Introduction
In trigonometry, finding the exact values of trigonometric functions for specific angles is a fundamental skill. While calculators can provide decimal approximations, understanding the underlying principles allows us to derive precise expressions involving radicals. This article delves into the process of determining the exact value of sin(105°). This problem often appears in precalculus and trigonometry courses and serves as a great example of applying trigonometric identities. We will explore the sum and difference identities which are crucial in breaking down the angle 105° into more manageable angles whose trigonometric values are well-known. By expressing 105° as a sum of two familiar angles, such as 60° and 45°, we can leverage the sum identity for sine to find the exact value. This method not only provides the correct answer but also enhances our understanding of trigonometric relationships and their applications. Understanding these concepts is vital for advanced mathematical studies and various scientific applications. The ability to manipulate trigonometric expressions and find exact values is a testament to a strong grasp of trigonometric principles.
Breaking Down the Problem
To find the exact value of sin(105°), we first recognize that 105° can be expressed as the sum of two angles whose sine and cosine values we know exactly. Common angles to consider are 30°, 45°, 60°, and their multiples. In this case, 105° can be written as 60° + 45°. This decomposition is key because we can then apply the sine addition formula. The sine addition formula states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). By identifying A as 60° and B as 45°, we can substitute these values into the formula and utilize the known trigonometric values for these angles. This approach allows us to avoid using a calculator and instead rely on our knowledge of the unit circle and special right triangles. The unit circle provides a visual representation of trigonometric values for various angles, while special right triangles (30-60-90 and 45-45-90) offer exact ratios for sine, cosine, and tangent. This method not only provides the answer but also reinforces our understanding of how trigonometric identities can simplify complex problems. The choice of breaking down 105° into 60° and 45° is strategic as these are standard angles for which we have readily available trigonometric values. Alternative decompositions might exist, but they may not lead to a straightforward solution using known exact values. This step-by-step approach showcases the power of trigonometric identities in simplifying problems and finding exact solutions.
Applying the Sine Addition Formula
Now that we have expressed 105° as 60° + 45°, we can apply the sine addition formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Let A = 60° and B = 45°. The next step involves substituting these values into the formula. We have sin(105°) = sin(60° + 45°) = sin(60°)cos(45°) + cos(60°)sin(45°). To proceed, we need to recall the exact values of sin(60°), cos(45°), cos(60°), and sin(45°). From the unit circle or special right triangles, we know that sin(60°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, and sin(45°) = √2/2. These values are fundamental and should be memorized for efficient problem-solving. Substituting these values into the equation, we get sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2). This equation now involves only simple arithmetic operations and radicals. The multiplication and addition of these terms will lead us to the exact value of sin(105°). This step-by-step substitution highlights the importance of knowing the exact trigonometric values for common angles and the application of trigonometric identities. The sine addition formula is a powerful tool, and its correct application is crucial for solving this type of problem. The formula allows us to break down a complex angle into simpler components, making the calculation more manageable.
Calculating the Exact Value
With the values substituted into the sine addition formula, we have sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2). The next step is to perform the multiplications. Multiplying the first term, (√3/2)(√2/2), gives us (√3 * √2) / (2 * 2) = √6 / 4. Multiplying the second term, (1/2)(√2/2), gives us √2 / 4. Now we add these two terms: sin(105°) = √6 / 4 + √2 / 4. Since both terms have the same denominator, we can combine them by adding the numerators: sin(105°) = (√6 + √2) / 4. This result represents the exact value of sin(105°). It is important to note that this expression cannot be simplified further as √6 and √2 are not like terms. The final value, (√6 + √2) / 4, is a precise representation, unlike a decimal approximation obtained from a calculator. This calculation demonstrates how combining trigonometric identities with algebraic manipulation allows us to find exact values. The ability to work with radicals and fractions is essential in this process. This step-by-step breakdown ensures clarity and accuracy in the calculation, reinforcing the importance of each step in the process.
Conclusion
In summary, we have successfully found the exact value of sin(105°) using the sine addition formula. We began by expressing 105° as the sum of 60° and 45°, two angles for which we know the exact trigonometric values. We then applied the sine addition formula, sin(A + B) = sin(A)cos(B) + cos(A)sin(B), substituting A = 60° and B = 45°. After substituting the known values of sin(60°), cos(45°), cos(60°), and sin(45°), we performed the necessary multiplications and additions to arrive at the final answer: sin(105°) = (√6 + √2) / 4. This result demonstrates the power of trigonometric identities in simplifying complex problems and finding exact solutions. The method used here is a valuable technique for evaluating trigonometric functions at various angles, particularly those that can be expressed as sums or differences of special angles. This approach not only provides the correct answer but also enhances our understanding of trigonometric relationships and their applications. Mastering these techniques is crucial for success in trigonometry and related fields. The ability to manipulate trigonometric expressions and find exact values is a testament to a strong grasp of trigonometric principles, which are fundamental in many areas of mathematics and science. The final answer, (√6 + √2) / 4, corresponds to option D in the original question.
Final Answer: The final answer is