Finding Theta Angle Given Tangent In A Right Angled Triangle
In trigonometry, determining angles within right-angled triangles is a fundamental concept with numerous applications in various fields, including engineering, physics, and navigation. This article will delve into a specific problem: finding the value of an angle θ in a right-angled triangle given the tangent of the angle. We will explore the underlying trigonometric principles, the step-by-step solution, and the importance of expressing the answer to the required degree of accuracy. This comprehensive guide aims to provide a clear understanding of the concepts involved and enhance your problem-solving skills in trigonometry.
Understanding the Problem: Tan θ = 28/47
The problem at hand presents us with a right-angled triangle where θ is one of the acute angles. We are given that the tangent of θ (tan θ) is equal to 28/47. Our task is to find the value of θ, expressed to one decimal place. To solve this, we need to understand the relationship between the tangent function and the sides of a right-angled triangle. In trigonometry, the tangent of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. Mathematically, this can be represented as:
tan θ = Opposite / Adjacent
In our case, tan θ = 28/47, which means that if we consider a right-angled triangle where θ is one of the angles, the side opposite to θ has a length proportional to 28, and the side adjacent to θ has a length proportional to 47. It's crucial to remember that these are ratios, not necessarily the actual lengths of the sides. The actual lengths could be any multiple of these values, as long as the ratio remains 28/47.
Before we proceed with finding the angle, it's beneficial to visualize the triangle. Imagine a right-angled triangle with one angle labeled as θ. The side directly across from θ is the "opposite" side, and the side next to θ (that is not the hypotenuse) is the "adjacent" side. The longest side, opposite the right angle, is the hypotenuse. The tangent function only involves the opposite and adjacent sides, which simplifies our problem-solving process. Understanding this fundamental relationship is key to tackling trigonometric problems effectively. Now, let's move on to the step-by-step solution to find the value of θ.
Step-by-Step Solution: Finding θ
Now that we understand the relationship between tan θ and the sides of the triangle, we can proceed to find the value of θ. The key to solving this problem is using the inverse tangent function, also known as the arctangent or tan⁻¹. The inverse tangent function essentially reverses the operation of the tangent function. If tan θ = x, then θ = tan⁻¹(x). In simpler terms, it tells us the angle whose tangent is a given value.
Step 1: Apply the Inverse Tangent Function
In our problem, we have tan θ = 28/47. To find θ, we need to apply the inverse tangent function to both sides of the equation:
θ = tan⁻¹(28/47)
This equation states that θ is the angle whose tangent is 28/47. Now, we need to calculate the value of tan⁻¹(28/47). This is where a calculator becomes essential. Most scientific calculators have a dedicated button for the inverse tangent function, usually labeled as tan⁻¹ or atan.
Step 2: Use a Calculator to Find tan⁻¹(28/47)
Before using the calculator, it's important to ensure that it is in the correct mode. Trigonometric functions can be calculated in degrees or radians. Since the problem asks for the answer to one decimal place, it is more common to provide the answer in degrees. Therefore, make sure your calculator is set to degree mode. The process of setting the mode varies slightly depending on the calculator model, but it usually involves pressing a "Mode" or "Settings" button and selecting "Deg" or "Degrees".
Once your calculator is in degree mode, enter the following sequence of keys:
- Press the inverse tangent function button (usually labeled as tan⁻¹ or atan). This might require pressing a "Shift" or "2nd" button first.
- Enter the fraction 28/47. This might involve using a division button or a fraction button, depending on your calculator.
- Press the equals (=) button to calculate the result.
The calculator should display a value close to 30.755 degrees. This is the angle whose tangent is 28/47. However, the problem requires the answer to be given to one decimal place.
Step 3: Round the Answer to 1 Decimal Place
Rounding a number to one decimal place involves looking at the digit in the second decimal place. If this digit is 5 or greater, we round up the digit in the first decimal place. If it is less than 5, we leave the digit in the first decimal place as it is.
