Question

find the exact value without a calculator. double - angle formulas: \\( \sin ( 2 \theta ) = 2 \sin \theta \cos \theta \\) \\( \cos ( 2 \theta ) = \cos ^ { 2 } \theta - \sin ^ { 2 } \theta \\) \\( \tan ( 2 \theta ) = \frac { 2 \tan \theta } { 1 - \tan ^ { 2 } \theta } \\) half - angle formulas: \\( \sin ( \frac { \theta } { 2 } ) = \pm \sqrt { \frac { 1 - \cos \theta } { 2 } } \\) \\( \cos ( \frac { \theta } { 2 } ) = \pm \sqrt { \frac { 1 + \cos \theta } { 2 } } \\) \\( \tan ( \frac { \theta } { 2 } ) = \pm \sqrt { \frac { 1 - \cos \theta } { 1 + \cos \theta } } \\)

Explanation:

Step1: Express \(195^{\circ}\) as a half - angle

Since \(195^{\circ}=\frac{390^{\circ}}{2}\), we use the half - angle formula for sine \(\sin(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{2}}\). Here \(\theta = 390^{\circ}\), and \(195^{\circ}\) is in the third quadrant where \(\sin\) is negative.

Step2: Find \(\cos(390^{\circ})\)

We know that \(\cos(390^{\circ})=\cos(360^{\circ}+30^{\circ})\). Using the identity \(\cos(A + 360^{\circ})=\cos A\), so \(\cos(390^{\circ})=\cos(30^{\circ})=\frac{\sqrt{3}}{2}\).

Step3: Substitute into the half - angle formula

Substitute \(\cos\theta=\frac{\sqrt{3}}{2}\) into \(\sin(\frac{\theta}{2})=-\sqrt{\frac{1 - \cos\theta}{2}}\) (negative because \(195^{\circ}\) is in QIII).

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Answer:

\(-\frac{\sqrt{2-\sqrt{3}}}{2}\)