Find Exact Value Of Cos(5π/6)cos(π/12) + Sin(5π/6)sin(π/12)
This article delves into the process of finding the exact value of the trigonometric expression: $\cos \left(\frac{5 \pi}{6}\right) \cos \left(\frac{\pi}{12}\right)+\sin \left(\frac{5 \pi}{6}\right) \sin \left(\frac{\pi}{12}\right)$. We will explore the underlying trigonometric identities and apply them step-by-step to arrive at the solution. This comprehensive guide aims to provide a clear understanding of the concepts involved and the methodology used, making it easier for students and enthusiasts to grasp the intricacies of trigonometric calculations.
Understanding the Core Trigonometric Identity
To effectively tackle this problem, the first step involves recognizing a fundamental trigonometric identity. The given expression closely resembles the cosine angle addition formula. This formula is a cornerstone of trigonometry and is expressed as follows:
However, our expression features a '+' sign between the cosine and sine products, which suggests we should consider a slightly modified version of this formula. By changing the sign within the cosine function, we arrive at the relevant identity:
This identity is crucial because it perfectly matches the structure of the expression we need to evaluate. By recognizing this match, we can significantly simplify the problem. We can see that if we let $A = \frac{5\pi}{6}$ and $B = \frac{\pi}{12}$, the given expression transforms into a more manageable form. This recognition is the critical first step in solving the problem, allowing us to move from a complex expression to a simpler cosine function.
By identifying the correct trigonometric identity, we lay the groundwork for a straightforward solution. This step underscores the importance of understanding and memorizing key trigonometric identities, as they serve as powerful tools for simplifying complex expressions and solving trigonometric equations. In the following sections, we will apply this identity to our specific problem and proceed to calculate the exact value of the expression.
Applying the Cosine Difference Identity
Having identified the appropriate trigonometric identity, which is $\cos(A - B) = \cos(A) \cos(B) + \sin(A) \sin(B)$, the next step is to apply it to our specific problem. The given expression is $\cos \left(\frac{5 \pi}{6}\right) \cos \left(\frac{\pi}{12}\right)+\sin \left(\frac{5 \pi}{6}\right) \sin \left(\frac{\pi}{12}\right)$. As discussed earlier, we can directly map this expression to the cosine difference identity by setting $A = \frac{5\pi}{6}$ and $B = \frac{\pi}{12}$. This substitution transforms our expression into a simpler form:
This transformation is a significant simplification because it combines the four trigonometric terms into a single cosine term. Now, our task is reduced to evaluating the cosine of the difference between the two angles. To do this, we first need to find the difference between the angles, which involves basic fraction arithmetic. The subtraction within the cosine function is the next step in our calculation.
To subtract the fractions, we need a common denominator. The least common multiple of 6 and 12 is 12, so we rewrite $\frac{5\pi}{6}$ with a denominator of 12. Multiplying both the numerator and the denominator by 2, we get $\frac{10\pi}{12}$. Now, we can subtract the fractions:
This simplifies to $\frac{3 \pi}{4}$. Therefore, our expression now becomes:
We have successfully reduced the original complex expression to the cosine of a single angle. The next step involves evaluating this cosine function, which will give us the exact value of the initial expression. This process highlights the power of trigonometric identities in simplifying complex calculations and making them more manageable.
Evaluating cos(3π/4)
After applying the cosine difference identity, we have simplified the original expression to $\cos\left(\frac{3 \pi}{4}\right)$. The next crucial step is to evaluate this cosine function to find its exact value. To do this, we need to understand the properties of the cosine function and its values at specific angles, particularly those related to the unit circle.
The angle $ rac{3 \pi}{4}$ is located in the second quadrant of the unit circle. In this quadrant, the cosine function is negative. We can relate $ rac{3 \pi}{4}$ to a reference angle in the first quadrant, which will help us determine its cosine value. The reference angle is the acute angle formed by the terminal side of the angle and the x-axis. For $ rac{3 \pi}{4}$, the reference angle is $\pi - \frac{3 \pi}{4} = \frac{\pi}{4}$.
We know that $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$. Since $ rac{3 \pi}{4}$ is in the second quadrant where cosine is negative, we have:
This is the exact value of the cosine function at $ rac{3 \pi}{4}$. Therefore, the exact value of the original expression $\cos \left(\frac{5 \pi}{6}\right) \cos \left(\frac{\pi}{12}\right)+\sin \left(\frac{5 \pi}{6}\right) \sin \left(\frac{\pi}{12}\right)$ is also $-\frac{\sqrt{2}}{2}$. This result concludes our calculation, providing a precise answer to the problem.
By understanding the unit circle and the properties of trigonometric functions in different quadrants, we can accurately evaluate trigonometric expressions at various angles. This knowledge is essential for solving a wide range of trigonometric problems and is a fundamental concept in mathematics.
Conclusion: The Exact Value and the Process
In conclusion, by applying the cosine difference identity and evaluating the resulting cosine function, we have successfully found the exact value of the given trigonometric expression. The initial problem was to find the value of $\cos \left(\frac{5 \pi}{6}\right) \cos \left(\frac{\pi}{12}\right)+\sin \left(\frac{5 \pi}{6}\right) \sin \left(\frac{\pi}{12}\right)$. Through a step-by-step process, we have determined that the exact value is $-\frac{\sqrt{2}}{2}$.
This solution involved several key steps. First, we recognized the structure of the expression and identified the appropriate trigonometric identity, which was the cosine difference identity: $\cos(A - B) = \cos(A) \cos(B) + \sin(A) \sin(B)$. This recognition was crucial because it allowed us to simplify the expression significantly.
Next, we applied the identity by substituting $A = \frac{5\pi}{6}$ and $B = \frac{\pi}{12}$, which transformed the expression into $\cos\left(\frac{5 \pi}{6} - \frac{\pi}{12}\right)$. We then performed the subtraction within the cosine function, finding a common denominator and simplifying the angle to $\frac{3 \pi}{4}$. This further reduced the problem to evaluating $\cos\left(\frac{3 \pi}{4}\right)$.
Finally, we evaluated the cosine function at this angle by understanding its position on the unit circle and its relationship to the reference angle. Since $\frac{3 \pi}{4}$ is in the second quadrant where cosine is negative, and the reference angle is $\frac{\pi}{4}$, we determined that $\cos\left(\frac{3 \pi}{4}\right) = -\frac{\sqrt{2}}{2}$. This gave us the exact value of the original expression.
This problem highlights the importance of understanding trigonometric identities, the unit circle, and the properties of trigonometric functions. By mastering these concepts, we can effectively solve complex trigonometric problems and find exact values, providing a solid foundation for further studies in mathematics and related fields.
Through this detailed exploration, we have not only found the solution but also reinforced the underlying principles and techniques involved in trigonometric calculations. This comprehensive approach is essential for developing a deep and lasting understanding of the subject matter.
