How do you find the exact values of cos(11pi/12) using the half angle formula?

Question

Victoria Adebayo · Accepted Answer

Find #cos ((11pi)/12)#

Ans: #- sqrt(2 + sqrt3)/2# Call #cos ((11pi)/12) = cos t# #cos 2t = cos ((22pi)/12) = cos ((11pi)/6) = cos ((pi)/6) = sqrt3/2# Apply the trig identity: #cos 2t = 2cos^2 t - 1 # #cos 2t = sqrt3/2 = 2cos^2 t - 1# #2cos^2 t = 1 + sqrt3/2 = (2 + sqrt3)/2# #cos^2 t = (2 + sqrt3)/4# #cos t = cos ((11pi)/12) = +- sqrt(2 + sqrt3)/2.# Since the arc ((11pi)/12) is located in Quadrant II, only the negative answer is accepted. #cos ((11pi)/12) = - sqrt(2 + sqrt3)/2# Check by calculator. #Arc ((11pi)/12) = 165# deg-> #cos ((11pi)/12) = cos 165 = - 0.97.# #- (sqrt(2 + sqrt3)/2) = - 0.97#. OK

Brooklyn Anderson · Answer

undefined To find the exact value of $\cos\left(\frac{11\pi}{12}\right)$ using the half-angle formula, we can first express $\frac{11\pi}{12}$ as a sum or difference of two angles whose cosine values are known. Since $\frac{11\pi}{12} = \frac{22\pi}{24} - \frac{\pi}{24}$, we can use the half-angle formula for cosine, which states that $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos(\theta)}{2}}$.

Now, we find the cosine of $\frac{\pi}{12}$ and $\frac{\pi}{24}$ using known values or angles that can be expressed in terms of these angles. Once we have these values, we substitute them into the half-angle formula to find the cosine of $\frac{11\pi}{12}$.

Let's denote $x = \frac{\pi}{12}$. Then, $\cos(2x) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$. By using the double-angle formula for cosine, we have $\cos(2x) = 2\cos^2(x) - 1$. Solving for $\cos(x)$, we get $\cos(x) = \sqrt{\frac{1 + \cos(2x)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}}$.

Now, we also need $\cos\left(\frac{\pi}{24}\right)$. Let $y = \frac{\pi}{24}$. Then, $\cos(2y) = \cos\left(\frac{\pi}{12}\right)$. Using the double-angle formula again, we have $\cos(2y) = 2\cos^2(y) - 1$. Solving for $\cos(y)$, we find $\cos(y) = \sqrt{\frac{1 + \cos(2y)}{2}} = \sqrt{\frac{1 + \cos\left(\frac{\pi}{12}\right)}{2}}$.

Finally, using the angle addition formula for cosine, $\cos\left(\frac{11\pi}{12}\right) = \cos\left(\frac{\pi}{12} + \frac{\pi}{24}\right) = \cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)$. Substitute the values of $\cos(x)$, $\cos(y)$, $\sin(x)$, and $\sin(y)$, and calculate to find the exact value of $\cos\left(\frac{11\pi}{12}\right)$.