Answer :
To find the exact value of [tex]\(\cos \left(195^{\circ}\right)\)[/tex], let's go through the following steps:
1. Convert 195 degrees to radians:
The angle in radians can be obtained using the formula [tex]\(\theta \text{ (in radians)} = \theta \text{ (in degrees)} \cdot \frac{\pi}{180}\)[/tex].
[tex]\[ 195^{\circ} = 195 \cdot \frac{\pi}{180} \approx 3.4033920413889427 \text{ radians} \][/tex]
2. Consider the angle's position on the unit circle:
[tex]\(195^{\circ}\)[/tex] is in the third quadrant, where the cosine values are negative.
3. Use angle addition formula for cosine:
[tex]\(195^{\circ}\)[/tex] can be written as [tex]\(180^{\circ} + 15^{\circ}\)[/tex].
We can use the angle addition formula [tex]\(\cos(180^{\circ} + x) = -\cos(x)\)[/tex].
Here, [tex]\(x = 15^{\circ}\)[/tex].
4. Evaluate [tex]\(\cos(15^{\circ})\)[/tex] using known trigonometric identities:
[tex]\(\cos(15^{\circ})\)[/tex] is given by:
[tex]\[ \cos(15^{\circ}) = \cos(45^{\circ} - 30^{\circ}) \][/tex]
Using the cosine subtraction identity:
[tex]\[ \cos(45^{\circ} - 30^{\circ}) = \cos(45^{\circ})\cos(30^{\circ}) + \sin(45^{\circ})\sin(30^{\circ}) \][/tex]
We know that [tex]\(\cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\)[/tex], [tex]\(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\)[/tex], and [tex]\(\sin(30^{\circ}) = \frac{1}{2}\)[/tex].
5. Substitute these values:
[tex]\[ \cos(15^{\circ}) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \][/tex]
[tex]\[ \cos(15^{\circ}) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \][/tex]
6. Apply the angle addition formula:
[tex]\[ \cos(195^{\circ}) = \cos(180^{\circ} + 15^{\circ}) = -\cos(15^{\circ}) = -\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) \][/tex]
Thus, the exact value of [tex]\(\cos \left(195^{\circ}\right)\)[/tex] is:
[tex]\[ \cos \left(195^{\circ}\right) = -\frac{\sqrt{6} + \sqrt{2}}{4} \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{\frac{-\sqrt{6}-\sqrt{2}}{4}} \][/tex]
1. Convert 195 degrees to radians:
The angle in radians can be obtained using the formula [tex]\(\theta \text{ (in radians)} = \theta \text{ (in degrees)} \cdot \frac{\pi}{180}\)[/tex].
[tex]\[ 195^{\circ} = 195 \cdot \frac{\pi}{180} \approx 3.4033920413889427 \text{ radians} \][/tex]
2. Consider the angle's position on the unit circle:
[tex]\(195^{\circ}\)[/tex] is in the third quadrant, where the cosine values are negative.
3. Use angle addition formula for cosine:
[tex]\(195^{\circ}\)[/tex] can be written as [tex]\(180^{\circ} + 15^{\circ}\)[/tex].
We can use the angle addition formula [tex]\(\cos(180^{\circ} + x) = -\cos(x)\)[/tex].
Here, [tex]\(x = 15^{\circ}\)[/tex].
4. Evaluate [tex]\(\cos(15^{\circ})\)[/tex] using known trigonometric identities:
[tex]\(\cos(15^{\circ})\)[/tex] is given by:
[tex]\[ \cos(15^{\circ}) = \cos(45^{\circ} - 30^{\circ}) \][/tex]
Using the cosine subtraction identity:
[tex]\[ \cos(45^{\circ} - 30^{\circ}) = \cos(45^{\circ})\cos(30^{\circ}) + \sin(45^{\circ})\sin(30^{\circ}) \][/tex]
We know that [tex]\(\cos(45^{\circ}) = \sin(45^{\circ}) = \frac{\sqrt{2}}{2}\)[/tex], [tex]\(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\)[/tex], and [tex]\(\sin(30^{\circ}) = \frac{1}{2}\)[/tex].
5. Substitute these values:
[tex]\[ \cos(15^{\circ}) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) \][/tex]
[tex]\[ \cos(15^{\circ}) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \][/tex]
6. Apply the angle addition formula:
[tex]\[ \cos(195^{\circ}) = \cos(180^{\circ} + 15^{\circ}) = -\cos(15^{\circ}) = -\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) \][/tex]
Thus, the exact value of [tex]\(\cos \left(195^{\circ}\right)\)[/tex] is:
[tex]\[ \cos \left(195^{\circ}\right) = -\frac{\sqrt{6} + \sqrt{2}}{4} \][/tex]
Therefore, the correct choice is:
[tex]\[ \boxed{\frac{-\sqrt{6}-\sqrt{2}}{4}} \][/tex]
