Trigonometry Examples

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

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

sin (2 x)

$\sin (2 x)$

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}

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

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}}

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

Step 3

Replace the known values in the equation.

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

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

Step 4

Simplify inside the radical.

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Step 4.1

Raise

4

$4$ to the power of

2

$2$ .

Adjacent

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

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

Step 4.2

One to any power is one.

Adjacent

= \sqrt{16 - 1 \cdot 1}

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

Step 4.3

Multiply

- 1

$- 1$ by

1

$1$ .

Adjacent

= \sqrt{16 - 1}

$= \sqrt{16 - 1}$

Step 4.4

Subtract

1

$1$ from

16

$16$ .

Adjacent

= \sqrt{15}

$= \sqrt{15}$

Adjacent

= \sqrt{15}

$= \sqrt{15}$

Step 5

Use the definition of sine to find the value of

sin (x)

$\sin (x)$ .

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

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

Step 6

Substitute in the known values.

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

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

Step 7

Apply the sine double-angle identity.

2 sin (x) cos (x)

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

Step 8

Use the definition of

s i n

$s i n$ to find the value of

sin (x)

$\sin (x)$ . In this case,

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

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

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

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

Step 9

Use the definition of

c o s

$c o s$ to find the value of

cos (x)

$\cos (x)$ . In this case,

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

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

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

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

Step 10

Substitute the values into

2 sin (x) cos (x)

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

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

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

Step 11

Evaluate

2 (\frac{1}{4}) \cdot \frac{\sqrt{15}}{4}

$2 (\frac{1}{4}) \cdot \frac{\sqrt{15}}{4}$ to find

sin (2 x)

$\sin (2 x)$ .

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Step 11.1

Cancel the common factor of

2

$2$ .

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Step 11.1.1

Factor

2

$2$ out of

4

$4$ .

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

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

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:

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