QuestionJuly 24, 2025

Use the formula for the cosine of the difference of two angles to find the exact value of the expression cos((pi )/(4)-(5pi )/(6)) Rewrite the expression using a sum or difference formula. Choose the correct answer below. A. sin(5pi )/(6)cos(5pi )/(6)-sin(pi )/(4)cos(pi )/(4) B. cos(pi )/(4)cos(5pi )/(6)+sin(pi )/(4)sin(5pi )/(6) C cos(pi )/(4)cos(5pi )/(6)-sin(pi )/(4)sin(5pi )/(6) D. sin(pi )/(4)cos(5pi )/(6)+cos(pi )/(4)sin(5pi )/(6) Find the exact value of the expression. cos((pi )/(4)-(5pi )/(6))=square

Use the formula for the cosine of the difference of two angles to find the exact value of the expression cos((pi )/(4)-(5pi )/(6)) Rewrite the expression using a sum or difference formula. Choose the correct answer below. A. sin(5pi )/(6)cos(5pi )/(6)-sin(pi )/(4)cos(pi )/(4) B. cos(pi )/(4)cos(5pi )/(6)+sin(pi )/(4)sin(5pi )/(6) C cos(pi )/(4)cos(5pi )/(6)-sin(pi )/(4)sin(5pi )/(6) D. sin(pi )/(4)cos(5pi )/(6)+cos(pi )/(4)sin(5pi )/(6) Find the exact value of the expression. cos((pi )/(4)-(5pi )/(6))=square
Use the formula for the cosine of the difference of two angles to find the exact value of the expression cos((pi )/(4)-(5pi )/(6)) Rewrite the expression using a sum or difference formula. Choose the correct answer below. A. sin(5pi )/(6)cos(5pi )/(6)-sin(pi )/(4)cos(pi )/(4) B. cos(pi )/(4)cos(5pi )/(6)+sin(pi )/(4)sin(5pi )/(6) C cos(pi )/(4)cos(5pi )/(6)-sin(pi )/(4)sin(5pi )/(6) D. sin(pi )/(4)cos(5pi )/(6)+cos(pi )/(4)sin(5pi )/(6) Find the exact value of the expression. cos((pi )/(4)-(5pi )/(6))=square

Solution
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Answer

\frac{-\sqrt{6} + \sqrt{2}}{4} Explanation 1. Apply the cosine difference formula Use ** \cos(a - b) = \cos a \cos b + \sin a \sin b **. Here, a = \frac{\pi}{4} and b = \frac{5\pi}{6}. 2. Substitute angle values Substitute a and b: \cos(\frac{\pi}{4})\cos(\frac{5\pi}{6}) + \sin(\frac{\pi}{4})\sin(\frac{5\pi}{6}). 3. Calculate trigonometric values \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}, \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}, \cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}, \sin(\frac{5\pi}{6}) = \frac{1}{2}. 4. Compute the expression \frac{\sqrt{2}}{2} \cdot (-\frac{\sqrt{3}}{2}) + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}. 5. Simplify the result Combine terms: \frac{-\sqrt{6} + \sqrt{2}}{4}.

Explanation

1. Apply the cosine difference formula<br /> Use **$ \cos(a - b) = \cos a \cos b + \sin a \sin b $**. Here, $a = \frac{\pi}{4}$ and $b = \frac{5\pi}{6}$.<br /><br />2. Substitute angle values<br /> Substitute $a$ and $b$: $\cos(\frac{\pi}{4})\cos(\frac{5\pi}{6}) + \sin(\frac{\pi}{4})\sin(\frac{5\pi}{6})$.<br /><br />3. Calculate trigonometric values<br /> $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$, $\cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}$, $\sin(\frac{5\pi}{6}) = \frac{1}{2}$.<br /><br />4. Compute the expression<br /> $\frac{\sqrt{2}}{2} \cdot (-\frac{\sqrt{3}}{2}) + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$.<br /><br />5. Simplify the result<br /> Combine terms: $\frac{-\sqrt{6} + \sqrt{2}}{4}$.
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