Finding The Exact Value Of Sin(arctan(-3/4)) A Step-by-Step Guide
This article delves into the process of determining the exact value of the trigonometric expression sin(arctan(-3/4)). This problem combines the concepts of inverse trigonometric functions and the fundamental trigonometric ratios. We will walk through the steps, providing clear explanations and utilizing diagrams to enhance understanding. By the end of this guide, you will not only be able to solve this specific problem but also grasp the underlying principles that can be applied to similar trigonometric challenges.
Understanding Inverse Trigonometric Functions
Before we dive into the solution, it's crucial to understand inverse trigonometric functions, particularly the arctangent function. The arctan function, denoted as arctan(x) or tan⁻¹(x), gives the angle whose tangent is x. In simpler terms, if tan(θ) = x, then arctan(x) = θ. However, it's important to note that the range of the arctangent function is restricted to (-π/2, π/2) or (-90°, 90°). This restriction is necessary to ensure that the arctangent function is a true inverse function, meaning it has a one-to-one correspondence between inputs and outputs. This means that for any given value of x, there is only one unique angle θ within the range (-π/2, π/2) that satisfies the equation tan(θ) = x. The arctangent function is fundamental in various fields, including calculus, physics, and engineering, for solving problems involving angles and trigonometric ratios. It allows us to find the angle when we know the ratio of the opposite side to the adjacent side in a right-angled triangle. Understanding the domain and range of arctangent is crucial for accurate calculations and interpretations in mathematical and real-world applications. Moreover, the arctangent function is closely related to other inverse trigonometric functions, such as arcsine and arccosine, which provide the angles whose sine and cosine are known, respectively. These functions together form a powerful toolkit for solving a wide range of trigonometric problems.
Visualizing the Problem with a Right Triangle
The key to solving this problem lies in visualizing the expression arctan(-3/4) as an angle in a right triangle. Let θ = arctan(-3/4). This implies that tan(θ) = -3/4. Since the tangent function is negative, θ must lie in the second or fourth quadrant. However, due to the range restriction of arctangent (-π/2 < θ < π/2), θ must be in the fourth quadrant. We can construct a right triangle in the fourth quadrant where the opposite side has a length of -3 and the adjacent side has a length of 4. The negative sign indicates the direction of the side along the y-axis, placing it below the x-axis, which is characteristic of the fourth quadrant. This visualization is crucial because it allows us to translate the abstract concept of an inverse trigonometric function into a concrete geometric representation. By drawing a right triangle, we can directly relate the tangent of the angle to the sides of the triangle. This approach simplifies the problem by allowing us to use the Pythagorean theorem to find the length of the hypotenuse, which is necessary to determine the sine of the angle. Furthermore, visualizing the angle in the correct quadrant is essential for determining the correct sign of the trigonometric functions. In this case, the fourth quadrant helps us understand why the sine of the angle will be negative. The right triangle visualization is a powerful tool that bridges the gap between trigonometry and geometry, making complex problems more accessible and intuitive.
Calculating the Hypotenuse
Now, to find the sine of θ, we need to determine the hypotenuse of our right triangle. Using the Pythagorean theorem (a² + b² = c²), where a and b are the lengths of the legs and c is the length of the hypotenuse, we can calculate the hypotenuse. In our case, a = 4 and b = -3. Plugging these values into the Pythagorean theorem, we get 4² + (-3)² = c², which simplifies to 16 + 9 = c². This gives us 25 = c², and taking the square root of both sides, we find that c = 5. It's important to note that we take the positive square root since the hypotenuse represents the distance from the origin to the point on the terminal side of the angle, and distances are always positive. The hypotenuse, therefore, is 5 units long. This calculation is a critical step in solving the problem because it provides the final side length needed to compute the sine of the angle. The Pythagorean theorem is a fundamental concept in geometry and trigonometry, and its application here demonstrates its importance in relating the sides of a right triangle. Once we have the length of the hypotenuse, we can easily determine the sine, cosine, and other trigonometric functions of the angle. In this specific problem, the hypotenuse allows us to connect the arctangent function to the sine function through the geometry of the right triangle. This connection is crucial for finding the exact value of the expression sin(arctan(-3/4)).
Finding the Sine Value
With the hypotenuse calculated, we can now find the sine of θ. Recall that sin(θ) is defined as the ratio of the opposite side to the hypotenuse. In our right triangle, the opposite side has a length of -3, and the hypotenuse has a length of 5. Therefore, sin(θ) = -3/5. This result is consistent with our earlier observation that θ lies in the fourth quadrant, where the sine function is negative. This step is the culmination of our geometric and trigonometric analysis, bringing together the understanding of inverse trigonometric functions, right triangles, and the Pythagorean theorem. By relating the arctangent function to the sides of a right triangle and then using the sine definition, we have successfully found the exact value of sin(arctan(-3/4)). The negative sign is crucial and correctly reflects the quadrant in which the angle lies, emphasizing the importance of understanding the properties of trigonometric functions in different quadrants. This process highlights the power of visualization and geometric interpretation in simplifying trigonometric problems. The exact value of sin(arctan(-3/4)) is -3/5, which completes our solution. This approach can be generalized to solve similar problems involving inverse trigonometric functions and their relationships to trigonometric ratios.
Therefore, sin(arctan(-3/4)) = -3/5.
Conclusion
In this detailed guide, we have successfully found the exact value of sin(arctan(-3/4)) by leveraging the understanding of inverse trigonometric functions, visualizing the problem with a right triangle, applying the Pythagorean theorem, and using the definition of sine. The key takeaway is the importance of geometric interpretation in solving trigonometric problems. By constructing a right triangle, we were able to translate the abstract concept of arctangent into a concrete visual representation, which simplified the calculation. This method can be applied to a variety of similar problems involving inverse trigonometric functions. Understanding the range restrictions of inverse trigonometric functions is crucial for determining the correct quadrant and sign of the result. The Pythagorean theorem is a fundamental tool for finding the missing side lengths in right triangles, which is essential for calculating trigonometric ratios. Finally, the definition of trigonometric functions as ratios of sides in a right triangle provides the direct link between the geometric representation and the trigonometric value. This step-by-step approach not only solves the specific problem but also reinforces the fundamental concepts of trigonometry and problem-solving strategies. The ability to visualize and manipulate trigonometric concepts geometrically is a valuable skill in mathematics and its applications in science and engineering. By mastering these techniques, you can confidently tackle a wide range of trigonometric challenges. We encourage you to practice more problems of this nature to solidify your understanding and build your problem-solving skills in trigonometry.