Trigonometry Examples

sin (x)

$\sin (x)$ ,

tan (x) = \frac{1}{2}

$\tan (x) = \frac{1}{2}$

Step 1

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

tan (x) = \frac{opposite}{adjacent}

$\tan (x) = \frac{opposite}{adjacent}$

Step 2

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

Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

Hypotenuse = \sqrt{{(1)}^{2} + {(2)}^{2}}

$Hypotenuse = \sqrt{{(1)}^{2} + {(2)}^{2}}$

Step 4

Simplify inside the radical.

Tap for more steps...

Step 4.1

One to any power is one.

Hypotenuse

= \sqrt{1 + {(2)}^{2}}

$= \sqrt{1 + {(2)}^{2}}$

Step 4.2

Raise

2

$2$ to the power of

2

$2$ .

Hypotenuse

= \sqrt{1 + 4}

$= \sqrt{1 + 4}$

Step 4.3

Add

1

$1$ and

4

$4$ .

Hypotenuse

= \sqrt{5}

$= \sqrt{5}$

Hypotenuse

= \sqrt{5}

$= \sqrt{5}$

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}{\sqrt{5}}

$\sin (x) = \frac{1}{\sqrt{5}}$

Step 7

Simplify the right side.

Tap for more steps...

Step 7.1

Multiply

\frac{1}{\sqrt{5}}

$\frac{1}{\sqrt{5}}$ by

\frac{\sqrt{5}}{\sqrt{5}}

$\frac{\sqrt{5}}{\sqrt{5}}$ .

sin (x) = \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}

$\sin (x) = \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 7.2

Combine and simplify the denominator.

Tap for more steps...

Step 7.2.1

Multiply

$\frac{1}{\sqrt{5}}$ by

$\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (x) = \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.2.2

Raise

$\sqrt{5}$ to the power of

$1$ .

$\sin (x) = \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.2.3

Raise

$\sqrt{5}$ to the power of

$1$ .

$\sin (x) = \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.2.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (x) = \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 7.2.5

Add

$1$ and

$1$ .

$\sin (x) = \frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 7.2.6

Rewrite

${\sqrt{5}}^{2}$ as

$5$ .

Tap for more steps...

Step 7.2.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{5}$ as

$5^{\frac{1}{2}}$ .

$\sin (x) = \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 7.2.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sin (x) = \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 7.2.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sin (x) = \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 7.2.6.4

Cancel the common factor of

$2$ .

Tap for more steps...

Step 7.2.6.4.1

Cancel the common factor.

$\sin (x) = \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 7.2.6.4.2

Rewrite the expression.

$\sin (x) = \frac{\sqrt{5}}{5}$

Step 7.2.6.5

Evaluate the exponent.

$\sin (x) = \frac{\sqrt{5}}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sin (x) = \frac{\sqrt{5}}{5}$

Decimal Form:

$\sin (x) = 0.44721359 \dots$

Enter YOUR Problem