Trigonometry Examples

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

Step 1

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.

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

Step 2

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.

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

Step 3

Replace the known values in the equation.

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

Step 4

Simplify inside the radical.

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Step 4.1

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

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

Step 4.2

One to any power is one.

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

Step 4.3

Multiply $- 1$ by $1$ .

Adjacent $= \sqrt{16 - 1}$

Step 4.4

Subtract $1$ from $16$ .

Adjacent $= \sqrt{15}$

Step 5

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

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

Step 6

Substitute in the known values.

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

Step 7

Apply the sine double-angle identity.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Cancel the common factor of $2$ .

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Step 11.1.1

Factor $2$ out of $4$ .

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

Step 11.1.2

Cancel the common factor.

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

Step 11.1.3

Rewrite the expression.

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

Step 11.2

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

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

Step 11.3

Multiply $4$ by $2$ .

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

Step 12

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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