Trigonometry Examples

$\frac{\cos (45)}{\sec (30) + \csc (60)}$

Step 1

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2}}{\sec (30) + \csc (60)}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2}{\sqrt{3}} + \csc (60)}$

Step 2.2

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} + \csc (60)}$

Step 2.3

Combine and simplify the denominator.

Tap for more steps...

Step 2.3.1

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}} + \csc (60)}$

Step 2.3.2

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} + \csc (60)}$

Step 2.3.3

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} + \csc (60)}$

Step 2.3.4

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}} + \csc (60)}$

Step 2.3.5

Add $1$ and $1$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}} + \csc (60)}$

Step 2.3.6

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

Tap for more steps...

Step 2.3.6.1

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} + \csc (60)}$

Step 2.3.6.2

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}} + \csc (60)}$

Step 2.3.6.3

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} + \csc (60)}$

Step 2.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.6.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3^{\frac{2}{2}}} + \csc (60)}$

Step 2.3.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3^{1}} + \csc (60)}$

Step 2.3.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3} + \csc (60)}$

Step 2.4

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3} + \frac{2}{\sqrt{3}}}$

Step 2.5

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

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

Step 2.6

Combine and simplify the denominator.

Tap for more steps...

Step 2.6.1

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}}$

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

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

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

Step 2.6.5

Add $1$ and $1$ .

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}}$

Step 2.6.6

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

Tap for more steps...

Step 2.6.6.1

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

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

Step 2.6.6.2

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

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

Step 2.6.6.3

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

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 2.6.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.6.6.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}}$

Step 2.6.6.4.2

Rewrite the expression.

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

Step 2.6.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3}}{3} + \frac{2 \sqrt{3}}{3}}$

Step 2.7

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{2}}{2}}{\frac{2 \sqrt{3} + 2 \sqrt{3}}{3}}$

Step 2.8

Add $2 \sqrt{3}$ and $2 \sqrt{3}$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Multiply $\frac{3}{4 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5

Combine and simplify the denominator.

Tap for more steps...

Step 5.1

Multiply $\frac{3}{4 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.2

Move $\sqrt{3}$ .

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

$\frac{\sqrt{2}}{2} \cdot \frac{3 \sqrt{3}}{4 {\sqrt{3}}^{1 + 1}}$

Step 5.6

Add $1$ and $1$ .

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

Step 5.7

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

Tap for more steps...

Step 5.7.1

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

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

Step 5.7.2

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

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

Step 5.7.3

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

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

Step 5.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.7.4.1

Cancel the common factor.

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

Step 5.7.4.2

Rewrite the expression.

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

Step 5.7.5

Evaluate the exponent.

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

Step 6

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{4}$ .

Tap for more steps...

Step 7.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{4}$ .

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

Step 7.2

Combine using the product rule for radicals.

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

Step 7.3

Multiply $2$ by $3$ .

$\frac{\sqrt{6}}{2 \cdot 4}$

Step 7.4

Multiply $2$ by $4$ .

$\frac{\sqrt{6}}{8}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6}}{8}$

Decimal Form:

$0.30618621 \dots$