Trigonometry Examples

$\csc (60) \sec (30) + \cot (45)$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{2}{\sqrt{3}} \sec (30) + \cot (45)$

Step 1.2

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

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \sec (30) + \cot (45)$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

$\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} \sec (30) + \cot (45)$

Step 1.3.2

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \sec (30) + \cot (45)$

Step 1.3.3

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \sec (30) + \cot (45)$

Step 1.3.4

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} \sec (30) + \cot (45)$

Step 1.3.5

Add $1$ and $1$ .

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} \sec (30) + \cot (45)$

Step 1.3.6

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

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Step 1.3.6.1

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

$\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \sec (30) + \cot (45)$

Step 1.3.6.2

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

$\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \sec (30) + \cot (45)$

Step 1.3.6.3

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

$\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \sec (30) + \cot (45)$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} \sec (30) + \cot (45)$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{3}}{3^{1}} \sec (30) + \cot (45)$

Step 1.3.6.5

Evaluate the exponent.

$\frac{2 \sqrt{3}}{3} \sec (30) + \cot (45)$

Step 1.4

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

$\frac{2 \sqrt{3}}{3} \cdot \frac{2}{\sqrt{3}} + \cot (45)$

Step 1.5

Cancel the common factor of $\sqrt{3}$ .

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Step 1.5.1

Factor $\sqrt{3}$ out of $2 \sqrt{3}$ .

$\frac{\sqrt{3} \cdot 2}{3} \cdot \frac{2}{\sqrt{3}} + \cot (45)$

Step 1.5.2

Cancel the common factor.

$\frac{\sqrt{3} \cdot 2}{3} \cdot \frac{2}{\sqrt{3}} + \cot (45)$

Step 1.5.3

Rewrite the expression.

$\frac{2}{3} \cdot 2 + \cot (45)$

Step 1.6

Combine $\frac{2}{3}$ and $2$ .

$\frac{2 \cdot 2}{3} + \cot (45)$

Step 1.7

Multiply $2$ by $2$ .

$\frac{4}{3} + \cot (45)$

Step 1.8

The exact value of $\cot (45)$ is $1$ .

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

Step 2

Simplify the expression.

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Step 2.1

Write $1$ as a fraction with a common denominator.

$\frac{4}{3} + \frac{3}{3}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{4 + 3}{3}$

Step 2.3

Add $4$ and $3$ .

$\frac{7}{3}$