If $$ \sin heta=\frac{2}{3}, 0

Question

If $$ \sin 	heta=\frac{2}{3}, 0<	heta<\frac{\pi}{2} $$, find the exact value of each of the following. $$ \begin{array}{lll}	ext { (a) } \sin (2 	heta) & 	ext { (b) } \cos (2 	heta) & 	ext { (c) } \sin \frac{	heta}{2}\end{array} $$ (d) $$ \cos \frac{	heta}{2} $$$$ ~ \begin{array}{l}	ext { (a) } \sin (2 	heta)=\frac{4 \sqrt{5}}{9} \ 	ext { (Type an exact answer, using radicals as needed.) } \ 	ext { (b) } \cos (2 	heta)=\frac{1}{9} \ 	ext { (Type an exact answer, using radicals as needed.) } \ 	ext { (c) } \sin \frac{	heta}{2}=\sqrt{\frac{3-\sqrt{5}}{6}} \ 	ext { (Type an exact answer, using radicals as needed.) } \ 	ext { (d) } \cos \frac{	heta}{2}=\square \ 	ext { (Type an exact answer, using radicals as needed.) }\end{array} $$

UpStudy AI · Accepted Answer

To find $$ \cos \frac{\theta}{2} $$ given that $$ \sin \theta = \frac{2}{3} $$ and $$ 0 < \theta < \frac{\pi}{2} $$, we can use the half-angle identity:

$$
\cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}}
$$

**Step 1: Find $$ \cos \theta $$**

Given $$ \sin \theta = \frac{2}{3} $$, we use the Pythagorean identity:

$$
\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{2}{3}\right)^2} = \sqrt{1 - \frac{4}{9}} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}
$$

**Step 2: Apply the Half-Angle Identity**

$$
\cos \frac{\theta}{2} = \sqrt{\frac{1 + \frac{\sqrt{5}}{3}}{2}} = \sqrt{\frac{3 + \sqrt{5}}{6}}
$$

Thus, the exact value is:

$$
\cos \frac{\theta}{2} = \sqrt{\frac{3 + \sqrt{5}}{6}}
$$

**Answer:**  
Problem (d) Answer:

$$
\cos \frac{\theta}{2} = \sqrt{\frac{\,3 + \sqrt{5}\,}{\,6\,}}
$$
$$ \cos \frac{\theta}{2} = \sqrt{\frac{3 + \sqrt{5}}{6}} $$

If , find the exact value of each of the following.

(d)

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