Find the exact value of the expression: cos(7pi/4 – 11pi/6)
The Correct Answer and Explanation is :
To find the exact value of the expression:
[
\cos\left(\frac{7\pi}{4} – \frac{11\pi}{6}\right)
]
Step 1: Compute the Angle Difference
We first find the difference between the angles:
[
\frac{7\pi}{4} – \frac{11\pi}{6}
]
To subtract these fractions, we need a common denominator. The least common denominator between 4 and 6 is 12. Rewriting both fractions:
[
\frac{7\pi}{4} = \frac{21\pi}{12}, \quad \frac{11\pi}{6} = \frac{22\pi}{12}
]
Now subtract:
[
\frac{21\pi}{12} – \frac{22\pi}{12} = \frac{-\pi}{12}
]
Step 2: Use the Cosine Identity
We use the identity for cosine of a negative angle:
[
\cos(-x) = \cos(x)
]
Thus,
[
\cos\left(\frac{-\pi}{12}\right) = \cos\left(\frac{\pi}{12}\right)
]
Step 3: Find the Exact Value of (\cos(\pi/12))
Using the cosine difference identity:
[
\cos(A – B) = \cos A \cos B + \sin A \sin B
]
Let:
[
A = \frac{\pi}{4}, \quad B = \frac{\pi}{6}
]
[
\cos\left(\frac{\pi}{12}\right) = \cos\left(\frac{\pi}{4} – \frac{\pi}{6}\right)
]
Using known values:
[
\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}
]
[
\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}, \quad \sin \frac{\pi}{6} = \frac{1}{2}
]
[
\cos \frac{\pi}{12} = \left(\frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} \right) + \left(\frac{\sqrt{2}}{2} \times \frac{1}{2} \right)
]
[
= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}
]
[
= \frac{\sqrt{6} + \sqrt{2}}{4}
]
Conclusion
[
\cos\left(\frac{7\pi}{4} – \frac{11\pi}{6}\right) = \frac{\sqrt{6} + \sqrt{2}}{4}
]
Now, I will generate an image representing the solution.
Here is the mathematical diagram illustrating the calculation of (\cos(7\pi/4 – 11\pi/6)). It includes the unit circle, labeled angles, the computed angle difference, the cosine identity, and the final exact result. Let me know if you need any modifications!