1}{2}}$. Analyzing ${\sin\left(\frac{4 \pi}{3}\right)}$ involves a similar approach to the previous evaluation, emphasizing the importance of understanding the unit circle, quadrant signs, and reference angles. The initial step is to recognize that the angle ${\frac{4 \pi}{3}}$ lies in the third quadrant. This is crucial because, as mentioned earlier, the signs of trigonometric functions vary depending on the quadrant. In the third quadrant, the sine function is negative, a critical piece of information for determining the final value. To find the value of ${\sin\left(\frac{4 \pi}{3}\right)}$, we express the angle in terms of its reference angle. By rewriting ${\frac{4 \pi}{3}}$ as ${\pi + \frac{\pi}{3}}$, we identify ${\frac{\pi}{3}}$ as the reference angle. The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis, and it helps simplify the evaluation by relating the trigonometric value of the angle to the trigonometric value of its reference angle. We then recall the sine value of the reference angle, ${\frac{\pi}{3}}$. This is a standard trigonometric value that is typically memorized or can be easily derived from a 30-60-90 triangle. The sine of ${\frac{\pi}{3}}$ is ${\frac{\sqrt{3}}{2}}$. However, since we know that sine is negative in the third quadrant, we must apply the appropriate sign. Therefore, ${\sin\left(\frac{4 \pi}{3}\right)}$ is equal to the negative of ${\sin\left(\frac{\pi}{3}\right)}$, which gives us ${-\frac{\sqrt{3}}{2}}$. This result clearly shows that ${\sin\left(\frac{4 \pi}{3}\right)}$ is not equal to either ${\frac{1}{2}}$ or ${-\frac{1}{2}}$. The evaluation underscores the significance of paying close attention to the quadrant in which the angle lies and using the reference angle to simplify the calculation. These steps are essential for accurate trigonometric evaluations and are applicable to a wide range of problems involving angles and trigonometric functions.

Investigating ${\cos\left(\frac{5 \pi}{3}\right)}$

Now, let's evaluate ${\cos\left(\frac{5 \pi}{3}\right)}$. The angle ${\frac{5 \pi}{3}}$ is in the fourth quadrant, where the x-coordinate is positive, and the y-coordinate is negative. The cosine function represents the x-coordinate on the unit circle. We can express ${\frac{5 \pi}{3}}$ as ${2 \pi - \frac{\pi}{3}}$. The reference angle is ${\frac{\pi}{3}}$, and ${\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}}$. Since cosine is positive in the fourth quadrant:

${ cos\left(\frac{5 \pi}{3}\right) = cos\left(\frac{\pi}{3}\right) = \frac{1}{2} }$

Hence, ${\cos\left(\frac{5 \pi}{3}\right)}$ is equal to ${\frac{1}{2}}$. Investigating ${\cos\left(\frac{5 \pi}{3}\right)}$ involves a similar methodical approach, focusing on the unit circle and the properties of the cosine function. The first crucial step is to identify the quadrant in which the angle ${\frac{5 \pi}{3}}$ lies. Recognizing that ${\frac{5 \pi}{3}}$ is in the fourth quadrant is essential because it determines the sign of the cosine function. In the fourth quadrant, the x-coordinate is positive, which means that cosine is positive in this region. This understanding is vital for arriving at the correct final value. The cosine function, by definition, represents the x-coordinate of a point on the unit circle corresponding to the given angle. This fundamental concept is key to evaluating cosine values. To determine the value of ${\cos\left(\frac{5 \pi}{3}\right)}$, we express the angle in terms of its reference angle. By rewriting ${\frac{5 \pi}{3}}$ as ${2 \pi - \frac{\pi}{3}}$, we can identify ${\frac{\pi}{3}}$ as the reference angle. The reference angle simplifies the evaluation by relating the cosine value of the angle to the cosine value of its reference angle. The cosine of the reference angle, ${\frac{\pi}{3}}$, is a standard trigonometric value that is typically memorized or can be easily derived from a 30-60-90 triangle. We know that ${\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}}$. Since we have already established that cosine is positive in the fourth quadrant, the value of ${\cos\left(\frac{5 \pi}{3}\right)}$ is the same as the cosine of its reference angle. Therefore, ${\cos\left(\frac{5 \pi}{3}\right) = \frac{1}{2}}$. This result confirms that ${\cos\left(\frac{5 \pi}{3}\right)}$ is indeed equal to ${\frac{1}{2}}$. The analysis highlights the importance of understanding the unit circle, identifying the correct quadrant, and using reference angles to simplify trigonometric evaluations. These skills are crucial for accurately determining trigonometric values and for solving more complex problems in mathematics and related fields. The step-by-step approach ensures clarity and precision in the evaluation process.

