Solving Cos(sin⁻¹(5/13) - Tan⁻¹(3/4)) A Step-by-Step Trigonometry Guide
Hey guys! Today, we're diving deep into a fascinating trigonometric problem: figuring out the value of cos(sin⁻¹(5/13) - tan⁻¹(3/4)). This looks a bit intimidating at first glance, but don't worry! We'll break it down step by step, making it super easy to understand. We'll be using some key trig identities and concepts, so buckle up and let's get started!
Understanding the Inverse Trigonometric Functions
Before we jump into the main problem, let's quickly recap inverse trigonometric functions. These functions, often denoted as sin⁻¹, cos⁻¹, and tan⁻¹, essentially answer the question: "What angle gives me this trigonometric ratio?" For instance, sin⁻¹(x) gives you the angle whose sine is x. It's crucial to remember that these functions have specific ranges for their outputs to ensure they are well-defined.
The inverse sine function, sin⁻¹(x), also known as arcsin(x), returns an angle in the range of [-π/2, π/2]. This means the output angle will always be between -90 degrees and 90 degrees. Similarly, the inverse tangent function, tan⁻¹(x), or arctan(x), returns an angle in the range of (-π/2, π/2), which is between -90 degrees and 90 degrees, but not including the endpoints. These restricted ranges are essential for the functions to be single-valued and to avoid ambiguity. When dealing with problems involving inverse trigonometric functions, it’s crucial to keep these ranges in mind to ensure the final answers are correct and fall within the defined intervals.
Understanding the domains and ranges of these inverse functions is paramount for solving complex trigonometric expressions. For example, consider sin⁻¹(5/13). This tells us we're looking for an angle whose sine is 5/13. Since 5/13 is a positive value, the angle will lie in the first quadrant (between 0 and π/2). Similarly, tan⁻¹(3/4) represents the angle whose tangent is 3/4, which also lies in the first quadrant. These initial observations help us visualize the angles and their relationships, which is crucial for applying trigonometric identities and simplifying the expression. By grasping these fundamental concepts, we can confidently approach and solve the problem at hand.
Breaking Down the Expression: sin⁻¹(5/13)
Let's start by tackling the first part of our problem: sin⁻¹(5/13). We'll call this angle A, so we have A = sin⁻¹(5/13). This means sin(A) = 5/13. Now, think of a right-angled triangle where the opposite side is 5 and the hypotenuse is 13. We can use the Pythagorean theorem (a² + b² = c²) to find the adjacent side. In this case, it's √(13² - 5²) = √(169 - 25) = √144 = 12. So, the adjacent side is 12.
Now that we have all three sides of the triangle, we can find cos(A) and tan(A). Cos(A) is the adjacent side divided by the hypotenuse, which is 12/13. Tan(A) is the opposite side divided by the adjacent side, which is 5/12. This step is crucial because we'll need cos(A) later when we use the cosine subtraction formula. Visualizing this right-angled triangle helps us easily determine these trigonometric ratios. By finding the missing side using the Pythagorean theorem, we've successfully transformed the inverse sine value into ratios we can use in trigonometric identities. This conversion is a key technique in solving these kinds of problems, allowing us to move from inverse functions to more manageable trigonometric ratios.
By establishing these relationships for angle A, we are laying the groundwork for using the cosine subtraction formula. Understanding the geometric representation of these trigonometric ratios in a right-angled triangle significantly simplifies the problem-solving process. It allows us to replace the inverse trigonometric function with standard trigonometric functions, making subsequent calculations more straightforward. This approach demonstrates the power of converting inverse trigonometric problems into geometric representations, which often leads to a clearer and more intuitive understanding of the solution.
Breaking Down the Expression: tan⁻¹(3/4)
Next up, let's handle the second part of our expression: tan⁻¹(3/4). We'll call this angle B, so B = tan⁻¹(3/4). This means tan(B) = 3/4. Again, we can think of a right-angled triangle, but this time, the opposite side is 3 and the adjacent side is 4. Using the Pythagorean theorem, we can find the hypotenuse: √(3² + 4²) = √(9 + 16) = √25 = 5.
Now that we have all the sides, we can find sin(B) and cos(B). Sin(B) is the opposite side divided by the hypotenuse, which is 3/5. Cos(B) is the adjacent side divided by the hypotenuse, which is 4/5. Just like before, this conversion is essential for applying the cosine subtraction formula. We've now successfully translated the inverse tangent value into usable trigonometric ratios. Visualizing the right triangle for angle B allows us to easily compute these ratios, which are crucial for the next stage of the problem.
By determining sin(B) and cos(B), we complete the transformation of the inverse tangent function into standard trigonometric functions. This step is crucial because it sets the stage for using the cosine subtraction formula, which involves the cosine and sine of both angles A and B. The ability to visualize these trigonometric ratios in a right-angled triangle not only simplifies the calculation process but also provides a clear geometric interpretation of the problem. This method highlights the importance of geometric reasoning in solving trigonometric problems, making it easier to understand and remember the steps involved.
Applying the Cosine Subtraction Formula
Now for the grand finale! We need to find cos(A - B), where A = sin⁻¹(5/13) and B = tan⁻¹(3/4). The cosine subtraction formula is our key here: cos(A - B) = cos(A)cos(B) + sin(A)sin(B). We've already found all these values! Remember:
- cos(A) = 12/13
- sin(A) = 5/13
- cos(B) = 4/5
- sin(B) = 3/5
Plugging these values into the formula, we get:
cos(A - B) = (12/13)(4/5) + (5/13)(3/5) = 48/65 + 15/65 = 63/65
And there you have it! The value of cos(sin⁻¹(5/13) - tan⁻¹(3/4)) is 63/65. This step-by-step approach showcases how breaking down a complex problem into smaller, manageable parts makes it much easier to solve. The cosine subtraction formula is a powerful tool in trigonometry, and applying it correctly here allows us to arrive at the final answer. By substituting the previously calculated trigonometric ratios into the formula, we efficiently compute the result, demonstrating the interconnectedness of different trigonometric concepts.
The successful application of the cosine subtraction formula underscores the importance of knowing and understanding key trigonometric identities. This formula allows us to express the cosine of the difference of two angles in terms of the cosines and sines of the individual angles, which is essential for solving problems involving inverse trigonometric functions. By mastering such formulas and knowing how to apply them, we can confidently tackle a wide range of trigonometric challenges and simplify complex expressions into manageable forms.
Conclusion: Mastering Trigonometric Challenges
So, guys, we've successfully navigated a pretty complex trigonometric problem! We started with cos(sin⁻¹(5/13) - tan⁻¹(3/4)) and, through careful steps and the use of trigonometric identities, arrived at the answer: 63/65. The key takeaways here are understanding inverse trigonometric functions, visualizing right-angled triangles, and mastering trigonometric identities like the cosine subtraction formula. By breaking down complex problems and applying these fundamental concepts, you can conquer any trigonometric challenge that comes your way! Keep practicing, and you'll become a trig whiz in no time!
Remember, the most challenging math problems become manageable when approached systematically. By first understanding the underlying concepts and then breaking the problem into smaller, more digestible steps, we can solve even the most complex equations. This approach not only helps in arriving at the correct solution but also deepens our understanding of the mathematical principles involved. So, keep exploring, keep practicing, and you'll find that trigonometry, like any other branch of mathematics, becomes more accessible and even enjoyable with consistent effort and a methodical approach.