Finding Trigonometric Values Given Csc X = 5 3

In trigonometry, understanding the relationships between trigonometric functions is crucial for solving a variety of problems. This article aims to provide a detailed explanation of how to find the values of other trigonometric functions, specifically sec(x), cos(x), cot(x), and tan(x), when the value of csc(x) is given. We will explore the fundamental trigonometric identities, apply these identities to the given problem, and discuss the step-by-step process of finding the required values. Additionally, we will delve into the importance of the given domain, $0^{\circ} < x < 90^{\circ}$, and its implications on the signs of the trigonometric functions. This comprehensive guide will not only help you solve this particular problem but also equip you with the knowledge to tackle similar problems in trigonometry.

The given problem states that csc(x) = 5/3, where x is an angle between 0° and 90°. Our goal is to find the values of the remaining trigonometric functions: sec(x), cos(x), cot(x), and tan(x). The domain $0^{\circ} < x < 90^{\circ}$ is significant because it restricts the angle x to the first quadrant. In the first quadrant, all trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) are positive. This information is crucial as it helps us determine the correct signs of the trigonometric values we are about to calculate. Understanding the relationship between trigonometric functions and their reciprocals is fundamental to solving this problem. Cosecant (csc) is the reciprocal of sine (sin), secant (sec) is the reciprocal of cosine (cos), and cotangent (cot) is the reciprocal of tangent (tan). These reciprocal relationships are key to navigating trigonometric identities and finding unknown values. The ability to visualize these relationships and apply them correctly is a cornerstone of trigonometric problem-solving.

Since csc(x) is the reciprocal of sin(x), we can easily find sin(x) by taking the reciprocal of the given value of csc(x). Given that csc(x) = 5/3, we have:

sin(x) = 1 / csc(x) = 1 / (5/3) = 3/5

This is a fundamental step, as it provides a starting point for finding other trigonometric values. The relationship between sine and cosecant is a direct reciprocal relationship, making this step straightforward. This calculation exemplifies the importance of understanding reciprocal identities in trigonometry. Knowing that sin(x) is the opposite side divided by the hypotenuse in a right-angled triangle, we can visualize a triangle where the opposite side is 3 and the hypotenuse is 5. This visualization can aid in understanding the relationships between different sides and angles, and it lays the groundwork for using the Pythagorean theorem in subsequent steps. By establishing sin(x), we are one step closer to unraveling the other trigonometric functions associated with angle x. Mastering this reciprocal relationship is essential for further trigonometric problem-solving.

To find cos(x), we can use the Pythagorean identity, which states:

sin²(x) + cos²(x) = 1

We already know sin(x) = 3/5, so we can substitute this value into the identity:

(3/5)² + cos²(x) = 1

9/25 + cos²(x) = 1

Subtracting 9/25 from both sides, we get:

cos²(x) = 1 - 9/25 = 16/25

Taking the square root of both sides:

cos(x) = ±√(16/25) = ±4/5

Since x is in the first quadrant ($0^{\circ} < x < 90^{\circ}$), cos(x) is positive. Therefore:

cos(x) = 4/5

The use of the Pythagorean identity is a cornerstone of trigonometric calculations. It allows us to relate sine and cosine, providing a pathway to find one if we know the other. This step demonstrates the power of these fundamental identities in simplifying complex trigonometric problems. By substituting the value of sin(x), we were able to isolate cos²(x) and subsequently find cos(x). The crucial consideration of the quadrant ensures that we choose the correct sign for cos(x). In the first quadrant, all trigonometric functions are positive, hence the choice of the positive root. This process highlights the importance of not just applying the identities but also understanding the context in which they are being applied. This level of understanding is vital for accurate and confident trigonometric problem-solving.

