Find the Trig Value sin(x)=1/4 , sin(2x)

$\sin (x) = \frac{1}{4}$ , $\sin (2 x)$

Use the definition of sine to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\sin (x) = \frac{opposite}{hypotenuse}$

Find the adjacent side of the unit circle triangle. Since the hypotenuse and opposite sides are known, use the Pythagorean theorem to find the remaining side.

$Adjacent = \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Replace the known values in the equation.

$Adjacent = \sqrt{{(4)}^{2} - {(1)}^{2}}$

Simplify $\sqrt{{(4)}^{2} - {(1)}^{2}}$ .

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Raise $4$ to the power of $2$ .

Adjacent $= \sqrt{16 - {(1)}^{2}}$

One to any power is one.

Adjacent $= \sqrt{16 - 1 \cdot 1}$

Multiply $- 1$ by $1$ .

Adjacent $= \sqrt{16 - 1}$

Subtract $1$ from $16$ .

Adjacent $= \sqrt{15}$

Use the definition of sine to find the value of $\sin (x)$ .

$\sin (x) = \frac{opposite}{hypotenuse}$

Substitute in the known values.

$\sin (x) = \frac{1}{4}$

Apply the sine double–angle identity.

$2 \sin (x) \cos (x)$

Use the definition of $s i n$ to find the value of $\sin (x)$ . In this case, $\sin (x) = \frac{1}{4}$ .

$\sin (x) = \frac{1}{4}$

Use the definition of $c o s$ to find the value of $\cos (x)$ . In this case, $\cos (x) = \frac{\sqrt{15}}{4}$ .

$\cos (x) = \frac{\sqrt{15}}{4}$

Substitute the values into $2 \sin (x) \cos (x)$ .

$\sin (2 x) = 2 (\frac{1}{4}) \cdot \frac{\sqrt{15}}{4}$

Evaluate $2 (\frac{1}{4}) \cdot \frac{\sqrt{15}}{4}$ to find $\sin (2 x)$ .

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Cancel the common factor of $2$ .

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Factor $2$ out of $4$ .

$\sin (2 x) = 2 (\frac{1}{4}) \cdot \frac{\sqrt{15}}{2 (2)}$

Cancel the common factor.

$\sin (2 x) = 2 (\frac{1}{4}) \cdot \frac{\sqrt{15}}{2 \cdot 2}$

Rewrite the expression.

$\sin (2 x) = \frac{1}{4} \cdot \frac{\sqrt{15}}{2}$

Multiply $\frac{1}{4}$ and $\frac{\sqrt{15}}{2}$ .

$\sin (2 x) = \frac{\sqrt{15}}{4 \cdot 2}$

Multiply $4$ by $2$ .

$\sin (2 x) = \frac{\sqrt{15}}{8}$

The result can be shown in multiple forms.

Exact Form:

$\sin (2 x) = \frac{\sqrt{15}}{8}$

Decimal Form:

$\sin (2 x) = 0.48412291 \dots$

Find the Trig Value sin(x)=1/4 , sin(2x)

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