Find The Exact Value Of Sin(π/12) Using Trigonometric Identities

In the realm of trigonometry, determining the exact values of trigonometric functions for specific angles is a fundamental skill. One such intriguing challenge is finding the exact value of $\sin(\frac$\pi}{12}). This article provides a detailed walkthrough of the process, leveraging trigonometric identities and formulas to arrive at the solution. Our primary focus will be on utilizing the sum and difference formulas, a cornerstone of trigonometric manipulation. We will explore how expressing $\frac{\pi{12}$ as a combination of more familiar angles allows us to apply these formulas effectively.

Expressing π/12 as a Sum or Difference

The initial and crucial step in solving this problem lies in expressing the angle $\frac{\pi}{12}$ as a sum or difference of angles whose sine and cosine values are known. Common angles that fit this criterion include $\frac{\pi}{6}$ (30 degrees), $\frac{\pi}{4}$ (45 degrees), and $\frac{\pi}{3}$ (60 degrees). These angles have well-defined trigonometric values that are easily recalled or derived from the unit circle. The objective is to manipulate these standard angles to obtain $\frac{\pi}{12}$ through addition or subtraction. After careful consideration, we can express $\frac{\pi}{12}$ as the difference between $\frac{\pi}{3}$ and $\frac{\pi}{4}$, that is, $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$. This decomposition is key because it allows us to employ the sine difference formula, a powerful tool for expanding trigonometric functions of angle differences. This formula states that $\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)$. By identifying A as $\frac{\pi}{3}$ and B as $\frac{\pi}{4}$, we can directly apply this formula to our problem. This step highlights the importance of recognizing suitable angle combinations and understanding the versatility of trigonometric identities in simplifying complex expressions. The ability to break down an unfamiliar angle into more manageable components is a fundamental technique in trigonometry, enabling us to solve a wide array of problems involving trigonometric functions. The sine difference formula is not just a mathematical tool; it's a bridge that connects unfamiliar angles to known trigonometric values, allowing us to navigate the complexities of trigonometric calculations with greater ease and precision. It's a testament to the power of mathematical identities in simplifying seemingly intricate problems.

Applying the Sine Difference Formula

With the angle $\frac{\pi}{12}$ artfully expressed as the difference $\frac{\pi}{3} - \frac{\pi}{4}$, we can now wield the sine difference formula to unravel its trigonometric value. This formula, a cornerstone of trigonometric identities, elegantly states that for any two angles A and B, $\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)$. To apply this formula to our specific case, we'll make the judicious substitutions of $A = \frac{\pi}{3}$ and $B = \frac{\pi}{4}$. This transforms our initial problem into a more manageable expression involving the sines and cosines of these well-known angles. By plugging in these values, we obtain the expression $\sin(\frac{\pi}{12}) = \sin(\frac{\pi}{3} - \frac{\pi}{4}) = \sin(\frac{\pi}{3})\cos(\frac{\pi}{4}) - \cos(\frac{\pi}{3})\sin(\frac{\pi}{4})$. This equation is a pivotal step in our journey, as it breaks down the problem into a form where we can directly substitute the known trigonometric values of $\frac{\pi}{3}$ and $\frac{\pi}{4}$. The beauty of this approach lies in its ability to transform a seemingly complex problem into a straightforward calculation. The sine difference formula acts as a powerful simplifying tool, allowing us to express the sine of a difference of angles in terms of the individual sines and cosines of those angles. This transformation is not just a mathematical manipulation; it's a strategic move that brings us closer to the solution by leveraging our knowledge of standard trigonometric values. It's a clear demonstration of how understanding and applying trigonometric identities can unlock the solutions to a wide range of problems, making them an indispensable part of any mathematician's toolkit. The formula's elegance and utility make it a central concept in trigonometry, and its application here perfectly illustrates its power in simplifying complex expressions and revealing the underlying mathematical structure.

