Trigonometry Examples

$\sin (θ) = \frac{\sqrt{7}}{6}$

Step 1

To find the value of $\csc (θ)$ , use the fact that $\frac{1}{\sin (θ)}$ then substitute in the known values.

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{1}{\frac{\sqrt{7}}{6}}$

Step 2

Simplify the result.

Tap for more steps...

Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = \frac{1}{\sin (θ)} = 1 (\frac{6}{\sqrt{7}})$

Step 2.2

Multiply $\frac{6}{\sqrt{7}}$ by $1$ .

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6}{\sqrt{7}}$

Step 2.3

Multiply $\frac{6}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 2.4

Combine and simplify the denominator.

Tap for more steps...

Step 2.4.1

Multiply $\frac{6}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.4.4

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

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 2.4.5

Add $1$ and $1$ .

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 2.4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Tap for more steps...

Step 2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 2.4.6.2

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

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 2.4.6.3

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

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.4.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.6.4.1

Cancel the common factor.

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.4.6.4.2

Rewrite the expression.

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{7}$

Step 2.4.6.5

Evaluate the exponent.

$\csc (θ) = \frac{1}{\sin (θ)} = \frac{6 \sqrt{7}}{7}$