Trigonometry Examples

${(4 \cos (30) - 6 \sin (120))}^{- 2}$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

${(4 \frac{\sqrt{3}}{2} - 6 \sin (120))}^{- 2}$

Step 1.2

Cancel the common factor of $2$ .

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Step 1.2.1

Factor $2$ out of $4$ .

${(2 (2) \frac{\sqrt{3}}{2} - 6 \sin (120))}^{- 2}$

Step 1.2.2

Cancel the common factor.

${(2 \cdot 2 \frac{\sqrt{3}}{2} - 6 \sin (120))}^{- 2}$

Step 1.2.3

Rewrite the expression.

${(2 \sqrt{3} - 6 \sin (120))}^{- 2}$

Step 1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

${(2 \sqrt{3} - 6 \sin (60))}^{- 2}$

Step 1.4

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

${(2 \sqrt{3} - 6 \frac{\sqrt{3}}{2})}^{- 2}$

Step 1.5

Cancel the common factor of $2$ .

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Step 1.5.1

Factor $2$ out of $- 6$ .

${(2 \sqrt{3} + 2 (- 3) \frac{\sqrt{3}}{2})}^{- 2}$

Step 1.5.2

Cancel the common factor.

${(2 \sqrt{3} + 2 \cdot - 3 \frac{\sqrt{3}}{2})}^{- 2}$

Step 1.5.3

Rewrite the expression.

${(2 \sqrt{3} - 3 \sqrt{3})}^{- 2}$

Step 2

Subtract $3 \sqrt{3}$ from $2 \sqrt{3}$ .

${(- \sqrt{3})}^{- 2}$

Step 3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4

Simplify the denominator.

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Step 4.1

Apply the product rule to $- \sqrt{3}$ .

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

Step 4.2

Raise $- 1$ to the power of $2$ .

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

Cancel the common factor of $2$ .

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Step 4.3.4.1

Cancel the common factor.

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

Step 4.3.4.2

Rewrite the expression.

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

Step 4.3.5

Evaluate the exponent.

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

Step 4.4

Multiply $3$ by $1$ .

$\frac{1}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$