Calculating ${\cos\left(\frac{\pi}{6}\right)}$

Next, let's calculate ${\cos\left(\frac{\pi}{6}\right)}$. The angle ${\frac{\pi}{6}}$ is in the first quadrant, where both x and y coordinates are positive. The cosine of ${\frac{\pi}{6}}$ is a standard value that can be derived from a 30-60-90 triangle. In such a triangle, the cosine of the 30-degree angle (or ${\frac{\pi}{6}}$ radians) is the ratio of the adjacent side to the hypotenuse, which is ${\frac{\sqrt{3}}{2}}$:

${ cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} }$

Thus, ${\cos\left(\frac{\pi}{6}\right)}$ is not equal to ${\frac{1}{2}}$ or ${-\frac{1}{2}}$. Calculating ${\cos\left(\frac{\pi}{6}\right)}$ is a fundamental exercise in trigonometry that highlights the importance of recognizing standard angles and their associated trigonometric values. The angle ${\frac{\pi}{6}}$, which is equivalent to 30 degrees, is a common angle encountered in various mathematical and scientific contexts. Its trigonometric values are well-defined and can be derived from geometric principles. Since ${\frac{\pi}{6}}$ lies in the first quadrant, both its sine and cosine values are positive. This is because in the first quadrant, both the x and y coordinates on the unit circle are positive, corresponding to the cosine and sine functions, respectively. The most straightforward way to calculate ${\cos\left(\frac{\pi}{6}\right)}$ is by considering a 30-60-90 triangle. This special right triangle has angles of 30 degrees (${\frac{\pi}{6}}$ radians), 60 degrees (${\frac{\pi}{3}}$ radians), and 90 degrees (${\frac{\pi}{2}}$ radians), and its sides have a specific ratio that allows us to easily determine the trigonometric values. In a 30-60-90 triangle, the side lengths are in the ratio 1:${\sqrt{3}}$:2, where 1 is the length of the side opposite the 30-degree angle, ${\sqrt{3}}$ is the length of the side opposite the 60-degree angle, and 2 is the length of the hypotenuse. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. For the 30-degree angle (${\frac{\pi}{6}}$ radians), the adjacent side has a length of ${\sqrt{3}}$, and the hypotenuse has a length of 2. Therefore, ${\cos\left(\frac{\pi}{6}\right)}$ is equal to ${\frac{\sqrt{3}}{2}}$. This value is a standard trigonometric value that is often memorized or quickly recalled in mathematical calculations. The result clearly shows that ${\cos\left(\frac{\pi}{6}\right)}$ is not equal to either ${\frac{1}{2}}$ or ${-\frac{1}{2}}$. The calculation reinforces the connection between geometry and trigonometry, demonstrating how the properties of special right triangles can be used to find trigonometric values of specific angles. This approach is fundamental for understanding and applying trigonometry in various fields.

Evaluating ${\cos\left(\frac{5 \pi}{6}\right)}$

Finally, let's evaluate ${\cos\left(\frac{5 \pi}{6}\right)}$. The angle ${\frac{5 \pi}{6}}$ is in the second quadrant, where the x-coordinate is negative, and the y-coordinate is positive. We can express ${\frac{5 \pi}{6}}$ as ${\pi - \frac{\pi}{6}}$. The reference angle is ${\frac{\pi}{6}}$, and we know ${\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}}$. Since cosine is negative in the second quadrant:

${ cos\left(\frac{5 \pi}{6}\right) = -cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} }$

