find the following.\n\n\\( \\tan \\frac { \\theta } { 2 } \\), given \\( \\sin \\theta = \\frac { 12 } { 13…

find the following.\n\n\\( \\tan \\frac { \\theta } { 2 } \\), given \\( \\sin \\theta = \\frac { 12 } { 13 }, 0 ^ { \\circ } < \\theta < 90 ^ { \\circ } \\)\n\n\\( \\tan \\frac { \\theta } { 2 } = \\square \\)\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expressio

find the following.\n\n\\( \\tan \\frac { \\theta } { 2 } \\), given \\( \\sin \\theta = \\frac { 12 } { 13 }, 0 ^ { \\circ } < \\theta < 90 ^ { \\circ } \\)\n\n\\( \\tan \\frac { \\theta } { 2 } = \\square \\)\n(simplify your answer, including any radicals. use integers or fractions for any numbers in the expressio

Answer

Explanation:

Step1: Find (\cos\theta)

Use the identity (\sin^{2}\theta+\cos^{2}\theta = 1). Given (\sin\theta=\frac{12}{13}), then (\cos\theta=\sqrt{1 - \sin^{2}\theta}=\sqrt{1-\left(\frac{12}{13}\right)^{2}}=\sqrt{\frac{169 - 144}{169}}=\sqrt{\frac{25}{169}}=\frac{5}{13}) (since (0^{\circ}<\theta<90^{\circ}), (\cos\theta>0)).

Step2: Use the half - angle formula for tangent

The half - angle formula for tangent is (\tan\frac{\theta}{2}=\frac{1-\cos\theta}{\sin\theta}). Substitute (\sin\theta=\frac{12}{13}) and (\cos\theta = \frac{5}{13}) into the formula: (\tan\frac{\theta}{2}=\frac{1-\frac{5}{13}}{\frac{12}{13}}=\frac{\frac{13 - 5}{13}}{\frac{12}{13}}=\frac{8}{12}=\frac{2}{3})

Answer:

(\frac{2}{3})