Answer:
[tex]\displaystyle \cos\left(-\frac{7\,\pi}{12}\right) = \frac{\sqrt{2} - \sqrt{6}}{4}[/tex].
Step-by-step explanation:
Convert the angle [tex]\displaystyle \left(-\frac{7\, \pi}{12}\right)[/tex] to degrees:
[tex]\displaystyle \left(-\frac{7\, \pi}{12}\right) = \left(-\frac{7\, \pi}{12}\right) \times \frac{180^\circ}{\pi} = -105^\circ[/tex].
Note, that [tex]\left(-105^\circ\right)[/tex] is the sum of two common angles: [tex]\left(-45^\circ\right)[/tex] and [tex]\left(-60^\circ\right)[/tex].
- [tex]\displaystyle \cos\left(-45^\circ\right) = \cos\left(45^\circ\right) = \frac{\sqrt{2}}{2}[/tex].
- [tex]\displaystyle \cos\left(-60^\circ\right) = \cos\left(60^\circ\right) = \frac{1}{2}[/tex].
- [tex]\displaystyle \sin\left(-45^\circ\right) = -\sin\left(45^\circ\right) = -\frac{\sqrt{2}}{2}[/tex].
- [tex]\displaystyle \sin\left(-60^\circ\right) = -\sin\left(60^\circ\right) = -\frac{\sqrt{3}}{2}[/tex].
By the sum-angle identity of cosine:
[tex]\cos(A + B) = \cos(A)\cdot \cos(B) - \sin(A) \cdot \sin(B)[/tex].
Apply the sum formula for cosine to find the exact value of [tex]\cos\left(-105^\circ \right)[/tex].
[tex]\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) \\ &= \frac{\sqrt{2}}{2} \times \frac{1}{2} - \left(-\frac{\sqrt{2}}{2}\right)\times \left(-\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{2} - \sqrt{6}}{4}\end{aligned}[/tex].
[tex]\displaystyle \left(-\frac{7\, \pi}{12}\right) = \left(-\frac{7\, \pi}{12}\right) \times \frac{180^\circ}{\pi} = -105^\circ[/tex]. In other words, [tex]\displaystyle \left(-\frac{7\, \pi}{12}\right)[/tex] and [tex]\left(-105^\circ\right)[/tex] correspond to the same angle. Therefore, the cosine of [tex]\displaystyle \left(-\frac{7\, \pi}{12}\right)\![/tex] would be equal to the cosine of [tex]\left(-105^\circ\right)\![/tex].
[tex]\displaystyle \cos\left(-\frac{7\,\pi}{12}\right) = \cos\left(-105^\circ\right) = \frac{\sqrt{2} - \sqrt{6}}{4}[/tex].