In our case, the calculator displayed 30.755 degrees. The digit in the second decimal place is 5, so we round up the digit in the first decimal place (7) to 8. Therefore, the value of θ, rounded to one decimal place, is 30.8 degrees.
Final Answer:
θ = 30.8°
This step-by-step solution demonstrates how to find the angle θ when given the value of tan θ. The key steps involve understanding the inverse tangent function, using a calculator correctly, and rounding the answer to the required degree of accuracy. This approach can be applied to similar problems involving other trigonometric functions like sine and cosine.
Importance of Accuracy and Rounding
In mathematics and particularly in practical applications, accuracy is paramount. When dealing with angles and trigonometric functions, even small differences in values can lead to significant errors in calculations and outcomes. Therefore, it is crucial to pay attention to the required degree of accuracy and round the final answer appropriately. In the given problem, we were asked to provide the answer to one decimal place. This instruction is not arbitrary; it reflects the level of precision expected in the result.
Rounding is a process of approximating a number to a specified number of decimal places or significant figures. It is essential to understand the rules of rounding to avoid introducing errors into the final answer. As we saw in the solution, we rounded the value 30.755 to 30.8 because the digit in the second decimal place (5) was equal to or greater than 5. Ignoring the rounding requirement and providing the answer as 30.755 would be technically incorrect, even though the difference seems small.
The importance of accuracy becomes even more evident in real-world applications. For example, in engineering, calculating angles is crucial for designing structures, machines, and systems. In navigation, accurate angle measurements are essential for determining direction and position. In these scenarios, even a fraction of a degree can make a significant difference. Therefore, understanding the importance of accuracy and rounding is a fundamental skill in mathematics and its applications.
Moreover, different contexts may require different levels of accuracy. Some situations might demand answers to two, three, or even more decimal places. The specific requirements should always be considered and followed to ensure the reliability and validity of the results. Paying close attention to accuracy and rounding is a hallmark of a careful and proficient problem solver.
Trigonometric Functions and Their Applications
The tangent function, along with sine and cosine, are the fundamental trigonometric functions. These functions establish a relationship between the angles of a right-angled triangle and the ratios of its sides. Understanding these functions is essential not only for solving mathematical problems but also for comprehending various phenomena in physics, engineering, and other scientific disciplines. Let's briefly explore the three main trigonometric functions:
- Sine (sin θ): Sine of an angle is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse. sin θ = Opposite / Hypotenuse
- Cosine (cos θ): Cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. cos θ = Adjacent / Hypotenuse
- Tangent (tan θ): As we discussed earlier, tangent of an angle is defined as the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. tan θ = Opposite / Adjacent
These three functions are interconnected, and their relationships can be expressed through various trigonometric identities. For example, the Pythagorean identity states that sin² θ + cos² θ = 1. This identity is a fundamental result in trigonometry and has numerous applications.
The applications of trigonometric functions are vast and diverse. In physics, they are used to analyze wave motion, oscillations, and projectile motion. In engineering, they are crucial for designing structures, analyzing forces, and understanding electrical circuits. In navigation, they are used for determining distances and directions. In computer graphics, they are used for creating 3D models and animations.
The inverse trigonometric functions, such as arcsine (sin⁻¹), arccosine (cos⁻¹), and arctangent (tan⁻¹), are used to find the angle when the ratio of the sides is known. As we saw in our problem, the inverse tangent function was essential for finding the value of θ. These functions are equally important in various applications, allowing us to determine angles from known side ratios.
Conclusion
In this article, we have explored a specific problem involving finding the value of an angle θ in a right-angled triangle given the tangent of the angle. We have discussed the underlying trigonometric principles, the step-by-step solution using the inverse tangent function, and the importance of accuracy and rounding. We have also briefly touched upon the broader applications of trigonometric functions in various fields. This comprehensive guide aims to provide a clear understanding of the concepts involved and enhance your problem-solving skills in trigonometry. Mastering these concepts is essential for success in mathematics and its applications in the real world. Remember to practice regularly and apply these principles to different problems to solidify your understanding and build confidence in your abilities.
