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If
cos theta=(1)/(6), then what is the positive value of
tan ((1)/(2)theta), in simplest radical form with a rational denominator?

If $\cos \theta=\frac{1}{6}$ , then what is the positive value of $\tan \frac{1}{2} \theta$ , in simplest radical form with a rational denominator?

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Find trigonometric ratios using multiple identities

Full solution

Q. If

\cos \theta=\frac{1}{6}

, then what is the positive value of

\tan \frac{1}{2} \theta

, in simplest radical form with a rational denominator?

Apply double angle formula: Use the double angle formula for tangent: $\tan(\frac{1}{2}\theta) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$ . Substitute $\cos(\theta) = \frac{1}{6}$ into the formula. $\tan(\frac{1}{2}\theta) = \pm\sqrt{\frac{1 - \frac{1}{6}}{1 + \frac{1}{6}}}$ .
Substitute $\cos(\theta):$ Simplify the expression inside the square root. $\newline$ $\tan\left(\frac{1}{2}\theta\right) = \pm\sqrt{\left(\frac{6}{6} - \frac{1}{6}\right) / \left(\frac{6}{6} + \frac{1}{6}\right)}.$ $\newline$ $\tan\left(\frac{1}{2}\theta\right) = \pm\sqrt{\left(\frac{5}{6}\right) / \left(\frac{7}{6}\right)}.$
Simplify expression: Simplify the fraction inside the square root by multiplying the numerator and denominator by $6$ . $\tan\left(\frac{1}{2}\theta\right) = \pm\sqrt{\frac{5}{7}}$ .
Simplify fraction: Since we are looking for the positive value, choose the positive square root. $\tan\left(\frac{1}{2}\theta\right) = \sqrt{\frac{5}{7}}$ .
Choose positive value: Rationalize the denominator. $\newline$ $\tan\left(\frac{1}{2}\theta\right) = \sqrt{\frac{5}{7}} \times \sqrt{\frac{7}{7}}.$ $\newline$ $\tan\left(\frac{1}{2}\theta\right) = \sqrt{\frac{35}{49}}.$
Rationalize denominator: Simplify the square root. $\tan\left(\frac{1}{2}\theta\right) = \frac{\sqrt{35}}{7}$ .

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