Finding Cos(A-B) Given Cos(A) And Cos(B) And Quadrant Information

In trigonometry, finding the cosine of the difference of two angles, denoted as cos(A - B), is a common task. This problem often arises in various mathematical and scientific contexts, such as physics, engineering, and computer graphics. When given the cosine values of individual angles A and B along with the quadrant information, we can employ trigonometric identities to determine cos(A - B). This article provides a comprehensive guide on how to solve such problems, utilizing the cosine subtraction formula and quadrant rules to arrive at the simplified fractional answer. This exploration will not only enhance your understanding of trigonometric identities but also refine your problem-solving skills in trigonometry. Mastering these techniques is crucial for tackling more complex trigonometric problems and applications in related fields. By the end of this article, you'll be equipped with the knowledge and skills to confidently solve similar trigonometric challenges.

In this problem, we are given that cos(A) = 11/61, with angle A in quadrant I, and cos(B) = 28/53, with angle B in quadrant IV. Our objective is to find the value of cos(A - B) as a simplified fraction. This task requires us to utilize the cosine subtraction formula, which is a fundamental trigonometric identity. The cosine subtraction formula states that cos(A - B) = cos(A)cos(B) + sin(A)sin(B). To apply this formula, we need to find the values of sin(A) and sin(B) in addition to the given cosine values. The quadrant information is crucial here because it helps us determine the signs of the sine values. In quadrant I, both sine and cosine are positive, while in quadrant IV, cosine is positive, and sine is negative. This understanding of quadrant rules is essential for accurately determining the signs of the trigonometric functions, which directly impacts the final result. By carefully considering these details, we can systematically solve the problem and arrive at the correct value of cos(A - B).

1. Find sin(A) using the Pythagorean Identity:

Since we know cos(A) = 11/61 and A is in quadrant I, we can use the Pythagorean identity sin²(A) + cos²(A) = 1 to find sin(A). This identity is a cornerstone of trigonometry, relating the sine and cosine of an angle. First, substitute the given value of cos(A) into the identity: sin²(A) + (11/61)² = 1. Simplifying, we get sin²(A) + 121/3721 = 1. To isolate sin²(A), subtract 121/3721 from both sides, resulting in sin²(A) = 1 - 121/3721 = 3600/3721. Now, take the square root of both sides to find sin(A). Since A is in quadrant I, sin(A) is positive. Therefore, sin(A) = √(3600/3721) = 60/61. This step demonstrates the importance of the Pythagorean identity in finding trigonometric function values and how quadrant information helps determine the correct sign.

2. Find sin(B) using the Pythagorean Identity:

Next, we find sin(B) using a similar approach. We are given cos(B) = 28/53, and B is in quadrant IV. Again, we use the Pythagorean identity sin²(B) + cos²(B) = 1. Substituting the given value of cos(B), we have sin²(B) + (28/53)² = 1. Simplifying, we get sin²(B) + 784/2809 = 1. Subtracting 784/2809 from both sides, we find sin²(B) = 1 - 784/2809 = 2025/2809. Taking the square root of both sides gives us sin(B) = ±√(2025/2809) = ±45/53. Since B is in quadrant IV, where sine is negative, we choose the negative value: sin(B) = -45/53. This step reinforces the application of the Pythagorean identity and highlights the significance of quadrant information in determining the correct sign of trigonometric functions.

3. Apply the Cosine Subtraction Formula:

Now that we have sin(A) = 60/61, cos(A) = 11/61, sin(B) = -45/53, and cos(B) = 28/53, we can use the cosine subtraction formula: cos(A - B) = cos(A)cos(B) + sin(A)sin(B). This formula is crucial for finding the cosine of the difference of two angles. Substitute the values we found into the formula: cos(A - B) = (11/61)(28/53) + (60/61)(-45/53). Multiplying the fractions, we get cos(A - B) = 308/3233 - 2700/3233. Combining the fractions, we have cos(A - B) = (308 - 2700)/3233 = -2392/3233. This step demonstrates the direct application of the cosine subtraction formula and the arithmetic operations required to obtain the final result.

4. Simplify the Fraction:

The final step is to simplify the fraction -2392/3233. To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both by it. The GCD of 2392 and 3233 is 1. Since the GCD is 1, the fraction is already in its simplest form. Therefore, the final answer is cos(A - B) = -2392/3233. This step ensures that the final answer is presented in the most reduced form, which is a standard practice in mathematical problem-solving. The ability to simplify fractions is a fundamental skill that enhances the clarity and precision of the solution.

Therefore, $\cos (A-B) = -\frac{2392}{3233}$

Solving trigonometric problems involving the difference of angles requires a solid understanding of trigonometric identities and quadrant rules. In this article, we tackled the problem of finding cos(A - B) given cos(A) = 11/61 with A in quadrant I, and cos(B) = 28/53 with B in quadrant IV. By systematically applying the Pythagorean identity and the cosine subtraction formula, we were able to arrive at the simplified fraction -2392/3233. The Pythagorean identity, sin²(x) + cos²(x) = 1, is essential for finding the sine of an angle when the cosine is known, or vice versa. This identity, combined with the quadrant rules, allows us to determine the correct signs of trigonometric functions. The cosine subtraction formula, cos(A - B) = cos(A)cos(B) + sin(A)sin(B), is the core formula for this type of problem. It allows us to express the cosine of the difference of two angles in terms of the cosines and sines of the individual angles. Understanding and applying these concepts accurately is crucial for solving a wide range of trigonometric problems. Practice with similar problems will further solidify your understanding and improve your problem-solving skills in trigonometry.

The techniques used to find cos(A - B) have practical applications in various fields. Trigonometric identities and angle subtraction formulas are essential in physics for analyzing wave interference, projectile motion, and alternating current circuits. For instance, in wave interference, understanding how waves combine involves analyzing the phase difference between them, which often requires the use of angle subtraction formulas. In engineering, these concepts are crucial for signal processing, control systems, and structural analysis. Engineers use trigonometric functions to model and analyze oscillatory systems, such as mechanical vibrations and electrical oscillations. In computer graphics, trigonometric functions are fundamental for rotations, transformations, and creating realistic animations. Understanding the relationships between angles and their trigonometric functions is vital for creating smooth and accurate movements in computer-generated images. Furthermore, navigation systems rely heavily on trigonometry for calculating distances and directions. Techniques similar to those used in this article are employed in GPS systems and other navigational tools. Mastering these trigonometric skills not only enhances your mathematical proficiency but also prepares you for a wide range of real-world applications in science, engineering, and technology.