Trigonometry Examples

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

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.

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

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.

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

Step 3

Replace the known values in the equation.

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

Step 4

Simplify inside the radical.

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Step 4.1

One to any power is one.

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

Step 4.2

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

Hypotenuse $= \sqrt{1 + 4}$

Step 4.3

Add $1$ and $4$ .

Hypotenuse $= \sqrt{5}$

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.

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

Step 7

Simplify the right side.

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Step 7.1

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

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

Step 7.2

Combine and simplify the denominator.

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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 $5$ .

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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 $2$ .

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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:

$\sin (x) = 0.44721359 \dots$