Finding The Exact Value Of Sin(π/12) A Step-by-Step Guide

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Introduction

In this article, we will delve into the process of finding the exact value of ${\sin \frac{\pi}{12}}$. This trigonometric value is not one of the standard angles (0, ${\frac{\pi}{6}}$, ${\frac{\pi}{4}}$, ${\frac{\pi}{3}}$, ${\frac{\pi}{2}}$) that we often memorize. Therefore, we need to employ trigonometric identities and angle manipulation techniques to determine its precise value. Understanding how to derive such values is crucial in various fields of mathematics, physics, and engineering.

This exploration not only enhances our understanding of trigonometric functions but also reinforces the application of fundamental trigonometric identities. We will break down the angle ${\frac{\pi}{12}}$ into more manageable angles for which we know the sine and cosine values, and then use these values to compute the sine of the original angle. This method showcases a powerful problem-solving approach in trigonometry, emphasizing the importance of recognizing patterns and applying appropriate formulas.

Background on Trigonometric Values

Before we dive into the solution, it’s essential to recap the trigonometric values of some standard angles. These values serve as the building blocks for finding the values of more complex angles. The sine, cosine, and tangent functions for angles like 0, ${\frac{\pi}{6}}$ (30°), ${\frac{\pi}{4}}$ (45°), ${\frac{\pi}{3}}$ (60°), and ${\frac{\pi}{2}}$ (90°) are commonly used and should be readily accessible. For instance:

• ${\sin(0) = 0}$
• ${\sin(\frac{\pi}{6}) = \frac{1}{2}}$
• ${\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}}$
• ${\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}}$
• ${\sin(\frac{\pi}{2}) = 1}$

Similarly, the cosine values for these angles are:

• ${\cos(0) = 1}$
• ${\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}}$
• ${\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}}$
• ${\cos(\frac{\pi}{3}) = \frac{1}{2}}$
• ${\cos(\frac{\pi}{2}) = 0}$

Knowing these values allows us to use trigonometric identities to find the values for other angles, including ${\frac{\pi}{12}}$.

Method 1: Using the Sine Difference Identity

One effective method to find the exact value of ${\sin \frac{\pi}{12}}$ is by expressing ${\frac{\pi}{12}}$ as the difference of two angles for which we know the sine and cosine values. We can write ${\frac{\pi}{12}}$ as:

$$\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$$

This decomposition is useful because both ${\frac{\pi}{3}}$ (60°) and ${\frac{\pi}{4}}$ (45°) are standard angles with known trigonometric values. Now, we can apply the sine difference identity:

$$\sin(A - B) = \sin A \cos B - \cos A \sin B$$

In our case, ${A = \frac{\pi}{3}}$ and ${B = \frac{\pi}{4}}$. Plugging these values into the identity, we get:

$$\sin \frac{\pi}{12} = \sin(\frac{\pi}{3} - \frac{\pi}{4}) = \sin \frac{\pi}{3} \cos \frac{\pi}{4} - \cos \frac{\pi}{3} \sin \frac{\pi}{4}$$

We know the values of each term on the right side:

• ${\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}}$
• ${\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}}$
• ${\cos \frac{\pi}{3} = \frac{1}{2}}$
• ${\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}}$

Substituting these values, we have:

$$\sin \frac{\pi}{12} = (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2}) - (\frac{1}{2})(\frac{\sqrt{2}}{2})$$

$$\sin \frac{\pi}{12} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$$

$$\sin \frac{\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}$$

Thus, the exact value of ${\sin \frac{\pi}{12}}$ is ${\frac{\sqrt{6} - \sqrt{2}}{4}}$. This method demonstrates the power of trigonometric identities in simplifying complex expressions.

Detailed Steps

1. Express ${\frac{\pi}{12}}$ as a difference of two known angles: ${\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}}$
2. Apply the sine difference identity: ${\sin(A - B) = \sin A \cos B - \cos A \sin B}$
3. Substitute the known values: ${\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}}$, ${\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}}$, ${\cos \frac{\pi}{3} = \frac{1}{2}}$, ${\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}}$
4. Simplify the expression: ${\sin \frac{\pi}{12} = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}}$

Method 2: Using the Half-Angle Identity

Another approach to find the exact value of ${\sin \frac{\pi}{12}}$ is by using the half-angle identity for sine. The half-angle identity is given by:

$$\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}$$

In our case, we want to find ${\sin \frac{\pi}{12}}$, so we can set ${\frac{x}{2} = \frac{\pi}{12}}$, which means ${x = \frac{\pi}{6}}$. The cosine of ${\frac{\pi}{6}}$ is a standard value that we know:

$$\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$$

Now, we can substitute this into the half-angle identity:

$$\sin \frac{\pi}{12} = \sin \frac{\frac{\pi}{6}}{2} = \pm \sqrt{\frac{1 - \cos \frac{\pi}{6}}{2}}$$

$$\sin \frac{\pi}{12} = \pm \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}}$$

Since ${\frac{\pi}{12}}$ is in the first quadrant, the sine value will be positive. Therefore, we take the positive square root:

$$\sin \frac{\pi}{12} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}}$$

To simplify this expression, we can multiply the numerator and denominator inside the square root by 2:

$$\sin \frac{\pi}{12} = \sqrt{\frac{2 - \sqrt{3}}{4}}$$

$$\sin \frac{\pi}{12} = \frac{\sqrt{2 - \sqrt{3}}}{2}$$

This result looks different from the one we obtained using the sine difference identity, but it is equivalent. To show this, we can manipulate the expression further. We multiply the numerator and denominator inside the square root by 2 again:

$$\sin \frac{\pi}{12} = \frac{\sqrt{2 - \sqrt{3}}}{2} = \frac{\sqrt{2(2 - \sqrt{3})}}{2\sqrt{2}} = \frac{\sqrt{4 - 2\sqrt{3}}}{2\sqrt{2}}$$

