Trigonometric Calculations Sin B, Sin D, Cos D, And Cos D + Sin D
This article delves into the realm of trigonometry, focusing on calculating trigonometric functions given specific values. We are given that cos B = 6/10 and tan D = 20/21, and our task is to find the values of sin B, sin D, cos D, and the sum cos D + sin D. This exploration will not only reinforce fundamental trigonometric identities but also demonstrate how to apply these concepts to solve practical problems. Trigonometry is a cornerstone of mathematics and physics, essential for understanding angles, triangles, and periodic phenomena. By working through these calculations, we enhance our grasp of these core principles, setting a solid foundation for more advanced studies in these fields. Let's embark on this journey of trigonometric discovery, unraveling the relationships between these functions and their applications.
To determine sin B, we utilize the fundamental Pythagorean trigonometric identity: sin² B + cos² B = 1. Given that cos B = 6/10, we can substitute this value into the identity to solve for sin B. This identity is a cornerstone of trigonometry, derived from the Pythagorean theorem applied to the unit circle. It establishes a fundamental relationship between the sine and cosine of an angle, allowing us to find one if we know the other. Mastering this identity is crucial for tackling a wide range of trigonometric problems. The identity not only helps in finding the sine or cosine of an angle but also serves as a building block for more complex trigonometric equations and proofs. Let's delve into the step-by-step calculation to find the precise value of sin B, ensuring each step is clearly articulated for a thorough understanding.
First, we substitute the given value of cos B into the identity:
sin² B + (6/10)² = 1
Next, we simplify the equation by squaring the fraction:
sin² B + 36/100 = 1
To isolate sin² B, we subtract 36/100 from both sides of the equation:
sin² B = 1 - 36/100
Now, we find a common denominator to combine the terms on the right side:
sin² B = 100/100 - 36/100
Subtracting the fractions gives us:
sin² B = 64/100
To find sin B, we take the square root of both sides of the equation:
sin B = ±√(64/100)
Simplifying the square root, we get:
sin B = ±8/10
Since we are not given any specific quadrant information for angle B, we consider both positive and negative values. However, in many practical contexts, we often consider the positive value for angles in the first quadrant (0 to 90 degrees). Therefore:
sin B = 8/10
Thus, the value of sin B is 8/10, which can be further simplified to 4/5. This calculation highlights the power of the Pythagorean identity in linking sine and cosine, enabling us to navigate trigonometric problems effectively. The ability to manipulate and apply this identity is a critical skill in trigonometry, paving the way for solving more intricate problems involving angles and triangles.
To determine sin D, given tan D = 20/21, we first need to find the hypotenuse of the right-angled triangle associated with angle D. We can use the relationship between tangent, sine, and cosine: tan D = sin D / cos D. However, since we don't know sin D or cos D directly, we use another approach. Recognizing that tangent is the ratio of the opposite side to the adjacent side in a right-angled triangle, we can consider a triangle where the opposite side is 20 and the adjacent side is 21. This understanding of tangent allows us to visualize the triangle and apply the Pythagorean theorem to find the hypotenuse. The Pythagorean theorem is a fundamental concept in geometry, linking the sides of a right-angled triangle, and it is essential for solving numerous trigonometric problems. Let's delve into the step-by-step calculation to find the precise value of sin D, ensuring each step is clearly articulated for a thorough understanding.
Using the Pythagorean theorem (a² + b² = c²), where a and b are the sides and c is the hypotenuse, we have:
c² = 20² + 21²
Squaring the numbers gives us:
c² = 400 + 441
Adding the values, we get:
c² = 841
Taking the square root of both sides, we find the hypotenuse:
c = √841 = 29
Now that we have the hypotenuse, we can find sin D using the definition: sin D = opposite / hypotenuse. In our triangle, the opposite side is 20, and the hypotenuse is 29. Therefore:
sin D = 20/29
Thus, the value of sin D is 20/29. This calculation showcases how we can use the tangent function along with the Pythagorean theorem to derive the sine of an angle. The ability to connect these trigonometric functions and geometric principles is a key skill in trigonometry, allowing us to tackle a variety of problems involving triangles and angles. By mastering these concepts, we can confidently navigate more complex trigonometric scenarios.
To determine cos D, given tan D = 20/21, we again refer to the right-angled triangle we established in part B. We found that the opposite side is 20, the adjacent side is 21, and the hypotenuse is 29. Cosine is defined as the ratio of the adjacent side to the hypotenuse. This definition is a fundamental concept in trigonometry, linking the cosine of an angle to the sides of a right-angled triangle. Understanding this relationship is crucial for solving various trigonometric problems. By applying this definition, we can easily calculate the value of cos D using the side lengths we've already determined. Let's proceed with the calculation to find the precise value of cos D, ensuring each step is clearly articulated for a thorough understanding.
Using the definition cos D = adjacent / hypotenuse, we substitute the values we have:
cos D = 21/29
Thus, the value of cos D is 21/29. This straightforward calculation underscores the importance of understanding the basic definitions of trigonometric functions. Knowing these definitions allows us to quickly and accurately find the values of sine, cosine, and tangent, given the sides of a right-angled triangle. The ability to apply these definitions is a cornerstone of trigonometry, enabling us to solve a wide range of problems involving angles and triangles. By mastering these fundamental concepts, we build a solid foundation for more advanced studies in mathematics and physics.
To find the sum cos D + sin D, we simply add the values we calculated in parts B and C. We found that sin D = 20/29 and cos D = 21/29. Adding these fractions is a straightforward arithmetic operation, but it's essential to ensure we have a common denominator before performing the addition. This simple addition demonstrates how trigonometric values can be combined to solve more complex problems. Understanding how to manipulate and combine these values is crucial for applications in various fields, including physics and engineering. Let's proceed with the calculation to find the precise value of cos D + sin D, ensuring each step is clearly articulated for a thorough understanding.
Adding the values, we have:
cos D + sin D = 21/29 + 20/29
Since the denominators are the same, we can add the numerators directly:
cos D + sin D = (21 + 20) / 29
Simplifying the numerator, we get:
cos D + sin D = 41/29
Thus, the value of cos D + sin D is 41/29. This final calculation demonstrates how individual trigonometric values can be combined to solve more complex expressions. The ability to manipulate and combine trigonometric functions is a critical skill in mathematics, enabling us to tackle a wide range of problems in various fields. By mastering these fundamental concepts, we build a solid foundation for advanced studies and applications.
In summary, we have successfully calculated sin B, sin D, cos D, and the sum cos D + sin D given the initial values of cos B and tan D. These calculations underscore the fundamental principles of trigonometry and the relationships between trigonometric functions. We utilized the Pythagorean identity to find sin B, applied the Pythagorean theorem and trigonometric definitions to find sin D and cos D, and finally, summed cos D and sin D to obtain the final result. This exercise not only reinforces the importance of these trigonometric concepts but also highlights their practical application in solving mathematical problems. Trigonometry is a powerful tool with applications spanning across various disciplines, including physics, engineering, and computer science. By mastering these fundamental concepts, we equip ourselves with the necessary skills to tackle more advanced problems and applications in these fields. The ability to understand and manipulate trigonometric functions is a valuable asset in any scientific or technical endeavor.
