Find the Exact Value sin(pi/3)cos(pi/6)-tan(pi/3)

$\sin (\frac{π}{3}) \cos (\frac{π}{6}) - \tan (\frac{π}{3})$

Step 1

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

$\frac{\sqrt{3}}{2} \cos (\frac{π}{6}) - \tan (\frac{π}{3})$

Step 2

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} - \tan (\frac{π}{3})$

Step 3

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

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

$\frac{\sqrt{3} \sqrt{3}}{2 \cdot 2} - \tan (\frac{π}{3})$

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

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

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

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

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

$\frac{{\sqrt{3}}^{1 + 1}}{2 \cdot 2} - \tan (\frac{π}{3})$

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

$\frac{{\sqrt{3}}^{2}}{4} - \tan (\frac{π}{3})$

Step 4

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$\frac{3}{4} - \tan (\frac{π}{3})$

Step 5

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

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.98205080 \dots$