Now, we can express the term inside the square root as a square of a binomial. Notice that ${4 - 2\sqrt{3} = (\sqrt{3} - 1)^2}$, so:

$$\sin \frac{\pi}{12} = \frac{\sqrt{(\sqrt{3} - 1)^2}}{2\sqrt{2}} = \frac{\sqrt{3} - 1}{2\sqrt{2}}$$

To rationalize the denominator, we multiply the numerator and denominator by ${\sqrt{2}}$:

$$\sin \frac{\pi}{12} = \frac{(\sqrt{3} - 1)\sqrt{2}}{2\sqrt{2}\sqrt{2}} = \frac{\sqrt{6} - \sqrt{2}}{4}$$

Thus, we arrive at the same result as before, ${\sin \frac{\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}}$. This method showcases the versatility of trigonometric identities and the different paths one can take to arrive at the same solution.

Detailed Steps

1. Apply the half-angle identity: ${\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}}$
2. Set ${x = \frac{\pi}{6}}$: ${\sin \frac{\pi}{12} = \sqrt{\frac{1 - \cos \frac{\pi}{6}}{2}}}$
3. Substitute the value of ${\cos \frac{\pi}{6}}$: ${\sin \frac{\pi}{12} = \sqrt{\frac{1 - \frac{\sqrt{3}}{2}}{2}}}$
4. Simplify the expression: ${\sin \frac{\pi}{12} = \frac{\sqrt{2 - \sqrt{3}}}{2}}$
5. Further simplification: ${\sin \frac{\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}}$

Comparison of Methods

Both the sine difference identity method and the half-angle identity method are effective in finding the exact value of ${\sin \frac{\pi}{12}}$. Each method offers a unique approach and highlights different aspects of trigonometric identities. Understanding both methods provides a more comprehensive grasp of trigonometry and enhances problem-solving skills.

The sine difference identity method directly breaks down the angle ${\frac{\pi}{12}}$ into a difference of two angles with known trigonometric values. This method is straightforward and involves direct substitution into a well-known identity. The steps are clear, and the simplification process is relatively simple. This method is particularly useful when the angle can be easily expressed as a sum or difference of standard angles.

On the other hand, the half-angle identity method utilizes the half-angle formula, which can be very powerful for finding trigonometric values of angles that are half of standard angles. While the initial expression obtained using this method (${\frac{\sqrt{2 - \sqrt{3}}}{2}}$) looks different from the final answer, the process of simplifying it to match the result from the sine difference identity method demonstrates algebraic manipulation skills and a deeper understanding of trigonometric relationships. This method requires more steps in simplification but showcases the interconnectedness of different trigonometric expressions.

Summary of the Comparison

• Sine Difference Identity Method:
  • Pros: Direct, straightforward, and easy to apply.
  • Cons: Requires recognizing the angle as a difference of two standard angles.
• Half-Angle Identity Method:
  • Pros: Versatile, applicable to a wide range of angles, and reinforces algebraic manipulation skills.
  • Cons: Requires more steps in simplification, and the initial expression may not be immediately recognizable.

In conclusion, the choice of method often depends on personal preference and the specific problem at hand. Both methods, however, underscore the importance of mastering trigonometric identities and algebraic simplification techniques.

Conclusion

In this article, we successfully found the exact value of ${\sin \frac{\pi}{12}}$ using two different methods: the sine difference identity and the half-angle identity. Both methods led us to the same result, ${\sin \frac{\pi}{12} = \frac{\sqrt{6} - \sqrt{2}}{4}}$. This exercise demonstrates the power and versatility of trigonometric identities in evaluating trigonometric functions for non-standard angles.

The sine difference identity method provided a direct and efficient way to break down the problem by expressing ${\frac{\pi}{12}}$ as the difference between ${\frac{\pi}{3}}$ and ${\frac{\pi}{4}}$, two angles with well-known trigonometric values. By applying the identity ${\sin(A - B) = \sin A \cos B - \cos A \sin B}$, we were able to substitute the known values and simplify the expression to obtain the final result.

The half-angle identity method, on the other hand, offered a different perspective. By using the formula ${\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}}$, with ${x = \frac{\pi}{6}}$, we arrived at an intermediate expression that required further simplification. This method highlighted the importance of algebraic manipulation and the ability to transform one expression into another equivalent form. The process of simplifying ${\frac{\sqrt{2 - \sqrt{3}}}{2}}$ to ${\frac{\sqrt{6} - \sqrt{2}}{4}}$ reinforced the understanding of square roots and rationalization techniques.

Importance of Trigonometric Skills

Mastering these techniques is essential for anyone studying mathematics, physics, engineering, or any field that relies on trigonometric functions. The ability to find exact values of trigonometric functions not only enhances problem-solving skills but also provides a deeper understanding of the underlying mathematical principles. The methods discussed in this article can be applied to a wide range of similar problems, making them valuable tools in a mathematical toolkit.

Furthermore, this exercise underscores the interconnectedness of different mathematical concepts. Trigonometry, algebra, and simplification techniques all play a crucial role in arriving at the final solution. The ability to seamlessly integrate these concepts is a hallmark of a proficient mathematician.

In summary, finding the exact value of ${\sin \frac{\pi}{12}}$ is more than just a mathematical problem; it is an opportunity to strengthen our understanding of trigonometry, improve our algebraic skills, and appreciate the beauty and coherence of mathematics.