Express as Sum: To find the value of cos((13 pi)/(12)), we can express (13 pi)/(12) as a sum or difference of angles whose cosine values we know from the unit circle. The angles we commonly use are multiples of pi/6, pi/4, and pi/3 because their trigonometric values are well known.Cosine of Sum: We can write (13 pi)/(12) as the sum of (9 pi)/(12) and (4 pi)/(12), which simplifies to (3 pi)/4 + pi/3. We know the cosine values for both (3 pi)/4 and pi/3.Substitute Values: Using the cosine of sum formula, cos(A + B) = cos(A)cos(B) - sin(A)sin(B), we can find the value of cos((13 pi)/(12)) by substituting A = (3 pi)/4 and B = pi/3.Perform Multiplication: The cosine of (3 pi)/4 is -√2/2 and the cosine of pi/3 is 1/2. The sine of (3 pi)/4 is √2/2 and the sine of pi/3 is √3/2.Simplify Expression: Substitute these values into the cosine of sum formula: cos((13 pi)/(12)) = cos((3 pi)/4 + pi/3) = cos((3 pi)/4)cos(pi/3) - sin((3 pi)/4)sin(pi/3).Combine Terms: Perform the multiplication: cos((13 pi)/(12)) = (-√2/2)(1/2) - (√2/2)(√3/2).Simplify the expression: cos((13 pi)/(12)) = -√2/4 - √6/4.Combine the terms: cos((13 pi)/(12)) = (-√2 - √6)/4.

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\cos\left(\frac{13\pi}{12}\right)=

Express as Sum: To find the value of $\cos\left(\frac{13 \pi}{12}\right)$ , we can express $\frac{13 \pi}{12}$ as a sum or difference of angles whose cosine values we know from the unit circle. The angles we commonly use are multiples of $\frac{\pi}{6}$ , $\frac{\pi}{4}$ , and $\frac{\pi}{3}$ because their trigonometric values are well known.
Cosine of Sum: We can write $(13 \pi)/(12)$ as the sum of $(9 \pi)/(12)$ and $(4 \pi)/(12)$ , which simplifies to $(3 \pi)/4 + \pi/3$ . We know the cosine values for both $(3 \pi)/4$ and $\pi/3$ .
Substitute Values: Using the cosine of sum formula, $\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)$ , we can find the value of $\cos\left(\frac{13 \pi}{12}\right)$ by substituting $A = \frac{3 \pi}{4}$ and $B = \frac{\pi}{3}$ .
Perform Multiplication: The cosine of $(3 \pi)/4$ is $-\sqrt{2}/2$ and the cosine of $\pi/3$ is $1/2$ . The sine of $(3 \pi)/4$ is $\sqrt{2}/2$ and the sine of $\pi/3$ is $\sqrt{3}/2$ .
Simplify Expression: Substitute these values into the cosine of sum formula: $\cos\left(\frac{13\pi}{12}\right) = \cos\left(\frac{3\pi}{4} + \frac{\pi}{3}\right) = \cos\left(\frac{3\pi}{4}\right)\cos\left(\frac{\pi}{3}\right) - \sin\left(\frac{3\pi}{4}\right)\sin\left(\frac{\pi}{3}\right)$ .
Combine Terms: Perform the multiplication: $\cos\left(\frac{$13$\pi}{$12$}\right) = \left(-\frac{\sqrt{$2$}}{$2$}\right)\left(\frac{$1$}{$2$}\right) - \left(\frac{\sqrt{$2$}}{$2$}\right)\left(\frac{\sqrt{$3$}}{$2$}\right).
Combine Terms: Perform the multiplication: $\cos\left(\frac{13 \pi}{12}\right) = \left(-\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)$. Simplify the expression: $\cos\left(\frac{13 \pi}{12}\right) = -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$.
Combine Terms: Perform the multiplication: $\cos\left(\frac{13 \pi}{12}\right) = \left(-\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)$. Simplify the expression: $\cos\left(\frac{13 \pi}{12}\right) = -\frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}$. Combine the terms: $\cos\left(\frac{13 \pi}{12}\right) = \frac{-\sqrt{2} - \sqrt{6}}{4}$.