Find the Exact Value (sin((19pi)/3))/(sec((17pi)/6))

$\frac{\sin (\frac{19 π}{3})}{\sec (\frac{17 π}{6})}$

Rewrite $\sec (\frac{17 π}{6})$ in terms of sines and cosines.

$\frac{\sin (\frac{19 π}{3})}{\frac{1}{\cos (\frac{17 π}{6})}}$

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (\frac{17 π}{6})}$ .

$\sin (\frac{19 π}{3}) \cos (\frac{17 π}{6})$

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (\frac{π}{3}) \cos (\frac{17 π}{6})$

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

$\frac{\sqrt{3}}{2} \cos (\frac{17 π}{6})$

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{\sqrt{3}}{2} \cos (\frac{5 π}{6})$

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 second quadrant.

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

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

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

Multiply $\frac{\sqrt{3}}{2} (- \frac{\sqrt{3}}{2})$ .

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $2$ by $2$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$- \frac{3}{4}$

The result can be shown in multiple forms.

Exact Form:

$- \frac{3}{4}$

Decimal Form:

$- 0.75$

Find the Exact Value (sin((19pi)/3))/(sec((17pi)/6))

Related Questions

Support LCMGCF.com by disabling your adblocker.