Substituting Known Trigonometric Values

Having skillfully applied the sine difference formula, our next pivotal step involves substituting the known trigonometric values for $\sin(\frac{\pi}{3})$, $\cos(\frac{\pi}{3})$, $\sin(\frac{\pi}{4})$, and $\cos(\frac{\pi}{4})$. These values are fundamental in trigonometry and are often derived from the unit circle or special right triangles (30-60-90 and 45-45-90 triangles). The sine of $\frac{\pi}{3}$ (60 degrees) is $\frac{\sqrt{3}}{2}$, and its cosine is $\frac{1}{2}$. For $\frac{\pi}{4}$ (45 degrees), both the sine and cosine are equal to $\frac{\sqrt{2}}{2}$. Armed with these values, we can now meticulously replace the trigonometric functions in our equation. This substitution transforms the equation $\sin(\frac{\pi}{12}) = \sin(\frac{\pi}{3})\cos(\frac{\pi}{4}) - \cos(\frac{\pi}{3})\sin(\frac{\pi}{4})$ into $\sin(\frac{\pi}{12}) = (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2}) - (\frac{1}{2})(\frac{\sqrt{2}}{2})$. This step is crucial as it converts the trigonometric expression into a purely arithmetic one, setting the stage for simplification and the final determination of the exact value. The accurate recall and substitution of these standard trigonometric values are essential for the successful solution of this problem. This process highlights the importance of memorizing or being able to quickly derive these key values, as they form the building blocks for solving a wide array of trigonometric problems. The substitution step is not merely a mechanical replacement of symbols; it's a strategic maneuver that bridges the gap between trigonometry and basic arithmetic, demonstrating the interconnectedness of different mathematical concepts. It showcases how a deep understanding of fundamental values can unlock the solutions to more complex problems, emphasizing the hierarchical nature of mathematical knowledge and the importance of mastering the basics.

Simplifying the Expression

With the trigonometric values elegantly substituted, our focus now shifts to simplifying the resulting expression. The equation we've arrived at is $\sin(\frac{\pi}{12}) = (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2}) - (\frac{1}{2})(\frac{\sqrt{2}}{2})$. The simplification process begins with performing the multiplications. Multiplying the fractions, we get $\sin(\frac{\pi}{12}) = \frac{\sqrt{3} \cdot \sqrt{2}}{4} - \frac{\sqrt{2}}{4}$. Recognizing that $\sqrt{3} \cdot \sqrt{2}$ can be combined under a single radical as $\sqrt{6}$, the equation further simplifies to $\sin(\frac{\pi}{12}) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$. Now, we observe that both terms share a common denominator of 4. This allows us to combine the fractions into a single term, resulting in $\sin(\frac{\pi}{12}) = \frac{\sqrt{6} - \sqrt{2}}{4}$. This final expression represents the exact value of $\sin(\frac{\pi}{12})$. The simplification process is a testament to the power of algebraic manipulation in extracting the most concise and informative form of a mathematical expression. Each step, from multiplying fractions to combining like terms, is a carefully executed maneuver aimed at revealing the underlying simplicity of the result. The ability to simplify expressions is a fundamental skill in mathematics, enabling us to transform complex equations into manageable forms and ultimately arrive at solutions. In this case, the simplification not only provides the exact value of $\sin(\frac{\pi}{12})$ but also showcases the elegance and beauty inherent in mathematical expressions when they are reduced to their simplest forms. It's a reminder that mathematics is not just about finding answers; it's about uncovering the underlying structure and relationships that govern the mathematical world.

Final Answer

Through a carefully orchestrated sequence of trigonometric manipulations and algebraic simplifications, we have successfully determined the exact value of $\sin(\frac{\pi}{12})$. Beginning with the strategic decomposition of $\frac{\pi}{12}$ into the difference $\frac{\pi}{3} - \frac{\pi}{4}$, we invoked the sine difference formula, a cornerstone of trigonometric identities. This formula elegantly transformed our problem into an expression involving the sines and cosines of well-known angles. The subsequent substitution of these standard trigonometric values paved the way for a purely arithmetic calculation. By simplifying the resulting expression, we artfully combined fractions and radicals to arrive at the final answer: $\sin(\frac{\pi}{12}) = \frac{\sqrt{6} - \sqrt{2}}{4}$. This result is not merely a numerical value; it's a testament to the power of trigonometric identities and the elegance of mathematical reasoning. The journey from the initial problem to the final answer highlights the importance of understanding and applying fundamental trigonometric principles. Each step, from the application of the sine difference formula to the simplification of the expression, showcases the interconnectedness of mathematical concepts and the ability of mathematical tools to unlock seemingly complex problems. This solution serves as a powerful example of how strategic problem-solving, combined with a solid foundation in trigonometric identities, can lead to precise and insightful results. The final answer, $\frac{\sqrt{6} - \sqrt{2}}{4}$, is a concise and elegant representation of the sine of $\frac{\pi}{12}$, encapsulating the beauty and precision of mathematics in a single expression. It stands as a testament to the power of mathematical tools and techniques in unraveling the intricacies of the trigonometric world.

Therefore, the exact value of $\sin(\frac{\pi}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$.