Finding Cos(x) And Tan(x) Given Sin(x) = 5/13
In this article, we will explore how to determine the values of cos(x) and tan(x) given that sin(x) = 5/13 and the angle x lies within the interval 0 < x < π/2. This problem falls under the domain of trigonometry, specifically dealing with trigonometric identities and the relationships between different trigonometric functions. We will utilize the fundamental Pythagorean trigonometric identity, sin²(x) + cos²(x) = 1, to find the value of cos(x). Subsequently, we will use the definition of the tangent function, tan(x) = sin(x) / cos(x), to calculate tan(x). The given interval for x helps us determine the sign of cos(x), as cosine is positive in the first quadrant (0 < x < π/2).
To find cos(x), we can use the fundamental trigonometric identity:
sin²(x) + cos²(x) = 1
We are given that sin(x) = 5/13. Substituting this value into the identity, we get:
(5/13)² + cos²(x) = 1
25/169 + cos²(x) = 1
Now, we need to isolate cos²(x). Subtract 25/169 from both sides of the equation:
cos²(x) = 1 - 25/169
To subtract the fractions, we need a common denominator, which is 169. So, we rewrite 1 as 169/169:
cos²(x) = 169/169 - 25/169
cos²(x) = (169 - 25) / 169
cos²(x) = 144/169
Now, we take the square root of both sides to find cos(x):
cos(x) = ±√(144/169)
cos(x) = ±(√144 / √169)
cos(x) = ±(12/13)
Since we are given that 0 < x < π/2, which means x lies in the first quadrant, where cosine is positive, we choose the positive value:
cos(x) = 12/13
Therefore, the value of cos(x) is 12/13. This result is consistent with the Pythagorean identity and the given interval for x. The cosine function represents the x-coordinate of a point on the unit circle, and in the first quadrant, the x-coordinate is always positive. Thus, our choice of the positive root is justified. Understanding the relationship between trigonometric functions and the unit circle is crucial for solving problems like this, as it provides a visual and geometric interpretation of the values.
Now that we have found cos(x), we can find tan(x) using the definition of the tangent function:
tan(x) = sin(x) / cos(x)
We are given that sin(x) = 5/13, and we found that cos(x) = 12/13. Substituting these values into the equation, we get:
tan(x) = (5/13) / (12/13)
To divide fractions, we multiply by the reciprocal of the divisor:
tan(x) = (5/13) * (13/12)
Notice that the 13 in the numerator and the 13 in the denominator cancel out:
tan(x) = 5/12
Therefore, the value of tan(x) is 5/12. This result is consistent with the definitions of sine, cosine, and tangent in a right-angled triangle. In the first quadrant, all trigonometric functions (sine, cosine, and tangent) are positive, which aligns with our calculated value for tan(x). The tangent function represents the ratio of the opposite side to the adjacent side in a right-angled triangle, and in this case, the ratio is 5/12. Understanding the geometric interpretations of trigonometric functions helps in visualizing and solving trigonometric problems effectively.
In summary, given that sin(x) = 5/13 and 0 < x < π/2, we have found the following:
- cos(x) = 12/13
- tan(x) = 5/12
These solutions were obtained by applying the fundamental Pythagorean trigonometric identity and the definition of the tangent function. The given interval for x played a crucial role in determining the correct sign for cos(x). Understanding trigonometric identities and the relationships between trigonometric functions is essential for solving various problems in mathematics, physics, and engineering. The ability to manipulate these identities and apply them in different contexts is a fundamental skill for anyone working with trigonometry. Furthermore, the geometric interpretation of trigonometric functions in the unit circle and right-angled triangles provides a deeper understanding of their behavior and properties.
This article demonstrated how to find cos(x) and tan(x) given the value of sin(x) and an interval for x. By using the Pythagorean identity and the definition of the tangent function, we were able to determine the values of these trigonometric functions. The problem highlights the importance of understanding trigonometric identities and their applications. It also emphasizes the significance of considering the given interval for the angle when determining the signs of trigonometric functions. Trigonometry is a fundamental branch of mathematics with applications in various fields, and mastering its concepts is crucial for success in these areas. The methods and techniques discussed in this article can be applied to solve a wide range of trigonometric problems, making it a valuable resource for students and professionals alike. The key takeaway is the interconnectedness of trigonometric functions and the power of trigonometric identities in solving problems.
