Calculating Tan(555°) A Step-by-Step Trigonometric Solution

In the realm of trigonometry, understanding how to calculate trigonometric functions for various angles is a fundamental skill. In this comprehensive guide, we will delve into the process of calculating the tangent of 555 degrees (tg555°). This problem might seem daunting at first, but by applying trigonometric identities and understanding the periodic nature of trigonometric functions, we can simplify the calculation and arrive at the correct answer. Our exploration will not only provide the solution but also enhance your understanding of trigonometric principles. The ability to calculate such values is crucial in various fields, including engineering, physics, and computer graphics, where angles and their trigonometric relationships are frequently encountered.

The task at hand is to calculate tg555°, which represents the tangent of an angle of 555 degrees. To solve this, we need to leverage our knowledge of trigonometric functions and their properties. One crucial property is the periodicity of trigonometric functions. Tangent, like sine and cosine, is a periodic function. This means that the value of the function repeats after a certain interval. For the tangent function, the period is 180 degrees. This periodicity allows us to simplify the angle by finding a coterminal angle within the standard range of 0 to 360 degrees. By understanding this periodicity, we can reduce the complexity of the problem and work with a more manageable angle. The concept of coterminal angles is key to simplifying trigonometric calculations involving angles outside the standard 0 to 360 degree range. This understanding forms the basis for our approach to solving for tg555°.

Before diving into the calculation, let's solidify our understanding of the key concepts involved: trigonometric functions and periodicity. Trigonometric functions, including sine, cosine, and tangent, relate angles of a triangle to the ratios of its sides. Tangent, specifically, is defined as the ratio of the sine of an angle to its cosine, or equivalently, the ratio of the opposite side to the adjacent side in a right-angled triangle. These functions are essential tools in mathematics and physics, used to model periodic phenomena such as waves and oscillations. Periodicity, on the other hand, refers to the repeating nature of these functions. For the tangent function, the period is 180 degrees, meaning that tg(θ) = tg(θ + 180n), where n is an integer. This property is crucial for simplifying calculations involving large angles. Understanding these concepts allows us to break down complex problems into simpler, more manageable steps. In the case of tg555°, the periodicity of the tangent function enables us to find an equivalent angle within the standard range, making the calculation straightforward.

The first step in calculating tg555° is to find a coterminal angle that lies within the range of 0° to 360°. Coterminal angles are angles that share the same terminal side when drawn in standard position. To find a coterminal angle, we can add or subtract multiples of 360° from the given angle until we obtain an angle within the desired range. In this case, we can subtract 360° from 555°: 555° - 360° = 195°. The resulting angle, 195°, is coterminal with 555° and falls within the standard range. This simplification is crucial because it allows us to work with a more familiar angle. Since the tangent function has a period of 180°, we can further simplify the angle by subtracting 180° from 195°: 195° - 180° = 15°. This gives us a reference angle of 15°, which is much easier to work with. Finding the coterminal angle is a fundamental step in simplifying trigonometric calculations, making it easier to determine the value of the trigonometric function.

As discussed earlier, the tangent function has a period of 180°, which means that tg(θ) = tg(θ + 180n), where n is an integer. This property is invaluable for simplifying trigonometric calculations. We've already found that 555° is coterminal with 195°, and we can express 195° as 180° + 15°. Therefore, tg555° = tg195° = tg(180° + 15°). Due to the periodicity of the tangent function, tg(180° + 15°) = tg15°. This simplification is a direct application of the periodic nature of the tangent function. By understanding and utilizing this property, we've reduced the problem to finding the tangent of 15 degrees. This step demonstrates the power of leveraging trigonometric identities and properties to simplify complex calculations. The periodicity of the tangent function allows us to focus on a smaller, more manageable angle, making the subsequent calculation more straightforward. This understanding is crucial for anyone working with trigonometric functions and their applications.

Now that we've simplified the problem to finding tg15°, we can use trigonometric identities to calculate this value. One common approach is to use the half-angle formula for tangent, or the tangent subtraction formula. We can express 15° as 45° - 30°, and then use the tangent subtraction formula: tg(A - B) = (tgA - tgB) / (1 + tgA * tgB). In this case, A = 45° and B = 30°. We know that tg45° = 1 and tg30° = 1/√3. Plugging these values into the formula, we get: tg15° = (1 - 1/√3) / (1 + 1 * 1/√3). To simplify this expression, we can multiply the numerator and denominator by √3: tg15° = (√3 - 1) / (√3 + 1). Next, we can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is (√3 - 1): tg15° = ((√3 - 1) * (√3 - 1)) / ((√3 + 1) * (√3 - 1)) This simplifies to: tg15° = (3 - 2√3 + 1) / (3 - 1) = (4 - 2√3) / 2 = 2 - √3. Therefore, tg15° = 2 - √3. This calculation demonstrates the application of trigonometric identities to find the exact value of trigonometric functions for specific angles. The use of the tangent subtraction formula and the rationalization of the denominator are common techniques in trigonometry. By mastering these techniques, one can solve a wide range of trigonometric problems.

Therefore, tg555° = tg15° = 2 - √3. This corresponds to option C) 2 - √3. By breaking down the problem into smaller, manageable steps, we were able to utilize trigonometric principles and identities to arrive at the correct solution. This process highlights the importance of understanding the properties of trigonometric functions, such as periodicity, and being able to apply relevant formulas. The ability to calculate trigonometric values for various angles is a fundamental skill in mathematics and has numerous applications in various fields. The solution not only provides the answer but also reinforces the understanding of trigonometric concepts and problem-solving techniques.

In conclusion, calculating tg555° involves understanding the periodicity of the tangent function, finding coterminal angles, and applying trigonometric identities. By following these steps, we determined that tg555° = 2 - √3. This exercise demonstrates the importance of a solid foundation in trigonometric principles and the ability to apply them effectively. The process of solving this problem not only provides the answer but also enhances our understanding of trigonometric functions and their properties. The skills and knowledge gained from this calculation are applicable to a wide range of mathematical and scientific problems, making it a valuable exercise in mastering trigonometry. Understanding how to manipulate angles and apply trigonometric identities is essential for success in advanced mathematics and related fields. This comprehensive guide has hopefully provided a clear and concise explanation of the steps involved in calculating tg555°, empowering you to tackle similar problems with confidence.