Precalculus Examples

$\sin (30) \cos (60) - \cos (30) \sin (60)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{1}{2} \cos (60) - \cos (30) \sin (60)$

Step 1.2

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$\frac{1}{2} \cdot \frac{1}{2} - \cos (30) \sin (60)$

Step 1.3

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 1.3.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{1}{2 \cdot 2} - \cos (30) \sin (60)$

Step 1.3.2

Multiply $2$ by $2$ .

$\frac{1}{4} - \cos (30) \sin (60)$

Step 1.4

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

$\frac{1}{4} - \frac{\sqrt{3}}{2} \sin (60)$

Step 1.5

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

$\frac{1}{4} - \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}$

Step 1.6

Multiply $- \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 1.6.1

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{1}{4} - \frac{\sqrt{3} \sqrt{3}}{2 \cdot 2}$

Step 1.6.2

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

$\frac{1}{4} - \frac{{\sqrt{3}}^{1} \sqrt{3}}{2 \cdot 2}$

Step 1.6.3

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

$\frac{1}{4} - \frac{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}{2 \cdot 2}$

Step 1.6.4

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

$\frac{1}{4} - \frac{{\sqrt{3}}^{1 + 1}}{2 \cdot 2}$

Step 1.6.5

Add $1$ and $1$ .

$\frac{1}{4} - \frac{{\sqrt{3}}^{2}}{2 \cdot 2}$

Step 1.6.6

Multiply $2$ by $2$ .

$\frac{1}{4} - \frac{{\sqrt{3}}^{2}}{4}$

Step 1.7

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

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Step 1.7.1

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

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

Step 1.7.2

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

$\frac{1}{4} - \frac{3^{\frac{1}{2} \cdot 2}}{4}$

Step 1.7.3

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

$\frac{1}{4} - \frac{3^{\frac{2}{2}}}{4}$

Step 1.7.4

Cancel the common factor of $2$ .

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Step 1.7.4.1

Cancel the common factor.

$\frac{1}{4} - \frac{3^{\frac{2}{2}}}{4}$

Step 1.7.4.2

Rewrite the expression.

$\frac{1}{4} - \frac{3^{1}}{4}$

Step 1.7.5

Evaluate the exponent.

$\frac{1}{4} - \frac{3}{4}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{1 - 3}{4}$

Step 2.2

Subtract $3$ from $1$ .

$\frac{- 2}{4}$

Step 2.3

Cancel the common factor of $- 2$ and $4$ .

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Step 2.3.1

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{4}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot - 1}{2 \cdot 2}$

Step 2.3.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 2}$

Step 2.3.2.3

Rewrite the expression.

$\frac{- 1}{2}$

Step 2.4

Move the negative in front of the fraction.

$- \frac{1}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$