Trigonometry Examples

$\tan^{9} (45) + \csc^{2} (60) + \sin (30) + \cos^{2} (60)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

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

$1^{9} + \csc^{2} (60) + \sin (30) + \cos^{2} (60)$

Step 1.2

One to any power is one.

$1 + \csc^{2} (60) + \sin (30) + \cos^{2} (60)$

Step 1.3

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

$1 + {(\frac{2}{\sqrt{3}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.4

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

$1 + {(\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5

Combine and simplify the denominator.

Tap for more steps...

Step 1.5.1

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

$1 + {(\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5.2

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

$1 + {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5.3

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

$1 + {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5.4

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

$1 + {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5.5

Add $1$ and $1$ .

$1 + {(\frac{2 \sqrt{3}}{{\sqrt{3}}^{2}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5.6

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

Tap for more steps...

Step 1.5.6.1

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

$1 + {(\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5.6.2

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

$1 + {(\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5.6.3

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

$1 + {(\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.6.4.1

Cancel the common factor.

$1 + {(\frac{2 \sqrt{3}}{3^{\frac{2}{2}}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5.6.4.2

Rewrite the expression.

$1 + {(\frac{2 \sqrt{3}}{3^{1}})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.5.6.5

Evaluate the exponent.

$1 + {(\frac{2 \sqrt{3}}{3})}^{2} + \sin (30) + \cos^{2} (60)$

Step 1.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 1.6.1

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

$1 + \frac{{(2 \sqrt{3})}^{2}}{3^{2}} + \sin (30) + \cos^{2} (60)$

Step 1.6.2

Apply the product rule to $2 \sqrt{3}$ .

$1 + \frac{2^{2} {\sqrt{3}}^{2}}{3^{2}} + \sin (30) + \cos^{2} (60)$

Step 1.7

Simplify the numerator.

Tap for more steps...

Step 1.7.1

Raise $2$ to the power of $2$ .

$1 + \frac{4 {\sqrt{3}}^{2}}{3^{2}} + \sin (30) + \cos^{2} (60)$

Step 1.7.2

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

Tap for more steps...

Step 1.7.2.1

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

$1 + \frac{4 {(3^{\frac{1}{2}})}^{2}}{3^{2}} + \sin (30) + \cos^{2} (60)$

Step 1.7.2.2

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

$1 + \frac{4 \cdot 3^{\frac{1}{2} \cdot 2}}{3^{2}} + \sin (30) + \cos^{2} (60)$

Step 1.7.2.3

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

$1 + \frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}} + \sin (30) + \cos^{2} (60)$

Step 1.7.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.7.2.4.1

Cancel the common factor.

$1 + \frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}} + \sin (30) + \cos^{2} (60)$

Step 1.7.2.4.2

Rewrite the expression.

$1 + \frac{4 \cdot 3^{1}}{3^{2}} + \sin (30) + \cos^{2} (60)$

Step 1.7.2.5

Evaluate the exponent.

$1 + \frac{4 \cdot 3}{3^{2}} + \sin (30) + \cos^{2} (60)$

Step 1.8

Raise $3$ to the power of $2$ .

$1 + \frac{4 \cdot 3}{9} + \sin (30) + \cos^{2} (60)$

Step 1.9

Multiply $4$ by $3$ .

$1 + \frac{12}{9} + \sin (30) + \cos^{2} (60)$

Step 1.10

Cancel the common factor of $12$ and $9$ .

Tap for more steps...

Step 1.10.1

Factor $3$ out of $12$ .

$1 + \frac{3 (4)}{9} + \sin (30) + \cos^{2} (60)$

Step 1.10.2

Cancel the common factors.

Tap for more steps...

Step 1.10.2.1

Factor $3$ out of $9$ .

$1 + \frac{3 \cdot 4}{3 \cdot 3} + \sin (30) + \cos^{2} (60)$

Step 1.10.2.2

Cancel the common factor.

