Question

if \\( \tan \theta = \frac { 2 } { 3 }, 0 < \theta < \frac { \pi } { 2 } \\), find \\( \sin \frac { \theta } { 2 } \\).

\\( \sin \frac { \theta } { 2 } = \square \\)
(simplify your answer, including any radicals. use integers or fractions for a

Explanation:

Step1: Find \(\cos\theta\)

Given \(\tan\theta=\frac{2}{3}=\frac{\text{opposite}}{\text{adjacent}}\). By the Pythagorean theorem, the hypotenuse \(r = \sqrt{2^{2}+3^{2}}=\sqrt{4 + 9}=\sqrt{13}\). Then \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{3}{\sqrt{13}}=\frac{3\sqrt{13}}{13}\).

Step2: Use the half - angle formula for sine

The half - angle formula for sine is \(\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}\). Since \(0\lt\theta\lt\frac{\pi}{2}\), then \(0\lt\frac{\theta}{2}\lt\frac{\pi}{4}\), and \(\sin\frac{\theta}{2}\gt0\).
Substitute \(\cos\theta=\frac{3\sqrt{13}}{13}\) into the formula:

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Answer:

\(\sqrt{\frac{13 - 3\sqrt{13}}{26}}\)