Trigonometry Examples

$\sin (θ) = \frac{1}{2}$ , $\tan (θ)$

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.

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

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.

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

Step 4

Simplify inside the radical.

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Step 4.1

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

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

Step 4.2

One to any power is one.

Adjacent $= \sqrt{4 - 1 \cdot 1}$

Step 4.3

Multiply $- 1$ by $1$ .

Adjacent $= \sqrt{4 - 1}$

Step 4.4

Subtract $1$ from $4$ .

Adjacent $= \sqrt{3}$

Step 5

Use the definition of tangent to find the value of $\tan (θ)$ .

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

Step 6

Substitute in the known values.

$\tan (θ) = \frac{1}{\sqrt{3}}$

Step 7

Simplify the right side.

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Step 7.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\tan (θ) = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7.2

Combine and simplify the denominator.

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Step 7.2.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\tan (θ) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\tan (θ) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\tan (θ) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\tan (θ) = \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.2.5

Add $1$ and $1$ .

$\tan (θ) = \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 7.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\tan (θ) = \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 7.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\tan (θ) = \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 7.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\tan (θ) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

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.

$\tan (θ) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.2.6.4.2

Rewrite the expression.

$\tan (θ) = \frac{\sqrt{3}}{3}$

Step 7.2.6.5

Evaluate the exponent.

$\tan (θ) = \frac{\sqrt{3}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\tan (θ) = \frac{\sqrt{3}}{3}$

Decimal Form:

$\tan (θ) = 0.57735026 \dots$

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