$1 + \frac{3 \cdot 4}{3 \cdot 3} + \sin (30) + \cos^{2} (60)$

Step 1.10.2.3

Rewrite the expression.

$1 + \frac{4}{3} + \sin (30) + \cos^{2} (60)$

Step 1.11

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$1 + \frac{4}{3} + \frac{1}{2} + \cos^{2} (60)$

Step 1.12

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

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

Step 1.13

Apply the product rule to $\frac{1}{2}$ .

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

Step 1.14

One to any power is one.

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

Step 1.15

Raise $2$ to the power of $2$ .

$1 + \frac{4}{3} + \frac{1}{2} + \frac{1}{4}$

Step 2

Find the common denominator.

Tap for more steps...

Step 2.1

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

$\frac{1}{1} + \frac{4}{3} + \frac{1}{2} + \frac{1}{4}$

Step 2.2

Multiply $\frac{1}{1}$ by $\frac{12}{12}$ .

$\frac{1}{1} \cdot \frac{12}{12} + \frac{4}{3} + \frac{1}{2} + \frac{1}{4}$

Step 2.3

Multiply $\frac{1}{1}$ by $\frac{12}{12}$ .

$\frac{12}{12} + \frac{4}{3} + \frac{1}{2} + \frac{1}{4}$

Step 2.4

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

$\frac{12}{12} + \frac{4}{3} \cdot \frac{4}{4} + \frac{1}{2} + \frac{1}{4}$

Step 2.5

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

$\frac{12}{12} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{1}{2} + \frac{1}{4}$

Step 2.6

Multiply $\frac{1}{2}$ by $\frac{6}{6}$ .

$\frac{12}{12} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{1}{2} \cdot \frac{6}{6} + \frac{1}{4}$

Step 2.7

Multiply $\frac{1}{2}$ by $\frac{6}{6}$ .

$\frac{12}{12} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{6}{2 \cdot 6} + \frac{1}{4}$

Step 2.8

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$\frac{12}{12} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{6}{2 \cdot 6} + \frac{1}{4} \cdot \frac{3}{3}$

Step 2.9

Multiply $\frac{1}{4}$ by $\frac{3}{3}$ .

$\frac{12}{12} + \frac{4 \cdot 4}{3 \cdot 4} + \frac{6}{2 \cdot 6} + \frac{3}{4 \cdot 3}$

Step 2.10

Multiply $3$ by $4$ .

$\frac{12}{12} + \frac{4 \cdot 4}{12} + \frac{6}{2 \cdot 6} + \frac{3}{4 \cdot 3}$

Step 2.11

Multiply $2$ by $6$ .

$\frac{12}{12} + \frac{4 \cdot 4}{12} + \frac{6}{12} + \frac{3}{4 \cdot 3}$

Step 2.12

Reorder the factors of $4 \cdot 3$ .

$\frac{12}{12} + \frac{4 \cdot 4}{12} + \frac{6}{12} + \frac{3}{3 \cdot 4}$

Step 2.13

Multiply $3$ by $4$ .

$\frac{12}{12} + \frac{4 \cdot 4}{12} + \frac{6}{12} + \frac{3}{12}$

Step 3

Combine the numerators over the common denominator.

$\frac{12 + 4 \cdot 4 + 6 + 3}{12}$

Step 4

Simplify the expression.

Tap for more steps...

Step 4.1

Multiply $4$ by $4$ .

$\frac{12 + 16 + 6 + 3}{12}$

Step 4.2

Add $12$ and $16$ .

$\frac{28 + 6 + 3}{12}$

Step 4.3

Add $28$ and $6$ .

$\frac{34 + 3}{12}$

Step 4.4

Add $34$ and $3$ .

$\frac{37}{12}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{37}{12}$

Decimal Form:

$3.08\overline{3}$

Mixed Number Form:

$3 \frac{1}{12}$