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diketahui $\cos x=\frac{3}{4}$ , maka nilai dari $\sin \frac{x}{2}$ adalah

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Factor by grouping

Full solution

Q. diketahui

\cos x=\frac{3}{4}

, maka nilai dari

\sin \frac{x}{2}

adalah

Use Half-Angle Identity: We know that $\cos(x) = \frac{3}{4}$ . To find $\sin(\frac{x}{2})$ , we can use the half-angle identity for sine, which is $\sin(\frac{x}{2}) = \pm\sqrt{(\frac{1 - \cos(x)}{2})}$ . We need to determine the sign (positive or negative) based on the quadrant in which $x$ lies. However, since we are not given the quadrant, we will assume $x$ is in a quadrant where sine is positive.
Plug in cos(x): First, we plug the value of $\cos(x)$ into the half-angle identity for sine: $\sin(\frac{x}{2}) = \pm\sqrt{(\frac{1 - \cos(x)}{2})} = \pm\sqrt{(\frac{1 - \frac{3}{4}}{2})}$ .
Simplify Expression: Now, we simplify the expression inside the square root: $(1 - \frac{3}{4}) = \frac{1}{4}$ . So, $\sin(\frac{x}{2}) = \pm\sqrt{(\frac{1}{4}/2)}$ .
Take Square Root: Next, we simplify the fraction inside the square root: $(\frac{1}{4})/2 = \frac{1}{8}$ . Therefore, $\sin(\frac{x}{2}) = \pm\sqrt{\frac{1}{8}}$ .
Rationalize Denominator: We take the square root of $\frac{1}{8}$ : $\sqrt{\frac{1}{8}} = \frac{\sqrt{1}}{\sqrt{8}} = \frac{1}{\sqrt{8}}$ . To rationalize the denominator, we multiply the numerator and the denominator by $\sqrt{8}$ : $\frac{1}{\sqrt{8}} \times \frac{\sqrt{8}}{\sqrt{8}} = \frac{\sqrt{8}}{8}$ .
Simplify $\sqrt{8}$ : We simplify $\sqrt{8}$ to $2\sqrt{2}$ , because $\sqrt{8} = \sqrt{4\cdot2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$ . So, $\sin\left(\frac{x}{2}\right) = \pm\left(\frac{2\sqrt{2}}{8}\right)$ .
Final Simplification: Finally, we simplify the fraction $\frac{2\sqrt{2}}{8}$ by dividing both the numerator and the denominator by $2$ : $\sin\left(\frac{x}{2}\right) = \pm\left(\frac{\sqrt{2}}{4}\right)$ .

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