Trigonometry Examples

$\frac{\cos (\frac{4 π}{3})}{\sin (\frac{4 π}{3})}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$\frac{- \cos (\frac{π}{3})}{\sin (\frac{4 π}{3})}$

Step 1.2

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{- \frac{1}{2}}{\sin (\frac{4 π}{3})}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$\frac{- \frac{1}{2}}{- \sin (\frac{π}{3})}$

Step 2.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{- \frac{1}{2}}{- \frac{\sqrt{3}}{2}}$

Step 3

Dividing two negative values results in a positive value.

$\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} \cdot \frac{2}{\sqrt{3}}$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2}{\sqrt{3}}$

Step 5.2

Rewrite the expression.

$\frac{1}{\sqrt{3}}$

Step 6

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

$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

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

$\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.2

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 7.3

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 7.4

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

$\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{\sqrt{3}}{3^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{\sqrt{3}}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{3}}{3}$

Decimal Form:

$0.57735026 \dots$