Find The Exact Value Of Sin(75 Degrees) A Step-by-Step Solution

In this comprehensive guide, we will explore how to find the exact value of sin(75°). This is a common problem in trigonometry and involves using trigonometric identities and angle sum formulas. We will break down the process step-by-step, ensuring a clear understanding of the methods involved. Trigonometric functions like sine, cosine, and tangent are fundamental in various fields, including physics, engineering, and computer graphics. Mastering these concepts is essential for anyone delving into these areas. This article aims to provide an in-depth explanation, making it easy for students and enthusiasts alike to grasp the concept of finding trigonometric values for non-standard angles. Let's dive into the detailed solution.

Understanding the Basics of Trigonometric Functions

Before we jump into finding the exact value of sin(75°), it's crucial to have a solid understanding of basic trigonometric functions and their properties. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions relate the angles of a right-angled triangle to the ratios of its sides. The sine of an angle is defined as the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side. These definitions form the foundation of trigonometry and are used extensively in solving various problems. Additionally, understanding the unit circle and the values of trigonometric functions for standard angles like 0°, 30°, 45°, 60°, and 90° is essential. These angles serve as building blocks for finding trigonometric values of other angles. Moreover, it's important to be familiar with trigonometric identities, which are equations that are always true for any value of the variables. These identities are invaluable tools for simplifying expressions and solving trigonometric equations. In the context of finding sin(75°), we will use the angle sum identity, which allows us to express the sine of a sum of two angles in terms of the sines and cosines of the individual angles. This approach will enable us to break down 75° into angles for which we know the exact trigonometric values, thereby allowing us to compute the exact value of sin(75°).

Breaking Down 75 Degrees into Known Angles

The key to finding the exact value of sin(75°) lies in expressing 75° as the sum of two angles for which we know the exact sine and cosine values. A common approach is to break 75° into 45° and 30°, as these angles are standard angles on the unit circle. We know that sin(45°) = √2/2, cos(45°) = √2/2, sin(30°) = 1/2, and cos(30°) = √3/2. These values are derived from the special right triangles, namely the 45-45-90 triangle and the 30-60-90 triangle. The 45-45-90 triangle has angles of 45°, 45°, and 90°, and its sides are in the ratio 1:1:√2. The 30-60-90 triangle has angles of 30°, 60°, and 90°, and its sides are in the ratio 1:√3:2. By understanding these triangles and their side ratios, we can easily recall the sine and cosine values for 30°, 45°, and 60°. Another way to think about this is by using the unit circle, where the coordinates of a point on the circle corresponding to an angle θ are (cos θ, sin θ). For the angles 30°, 45°, and 60°, we can visualize their corresponding points on the unit circle and determine their cosine and sine values. Breaking down 75° into 45° and 30° is a strategic step that allows us to use the angle sum formula for sine, which states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). By substituting A = 45° and B = 30°, we can express sin(75°) in terms of the sines and cosines of 45° and 30°, which we already know. This technique highlights the importance of recognizing how angles can be decomposed into simpler, known angles to facilitate trigonometric calculations.

Applying the Angle Sum Formula for Sine

To find the exact value of sin(75°), we will use the angle sum formula for sine, which is given by sin(A + B) = sin(A)cos(B) + cos(A)sin(B). As we discussed earlier, we can express 75° as the sum of 45° and 30°. Therefore, we can substitute A = 45° and B = 30° into the formula. This gives us sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°). Now, we need to substitute the known values of the trigonometric functions for 45° and 30°. We know that sin(45°) = √2/2, cos(45°) = √2/2, sin(30°) = 1/2, and cos(30°) = √3/2. Plugging these values into the equation, we get sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2). This equation now involves simple multiplication and addition of fractions, which we can easily compute. The first term, (√2/2)(√3/2), multiplies the numerators and the denominators separately, resulting in (√2 * √3) / (2 * 2) = √6 / 4. The second term, (√2/2)(1/2), similarly multiplies the numerators and the denominators, giving us √2 / 4. Thus, we have sin(75°) = √6 / 4 + √2 / 4. This is a significant step as we have expressed sin(75°) in terms of exact values involving square roots and integers. The next step is to combine these terms to obtain a simplified expression for sin(75°). This process illustrates the power of trigonometric identities in simplifying complex trigonometric expressions and finding exact values.

Simplifying the Expression

After applying the angle sum formula, we have the expression sin(75°) = √6 / 4 + √2 / 4. To simplify this expression, we notice that both terms have a common denominator of 4. Therefore, we can combine the numerators over the common denominator, which gives us sin(75°) = (√6 + √2) / 4. This is the exact value of sin(75°) in a simplified form. The expression (√6 + √2) / 4 is an irrational number, but it represents the precise value of sin(75°). It's important to note that this value cannot be further simplified in terms of elementary radicals. However, it is a closed-form expression, which means it is an exact representation of the value without any approximation. The process of simplifying expressions like this is a fundamental skill in mathematics, particularly in algebra and trigonometry. It involves identifying common factors, combining like terms, and using algebraic manipulations to arrive at the most concise form. In this case, the simplification was straightforward due to the common denominator. However, in other situations, simplification might involve more complex techniques, such as rationalizing the denominator or using trigonometric identities to further reduce the expression. The final result, (√6 + √2) / 4, is the exact value of sin(75°), and it matches one of the options provided. This demonstrates the effectiveness of using the angle sum formula and simplifying techniques to solve trigonometric problems.

Final Answer and Conclusion

In conclusion, the exact value of sin(75°) is (√6 + √2) / 4. We arrived at this answer by breaking down 75° into the sum of 45° and 30°, applying the angle sum formula for sine, and simplifying the resulting expression. This process highlights the importance of understanding basic trigonometric functions, standard angles, and trigonometric identities. The angle sum formula, sin(A + B) = sin(A)cos(B) + cos(A)sin(B), is a powerful tool for finding the sine of the sum of two angles. By strategically choosing angles for A and B, we can express the sine of a non-standard angle like 75° in terms of the sines and cosines of standard angles, which we know exactly. The ability to manipulate trigonometric expressions and simplify them is a crucial skill in mathematics and its applications. This problem serves as a good example of how these skills can be applied to find exact values of trigonometric functions. The final answer, (√6 + √2) / 4, is a precise representation of sin(75°) and is the correct option among the choices provided. Understanding this process not only helps in solving similar problems but also builds a strong foundation in trigonometry, which is essential for various advanced topics in mathematics and other scientific disciplines. Mastering these concepts and techniques will undoubtedly be beneficial for anyone pursuing studies or careers in STEM fields.

Correct Answer: (√2 + √6) / 4