Question

Find an exact value.

Cos(-7pi/12)
A. Sqrt 6 + sqrt 2
B. Sqrt 6 - sqrt 2 / 4
C. Sqrt 2 - sqrt 6
Sqrt 2 - sqrt 6 / 4

Answer 1

`\displaystyle \cos\left(-(7\,\pi)/(12)\right) = (√(2) - √(6))/(4)`.

Explanation:

Convert the angle
`\displaystyle \left(-(7\, \pi)/(12)\right)` to degrees:

`\displaystyle \left(-(7\, \pi)/(12)\right) = \left(-(7\, \pi)/(12)\right) * (180^\circ)/(\pi) = -105^\circ`.

Note, that
`\left(-105^\circ\right)` is the sum of two common angles:
`\left(-45^\circ\right)` and
`\left(-60^\circ\right)`.

• 
  `\displaystyle \cos\left(-45^\circ\right) = \cos\left(45^\circ\right) = (√(2))/(2)`.
• 
  `\displaystyle \cos\left(-60^\circ\right) = \cos\left(60^\circ\right) = (1)/(2)`.

• 
  `\displaystyle \sin\left(-45^\circ\right) = -\sin\left(45^\circ\right) = -(√(2))/(2)`.
• 
  `\displaystyle \sin\left(-60^\circ\right) = -\sin\left(60^\circ\right) = -(√(3))/(2)`.

By the sum-angle identity of cosine:

`\cos(A + B) = \cos(A)\cdot \cos(B) - \sin(A) \cdot \sin(B)`.

Apply the sum formula for cosine to find the exact value of
`\cos\left(-105^\circ \right)`.

`\begin{aligned}\cos\left(-105^\circ \right) &= \cos\left(\left(-45^\circ\right) + \left(-60^\circ\right)\right) \\ &= \cos\left(-45^\circ\right) \cdot \cos\left(-60^\circ\right)\right) - \sin\left(-45^\circ\right) \cdot \sin\left(-60^\circ\right)\right) \\ &= (√(2))/(2) * (1)/(2) - \left(-(√(2))/(2)\right)* \left(-(√(3))/(2)\right) = (√(2) - √(6))/(4)\end{aligned}`.

`\displaystyle \left(-(7\, \pi)/(12)\right) = \left(-(7\, \pi)/(12)\right) * (180^\circ)/(\pi) = -105^\circ`. In other words,
`\displaystyle \left(-(7\, \pi)/(12)\right)` and
`\left(-105^\circ\right)` correspond to the same angle. Therefore, the cosine of
`\displaystyle \left(-(7\, \pi)/(12)\right)\!` would be equal to the cosine of
`\left(-105^\circ\right)\!`.

`\displaystyle \cos\left(-(7\,\pi)/(12)\right) = \cos\left(-105^\circ\right) = (√(2) - √(6))/(4)`.