Secant (sec) is the reciprocal of cosine (cos). Since we have found that cos(x) = 4/5, we can find sec(x) by taking the reciprocal:

sec(x) = 1 / cos(x) = 1 / (4/5) = 5/4

This step reinforces the reciprocal relationships between trigonometric functions. Secant and cosine are inversely related, so finding sec(x) is a straightforward process once cos(x) is known. This direct application of the reciprocal relationship underscores the importance of memorizing these fundamental connections. The calculation here is simple but crucial, as it directly leverages the previously calculated value of cos(x). By understanding and applying this reciprocal identity, we efficiently determine the value of sec(x). This step further illustrates how interconnected trigonometric functions are and how knowing one value can lead to the determination of others. Mastery of these reciprocal relationships enhances one's ability to navigate trigonometric problems effectively.

Tangent (tan) can be found using the identity:

tan(x) = sin(x) / cos(x)

We know sin(x) = 3/5 and cos(x) = 4/5, so we substitute these values:

tan(x) = (3/5) / (4/5) = 3/4

Alternatively, one can visualize a right-angled triangle and understand that tan(x) is the ratio of the opposite side to the adjacent side. The identity tan(x) = sin(x) / cos(x) is a fundamental relationship in trigonometry, and this step directly applies this identity. By substituting the known values of sin(x) and cos(x), we easily calculate tan(x). This calculation underscores the power of these fundamental identities in connecting different trigonometric functions. The result, tan(x) = 3/4, provides another piece of the puzzle, illustrating how trigonometric functions are intertwined. The ability to remember and apply these identities efficiently is crucial for solving trigonometric problems. Understanding the geometrical interpretation of tangent as the ratio of sides in a right-angled triangle can also provide a visual aid in remembering and applying this function.

Cotangent (cot) is the reciprocal of tangent (tan). Since we found that tan(x) = 3/4, we can find cot(x) by taking the reciprocal:

cot(x) = 1 / tan(x) = 1 / (3/4) = 4/3

This final step demonstrates the consistent application of reciprocal relationships in trigonometry. The connection between cotangent and tangent is a direct inverse relationship, making this calculation simple and straightforward. Knowing that cotangent is the reciprocal of tangent allows for a quick determination of its value once tan(x) is known. This step emphasizes the importance of memorizing and understanding these reciprocal relationships. The result, cot(x) = 4/3, completes the set of trigonometric values for the given angle x. This calculation reinforces the interconnectedness of trigonometric functions and the ease with which values can be found once key relationships are understood. The ability to efficiently apply reciprocal identities is a hallmark of proficiency in trigonometry.

Given csc(x) = 5/3 and $0^{\circ} < x < 90^{\circ}$, we have found the following values:

• sin(x) = 3/5
• cos(x) = 4/5
• sec(x) = 5/4
• tan(x) = 3/4
• cot(x) = 4/3

This summary consolidates all the calculated values, providing a clear overview of the solution. Each trigonometric function's value has been determined step-by-step, highlighting the interconnectedness of these functions. This comprehensive solution demonstrates the importance of understanding reciprocal and Pythagorean identities in trigonometric problem-solving. The consistent application of these fundamental principles allows for the efficient determination of unknown trigonometric values. This summary serves as a testament to the power of trigonometric identities in simplifying complex calculations. The clarity and completeness of the solution underscore the importance of a systematic approach to trigonometric problems.

In conclusion, we have successfully found the values of sec(x), cos(x), cot(x), and tan(x), given that csc(x) = 5/3 and $0^{\circ} < x < 90^{\circ}$. This process involved understanding the reciprocal relationships between trigonometric functions and applying the Pythagorean identity. The steps included finding sin(x), using the Pythagorean identity to find cos(x), and then using reciprocal identities to find sec(x), tan(x), and cot(x). This example illustrates the importance of understanding and applying fundamental trigonometric identities and relationships. The ability to navigate these identities and relationships is crucial for solving a wide range of trigonometric problems. Mastering these concepts not only allows for the successful solution of problems like this but also builds a strong foundation for more advanced topics in mathematics and physics. The systematic approach demonstrated in this article can be applied to various trigonometric problems, making it a valuable tool for students and practitioners alike. The knowledge gained from solving this problem enhances one's overall understanding of trigonometric functions and their applications.