Thus, ${\cos\left(\frac{5 \pi}{6}\right)}$ is not equal to ${\frac{1}{2}}$ or ${-\frac{1}{2}}$. Evaluating ${\cos\left(\frac{5 \pi}{6}\right)}$ requires a thorough understanding of the unit circle, quadrant signs, and reference angles, similar to the other trigonometric evaluations we have performed. The first critical step is to identify that the angle ${\frac{5 \pi}{6}}$ lies in the second quadrant. This is essential because the sign of the cosine function in the second quadrant is negative, a key factor in determining the final value. In the second quadrant, the x-coordinate is negative, which directly corresponds to the cosine function's value. To evaluate ${\cos\left(\frac{5 \pi}{6}\right)}$, we express the angle in terms of its reference angle. Rewriting ${\frac{5 \pi}{6}}$ as ${\pi - \frac{\pi}{6}}$ allows us to identify ${\frac{\pi}{6}}$ as the reference angle. The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis, simplifying the evaluation by relating the trigonometric value of the angle to that of its reference angle. We recall the cosine value of the reference angle, ${\frac{\pi}{6}}$, which is a standard trigonometric value. As we discussed earlier, ${\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}}$. This value is derived from the properties of a 30-60-90 triangle, where the cosine of the 30-degree angle (${\frac{\pi}{6}}$ radians) is the ratio of the adjacent side to the hypotenuse. Given that cosine is negative in the second quadrant, we apply the negative sign to the cosine value of the reference angle. Therefore, ${\cos\left(\frac{5 \pi}{6}\right)}$ is equal to the negative of ${\cos\left(\frac{\pi}{6}\right)}$, which gives us ${-\frac{\sqrt{3}}{2}}$. This result clearly demonstrates that ${\cos\left(\frac{5 \pi}{6}\right)}$ is not equal to either ${\frac{1}{2}}$ or ${-\frac{1}{2}}$. The evaluation emphasizes the importance of considering the quadrant in which the angle lies and using the reference angle to simplify the calculation. These steps are crucial for accurate trigonometric evaluations and are applicable to a wide range of problems involving angles and trigonometric functions. The step-by-step approach ensures clarity and precision in the evaluation process, reinforcing the fundamental principles of trigonometry.

Conclusion

In conclusion, after evaluating all the given trigonometric expressions, we find that only ${\sin\left(\frac{7 \pi}{6}\right)}$ and ${\cos\left(\frac{5 \pi}{3}\right)}$ are equal to ${-\frac{1}{2}}$ and ${\frac{1}{2}}$, respectively. This exercise demonstrates the importance of understanding the unit circle, reference angles, and the signs of trigonometric functions in different quadrants. The ability to accurately evaluate trigonometric functions is fundamental in many areas of mathematics and its applications. This comprehensive exploration underscores the critical role that trigonometric functions play in various mathematical and scientific domains. Through a detailed evaluation of several expressions, we have identified that ${\sin\left(\frac{7 \pi}{6}\right)}$ equals ${-\frac{1}{2}}$, and ${\cos\left(\frac{5 \pi}{3}\right)}$ equals ${\frac{1}{2}}$. These findings not only answer the specific question posed but also highlight the broader principles of trigonometric evaluations. The unit circle, with its representation of angles and corresponding trigonometric values, serves as a cornerstone for understanding these functions. By visualizing angles on the unit circle, one can readily determine the signs and magnitudes of sine and cosine values. Reference angles further simplify the process, allowing us to relate trigonometric values of angles in different quadrants to those in the first quadrant, where values are more easily memorized or calculated. Moreover, the signs of trigonometric functions in each quadrant are crucial for accurate evaluations. For instance, sine is positive in the first and second quadrants and negative in the third and fourth, while cosine is positive in the first and fourth quadrants and negative in the second and third. Mastery of these concepts is not just an academic exercise; it is a fundamental skill that underpins many applications of mathematics. From physics, where trigonometric functions are used to describe oscillations and waves, to engineering, where they are essential for analyzing structures and circuits, the ability to accurately evaluate trigonometric functions is indispensable. Therefore, a thorough understanding of these functions, coupled with the ability to apply them in various contexts, is a valuable asset for students and professionals alike. This exploration serves as a testament to the enduring importance of trigonometry and its pervasive influence across various scientific and technical disciplines. The skills and insights gained from this analysis will undoubtedly contribute to a deeper appreciation of mathematics and its practical applications.

Final Answer: The final answer is:

• ${\sin\left(\frac{7 \pi}{6}\right)}$
• ${\cos\left(\frac{5 \pi}{3}\right)}$