Solving For Tan(x+y) Given Cos X And Cos Y Trigonometry Problem
In trigonometry, determining the values of trigonometric functions for the sum or difference of angles is a fundamental concept. This article delves into a specific problem where we are given the cosine values of two acute angles, x and y, and are tasked with finding the tangent of their sum, tan(x + y). We will explore the necessary trigonometric identities, step-by-step calculations, and the underlying principles that govern this type of problem. This article aims to provide a comprehensive understanding of how to solve such problems and the reasoning behind each step.
Problem Statement
Given that cos x = 4/5 and cos y = 24/25, where x and y are both acute angles, what is the value of tan(x + y)? This problem requires us to use trigonometric identities to find the value of tan(x + y) based on the given information. The key here is to understand and apply the compound angle formula for the tangent function and to derive the necessary sine and tangent values for angles x and y from their cosine values.
Understanding the Fundamentals of Trigonometry
Before diving into the solution, it's essential to recap some fundamental trigonometric concepts. Trigonometric functions, such as sine (sin), cosine (cos), and tangent (tan), relate the angles of a right-angled triangle to the ratios of its sides. The relationships are defined as follows:
- sin(θ) = Opposite / Hypotenuse
- cos(θ) = Adjacent / Hypotenuse
- tan(θ) = Opposite / Adjacent
These definitions form the bedrock of trigonometry, allowing us to analyze and solve a wide range of problems involving angles and triangles. Moreover, the Pythagorean identity, sin²(θ) + cos²(θ) = 1, is a cornerstone in establishing relationships between sine and cosine, enabling the determination of one if the other is known. This identity is particularly crucial in solving problems like the one we're addressing.
The Importance of Acute Angles
In the problem statement, it's mentioned that x and y are acute angles. An acute angle is an angle that measures less than 90 degrees. This piece of information is crucial because it restricts the possible values of trigonometric functions. In the first quadrant (0 to 90 degrees), all trigonometric functions (sine, cosine, and tangent) are positive. This simplifies our calculations as we don't need to worry about the signs of the trigonometric values, which would be a concern if the angles were in other quadrants.
Compound Angle Formulas
Compound angle formulas are trigonometric identities that express trigonometric functions of sums or differences of angles in terms of trigonometric functions of those angles individually. For the tangent function, the compound angle formula for the sum of two angles is:
tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x)tan(y))
This formula is pivotal in solving our problem because it allows us to express tan(x + y) in terms of tan(x) and tan(y), which we can derive from the given cosine values.
Step-by-Step Solution
1. Finding sin(x) and sin(y) Using the Pythagorean Identity
We are given cos x = 4/5 and cos y = 24/25. To use the compound angle formula for tangent, we first need to find sin(x) and sin(y). We can use the Pythagorean identity, sin²(θ) + cos²(θ) = 1.
For angle x:
sin²(x) + (4/5)² = 1 sin²(x) + 16/25 = 1 sin²(x) = 1 - 16/25 sin²(x) = 9/25 sin(x) = ±√(9/25) sin(x) = ±3/5
Since x is an acute angle, sin(x) is positive. Therefore, sin(x) = 3/5.
For angle y:
sin²(y) + (24/25)² = 1 sin²(y) + 576/625 = 1 sin²(y) = 1 - 576/625 sin²(y) = 49/625 sin(y) = ±√(49/625) sin(y) = ±7/25
Similarly, since y is an acute angle, sin(y) is positive. Therefore, sin(y) = 7/25.
2. Calculating tan(x) and tan(y)
Now that we have sin(x), cos(x), sin(y), and cos(y), we can calculate tan(x) and tan(y) using the identity tan(θ) = sin(θ) / cos(θ).
For angle x:
tan(x) = sin(x) / cos(x) tan(x) = (3/5) / (4/5) tan(x) = 3/4
For angle y:
tan(y) = sin(y) / cos(y) tan(y) = (7/25) / (24/25) tan(y) = 7/24
3. Applying the Compound Angle Formula for tan(x + y)
Now we have all the necessary components to find tan(x + y). Using the formula:
tan(x + y) = (tan(x) + tan(y)) / (1 - tan(x)tan(y))
Substitute the values of tan(x) and tan(y):
tan(x + y) = ((3/4) + (7/24)) / (1 - (3/4)(7/24))
4. Simplifying the Expression
First, simplify the numerator:
(3/4) + (7/24) = (18/24) + (7/24) = 25/24
Next, simplify the denominator:
1 - (3/4)(7/24) = 1 - (21/96) = 1 - (7/32) = (32/32) - (7/32) = 25/32
Now, divide the numerator by the denominator:
tan(x + y) = (25/24) / (25/32) tan(x + y) = (25/24) * (32/25)
The 25s cancel out:
tan(x + y) = 32/24
Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 8:
tan(x + y) = (32 ÷ 8) / (24 ÷ 8) tan(x + y) = 4/3
Therefore, the value of tan(x + y) is 4/3.
