If $\sin θ = \frac{2}{5}, 0 < θ < \frac{π}{2}$ , find the exact value of each of the following.
(a) $\sin (2 θ)$
(b) $\cos (2 θ)$
(c) $\sin \frac{θ}{2}$
(d) $\cos \frac{θ}{2}$
(a) $\sin (2 θ) = ◻$
(Type an exact answer, using radicals as needed.)

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Final Answer: $\sin (2 θ) = \frac{4 \sqrt{21}}{25}$

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Step 1 ：We are given that $\sin θ = \frac{2}{5}$ and $0 < θ < \frac{π}{2}$ . We are asked to find the value of $\sin (2 θ)$ .

Step 2 ：We can use the double angle formula for sine, which is $\sin (2 θ) = 2 \sin θ \cos θ$ . However, we only know the value of $\sin θ$ , not $\cos θ$ .

Step 3 ：We can find $\cos θ$ using the Pythagorean identity $\sin^{2} θ + \cos^{2} θ = 1$ . Solving for $\cos θ$ , we get $\cos θ = \sqrt{1 - \sin^{2} θ} = \sqrt{1 - {(\frac{2}{5})}^{2}} = \sqrt{1 - \frac{4}{25}} = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5}$ .

Step 4 ：Now that we have the value of $\cos θ$ , we can substitute it into the double angle formula to find $\sin (2 θ)$ . Substituting $\sin θ = \frac{2}{5}$ and $\cos θ = \frac{\sqrt{21}}{5}$ into the formula, we get $\sin (2 θ) = 2 \cdot \frac{2}{5} \cdot \frac{\sqrt{21}}{5} = \frac{4 \sqrt{21}}{25}$ .

Step 5 ：Final Answer: $\sin (2 θ) = \frac{4 \sqrt{21}}{25}$

If sin⁡θ=25,0<θ<π2, find the exact value of each of the following.(a) sin⁡(2θ)(b) cos⁡(2θ)(c) sin⁡θ2(d) cos⁡θ2(a) sin⁡(2θ)=◻(Type an exact answer, using radicals as needed.)

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Popular Problems

If $\sin θ = \frac{2}{5}, 0 < θ < \frac{π}{2}$ , find the exact value of each of the following.
(a) $\sin (2 θ)$
(b) $\cos (2 θ)$
(c) $\sin \frac{θ}{2}$
(d) $\cos \frac{θ}{2}$
(a) $\sin (2 θ) = ◻$
(Type an exact answer, using radicals as needed.)