Trigonometry Examples

$2 \sec (60) \csc (300)$

Step 1

The exact value of $\sec (60)$ is $2$ .

$2 \cdot 2 \csc (300)$

Step 2

Multiply $2$ by $2$ .

$4 \csc (300)$

Step 3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the fourth quadrant.

$4 (- \csc (60))$

Step 4

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

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

Step 5

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

$4 (- (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}))$

Step 6

Combine and simplify the denominator.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Add $1$ and $1$ .

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

Step 6.6

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

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Step 6.6.1

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

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

Step 6.6.2

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

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

Step 6.6.3

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

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

Step 6.6.4

Cancel the common factor of $2$ .

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Step 6.6.4.1

Cancel the common factor.

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

Step 6.6.4.2

Rewrite the expression.

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

Step 6.6.5

Evaluate the exponent.

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

Step 7

Multiply $4 (- \frac{2 \sqrt{3}}{3})$ .

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Step 7.1

Multiply $- 1$ by $4$ .

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

Step 7.2

Combine $- 4$ and $\frac{2 \sqrt{3}}{3}$ .

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

Step 7.3

Multiply $2$ by $- 4$ .

$\frac{- 8 \sqrt{3}}{3}$

Step 8

Move the negative in front of the fraction.

$- \frac{8 \sqrt{3}}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{8 \sqrt{3}}{3}$

Decimal Form:

$- 4.61880215 \dots$