Alternative Approaches and Insights
While the step-by-step solution provides a clear and methodical approach, it's worth noting alternative methods and insights that can deepen our understanding. One such method involves directly using right-angled triangles to visualize and compute the trigonometric ratios.
Using Right-Angled Triangles
Since x and y are acute angles, we can represent them within right-angled triangles. For angle x, given cos x = 4/5, we can construct a right-angled triangle where the adjacent side is 4 and the hypotenuse is 5. Using the Pythagorean theorem (a² + b² = c²), the opposite side can be calculated as √(5² - 4²) = √9 = 3. This directly gives us sin x = 3/5 and tan x = 3/4.
Similarly, for angle y, given cos y = 24/25, we construct a right-angled triangle with the adjacent side as 24 and the hypotenuse as 25. The opposite side is √(25² - 24²) = √49 = 7, yielding sin y = 7/25 and tan y = 7/24. This visual approach reinforces the fundamental definitions of trigonometric functions and can be particularly helpful for learners who benefit from geometric representations.
Implications and Applications
The ability to compute trigonometric functions of compound angles has wide-ranging applications in various fields, including physics, engineering, and computer graphics. For example, in physics, understanding how angles combine is crucial in analyzing projectile motion, wave interference, and oscillations. In engineering, particularly in structural and mechanical design, determining resultant forces and displacements often involves trigonometric calculations with compound angles. In computer graphics, rotations and transformations are frequently represented using trigonometric functions, making compound angle formulas essential for manipulating objects in 3D space.
Common Mistakes and How to Avoid Them
When solving trigonometric problems, several common mistakes can lead to incorrect answers. Recognizing these pitfalls and understanding how to avoid them is vital for mastering the subject. Here are some frequent errors and strategies to prevent them:
Sign Errors
One of the most common mistakes is making sign errors, particularly when dealing with angles in different quadrants. Remember that the signs of trigonometric functions vary depending on the quadrant. For example, sine is positive in the first and second quadrants but negative in the third and fourth. Cosine is positive in the first and fourth quadrants but negative in the second and third. Tangent is positive in the first and third quadrants and negative in the second and fourth. Always consider the quadrant in which the angle lies and ensure the trigonometric function has the correct sign. In our problem, since both angles are acute (less than 90 degrees), all trigonometric functions are positive, simplifying the calculations.
Incorrectly Applying Trigonometric Identities
Another common mistake is misapplying trigonometric identities. For example, incorrectly using the compound angle formula or the Pythagorean identity can lead to wrong results. It’s essential to memorize the identities correctly and understand when and how to apply them. Practice using these identities in various problems to reinforce your understanding.
Arithmetic Errors
Arithmetic errors, such as mistakes in fractions or simplification, can also lead to incorrect answers. Careless mistakes during calculations can easily occur, especially when dealing with complex expressions. Double-check each step of your calculations, and if possible, use a calculator to verify your results. Organizing your work clearly and writing down each step can also help minimize errors.
Not Simplifying the Final Answer
Sometimes, students arrive at the correct answer but fail to simplify it to its simplest form. For example, a fraction should be reduced to its lowest terms. In our solution, we simplified 32/24 to 4/3. Always ensure your final answer is simplified as much as possible.
Overcomplicating the Problem
Some students may try to overcomplicate the problem by introducing unnecessary steps or methods. It’s crucial to choose the most efficient approach and avoid making the problem more complex than it needs to be. Start by identifying the given information and the desired outcome, and then select the appropriate trigonometric identities and formulas to solve the problem systematically.
Practice Problems
To solidify your understanding, working through additional practice problems is essential. Here are a few problems similar to the one we solved, which you can try on your own:
- Given cos a = 12/13 and cos b = 3/5, where a and b are acute angles, find tan(a + b).
- If sin x = 5/13 and sin y = 8/17, where x and y are acute angles, find tan(x + y).
- Given cos p = 15/17 and cos q = 7/25, where p and q are acute angles, find tan(p + q).
Solving these problems will give you valuable practice in applying trigonometric identities and techniques. Remember to follow a step-by-step approach, carefully consider the signs of the trigonometric functions, and double-check your calculations.
Conclusion
In conclusion, finding the tangent of the sum of two angles given their cosine values involves a systematic application of trigonometric identities. We started by using the Pythagorean identity to find the sine values, then calculated the tangent values for each angle, and finally applied the compound angle formula for the tangent function. The solution tan(x + y) = 4/3 demonstrates the power and utility of trigonometric principles. By understanding these concepts and practicing regularly, you can confidently tackle a wide range of trigonometric problems.
This detailed exploration not only solves the specific problem but also reinforces fundamental trigonometric concepts and provides insights into their broader applications. The step-by-step approach, alternative methods, common mistakes to avoid, and practice problems collectively enhance understanding and mastery of trigonometry.
