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Sagot :
Certainly! Let's solve the given problem using the half-angle identities:
We want to find the exact value of [tex]\(\sin \left(\frac{7 \pi}{12}\right)\)[/tex].
1. Double the Angle:
Start by letting [tex]\(\theta = \frac{7 \pi}{12}\)[/tex]. Then we can express [tex]\(\theta\)[/tex] as half of a larger angle:
[tex]\[ \theta = \frac{7 \pi}{12} = \frac{1}{2} \left(\frac{7 \pi}{6}\right) \][/tex]
Thus, we need to find [tex]\(\theta = \frac{7 \pi}{12}\)[/tex], where the larger angle is [tex]\( \frac{7 \pi}{6} \)[/tex].
2. Using the Cosine of the Larger Angle:
To use the half-angle identity, we need the cosine of the double angle. The double angle here is:
[tex]\[ 2\theta = \frac{7 \pi}{6} \][/tex]
3. Determine [tex]\(\cos \left(\frac{7 \pi}{6}\right)\)[/tex]:
On the unit circle, [tex]\(\frac{7 \pi}{6}\)[/tex] is in the third quadrant where the cosine value is negative. Given that [tex]\(\frac{7 \pi}{6} = \pi + \frac{\pi}{6}\)[/tex], the cosine value is the same as [tex]\(\cos\left(\frac{\pi}{6}\right)\)[/tex] but negative, so:
[tex]\[ \cos \left(\frac{7 \pi}{6}\right) = - \cos \left(\frac{\pi}{6}\right) = - \frac{\sqrt{3}}{2} \][/tex]
4. Half-angle Identity for Sine:
The half-angle identity for sine is given by:
[tex]\[ \sin \left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos(x)}{2}} \][/tex]
Here, [tex]\(x = \frac{7 \pi}{6}\)[/tex], so the identity becomes:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{1 - \cos \left(\frac{7 \pi}{6}\right)}{2}} \][/tex]
5. Plug in the Value of Cosine:
Now, substitute [tex]\(\cos \left(\frac{7 \pi}{6}\right) = -\frac{\sqrt{3}}{2}\)[/tex] into the half-angle identity:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{1 - \left(-\frac{\sqrt{3}}{2}\right)}{2}} \][/tex]
Simplify within the square root:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \][/tex]
6. Simplify Inside the Square Root:
Combine the terms inside the fraction:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{2}} \][/tex]
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}} \][/tex]
Multiply the denominator by 2:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{2 + \sqrt{3}}{4}} \][/tex]
7. Final Simplification:
Simplify the square root:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2} \][/tex]
8. Numerical Result:
[tex]\(\sin \left(\frac{7 \pi}{12}\right) = 0.9659258262890683\)[/tex]
Thus, the exact value of [tex]\(\sin \left(\frac{7 \pi}{12}\right)\)[/tex] is:
[tex]\[ \boxed{\frac{\sqrt{2 + \sqrt{3}}}{2}} \][/tex]
And the numerical approximation is:
[tex]\[ 0.9659258262890683 \][/tex]
We want to find the exact value of [tex]\(\sin \left(\frac{7 \pi}{12}\right)\)[/tex].
1. Double the Angle:
Start by letting [tex]\(\theta = \frac{7 \pi}{12}\)[/tex]. Then we can express [tex]\(\theta\)[/tex] as half of a larger angle:
[tex]\[ \theta = \frac{7 \pi}{12} = \frac{1}{2} \left(\frac{7 \pi}{6}\right) \][/tex]
Thus, we need to find [tex]\(\theta = \frac{7 \pi}{12}\)[/tex], where the larger angle is [tex]\( \frac{7 \pi}{6} \)[/tex].
2. Using the Cosine of the Larger Angle:
To use the half-angle identity, we need the cosine of the double angle. The double angle here is:
[tex]\[ 2\theta = \frac{7 \pi}{6} \][/tex]
3. Determine [tex]\(\cos \left(\frac{7 \pi}{6}\right)\)[/tex]:
On the unit circle, [tex]\(\frac{7 \pi}{6}\)[/tex] is in the third quadrant where the cosine value is negative. Given that [tex]\(\frac{7 \pi}{6} = \pi + \frac{\pi}{6}\)[/tex], the cosine value is the same as [tex]\(\cos\left(\frac{\pi}{6}\right)\)[/tex] but negative, so:
[tex]\[ \cos \left(\frac{7 \pi}{6}\right) = - \cos \left(\frac{\pi}{6}\right) = - \frac{\sqrt{3}}{2} \][/tex]
4. Half-angle Identity for Sine:
The half-angle identity for sine is given by:
[tex]\[ \sin \left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos(x)}{2}} \][/tex]
Here, [tex]\(x = \frac{7 \pi}{6}\)[/tex], so the identity becomes:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{1 - \cos \left(\frac{7 \pi}{6}\right)}{2}} \][/tex]
5. Plug in the Value of Cosine:
Now, substitute [tex]\(\cos \left(\frac{7 \pi}{6}\right) = -\frac{\sqrt{3}}{2}\)[/tex] into the half-angle identity:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{1 - \left(-\frac{\sqrt{3}}{2}\right)}{2}} \][/tex]
Simplify within the square root:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} \][/tex]
6. Simplify Inside the Square Root:
Combine the terms inside the fraction:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{3}}{2}}{2}} \][/tex]
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{\frac{2 + \sqrt{3}}{2}}{2}} \][/tex]
Multiply the denominator by 2:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{2 + \sqrt{3}}{4}} \][/tex]
7. Final Simplification:
Simplify the square root:
[tex]\[ \sin \left(\frac{7 \pi}{12}\right) = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2} \][/tex]
8. Numerical Result:
[tex]\(\sin \left(\frac{7 \pi}{12}\right) = 0.9659258262890683\)[/tex]
Thus, the exact value of [tex]\(\sin \left(\frac{7 \pi}{12}\right)\)[/tex] is:
[tex]\[ \boxed{\frac{\sqrt{2 + \sqrt{3}}}{2}} \][/tex]
And the numerical approximation is:
[tex]\[ 0.9659258262890683 \][/tex]
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