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Sagot :
To determine the exact value of [tex]\(\cos \left(\frac{5 \pi}{6}+\frac{7 \pi}{4}\right)\)[/tex], we can use the sum identity for cosine, which states:
[tex]\[ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \][/tex]
Given [tex]\(a = \frac{5 \pi}{6}\)[/tex] and [tex]\(b = \frac{7 \pi}{4}\)[/tex], let's compute each component step-by-step.
1. Calculate [tex]\(\cos \left(\frac{5 \pi}{6}\right)\)[/tex]:
The angle [tex]\(\frac{5 \pi}{6}\)[/tex] is in the second quadrant. The reference angle is [tex]\(\pi - \frac{5 \pi}{6} = \frac{\pi}{6}\)[/tex].
In the second quadrant, cosine is negative:
[tex]\[ \cos \left(\frac{5 \pi}{6}\right) = -\cos \left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} \][/tex]
2. Calculate [tex]\(\cos \left(\frac{7 \pi}{4}\right)\)[/tex]:
The angle [tex]\(\frac{7 \pi}{4}\)[/tex] is in the fourth quadrant. The reference angle is [tex]\(2\pi - \frac{7 \pi}{4} = \frac{\pi}{4}\)[/tex].
In the fourth quadrant, cosine is positive:
[tex]\[ \cos \left(\frac{7 \pi}{4}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \][/tex]
3. Calculate [tex]\(\sin \left(\frac{5 \pi}{6}\right)\)[/tex]:
The angle [tex]\(\frac{5 \pi}{6}\)[/tex] is in the second quadrant. The reference angle is [tex]\(\pi - \frac{5 \pi}{6} = \frac{\pi}{6}\)[/tex].
In the second quadrant, sine is positive:
[tex]\[ \sin \left(\frac{5 \pi}{6}\right) = \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \][/tex]
4. Calculate [tex]\(\sin \left(\frac{7 \pi}{4}\right)\)[/tex]:
The angle [tex]\(\frac{7 \pi}{4}\)[/tex] is in the fourth quadrant. The reference angle is [tex]\(2\pi - \frac{7 \pi}{4} = \frac{\pi}{4}\)[/tex].
In the fourth quadrant, sine is negative:
[tex]\[ \sin \left(\frac{7 \pi}{4}\right) = -\sin \left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \][/tex]
5. Compute [tex]\(\cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right)\)[/tex]:
[tex]\[ \cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right) = \cos \left(\frac{5 \pi}{6}\right) \cos \left(\frac{7 \pi}{4}\right) - \sin \left(\frac{5 \pi}{6}\right) \sin \left(\frac{7 \pi}{4}\right) \][/tex]
Substitute the values found above:
[tex]\[ \cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right) = \left(-\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \left(\frac{1}{2}\right) \left(-\frac{\sqrt{2}}{2}\right) \][/tex]
Simplify the expression:
[tex]\[ \cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right) = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \][/tex]
Combine the terms:
[tex]\[ \cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right) = \frac{-\sqrt{6} + \sqrt{2}}{4} \][/tex]
Numerically, this result equates to approximately [tex]\(-0.2588\)[/tex].
Therefore, the exact value of [tex]\(\cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right)\)[/tex] is
[tex]\[ \frac{-\sqrt{6} + \sqrt{2}}{4} \approx -0.2588 \][/tex]
[tex]\[ \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \][/tex]
Given [tex]\(a = \frac{5 \pi}{6}\)[/tex] and [tex]\(b = \frac{7 \pi}{4}\)[/tex], let's compute each component step-by-step.
1. Calculate [tex]\(\cos \left(\frac{5 \pi}{6}\right)\)[/tex]:
The angle [tex]\(\frac{5 \pi}{6}\)[/tex] is in the second quadrant. The reference angle is [tex]\(\pi - \frac{5 \pi}{6} = \frac{\pi}{6}\)[/tex].
In the second quadrant, cosine is negative:
[tex]\[ \cos \left(\frac{5 \pi}{6}\right) = -\cos \left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2} \][/tex]
2. Calculate [tex]\(\cos \left(\frac{7 \pi}{4}\right)\)[/tex]:
The angle [tex]\(\frac{7 \pi}{4}\)[/tex] is in the fourth quadrant. The reference angle is [tex]\(2\pi - \frac{7 \pi}{4} = \frac{\pi}{4}\)[/tex].
In the fourth quadrant, cosine is positive:
[tex]\[ \cos \left(\frac{7 \pi}{4}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \][/tex]
3. Calculate [tex]\(\sin \left(\frac{5 \pi}{6}\right)\)[/tex]:
The angle [tex]\(\frac{5 \pi}{6}\)[/tex] is in the second quadrant. The reference angle is [tex]\(\pi - \frac{5 \pi}{6} = \frac{\pi}{6}\)[/tex].
In the second quadrant, sine is positive:
[tex]\[ \sin \left(\frac{5 \pi}{6}\right) = \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \][/tex]
4. Calculate [tex]\(\sin \left(\frac{7 \pi}{4}\right)\)[/tex]:
The angle [tex]\(\frac{7 \pi}{4}\)[/tex] is in the fourth quadrant. The reference angle is [tex]\(2\pi - \frac{7 \pi}{4} = \frac{\pi}{4}\)[/tex].
In the fourth quadrant, sine is negative:
[tex]\[ \sin \left(\frac{7 \pi}{4}\right) = -\sin \left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \][/tex]
5. Compute [tex]\(\cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right)\)[/tex]:
[tex]\[ \cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right) = \cos \left(\frac{5 \pi}{6}\right) \cos \left(\frac{7 \pi}{4}\right) - \sin \left(\frac{5 \pi}{6}\right) \sin \left(\frac{7 \pi}{4}\right) \][/tex]
Substitute the values found above:
[tex]\[ \cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right) = \left(-\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \left(\frac{1}{2}\right) \left(-\frac{\sqrt{2}}{2}\right) \][/tex]
Simplify the expression:
[tex]\[ \cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right) = -\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} \][/tex]
Combine the terms:
[tex]\[ \cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right) = \frac{-\sqrt{6} + \sqrt{2}}{4} \][/tex]
Numerically, this result equates to approximately [tex]\(-0.2588\)[/tex].
Therefore, the exact value of [tex]\(\cos \left(\frac{5 \pi}{6} + \frac{7 \pi}{4}\right)\)[/tex] is
[tex]\[ \frac{-\sqrt{6} + \sqrt{2}}{4} \approx -0.2588 \][/tex]
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