Algebra Examples

$K = \frac{(\sin (30 °) + \sec (60 °)) (\tan (45 °) + \sec^{2} (45 °))}{5 \cos (37 °)}$

Step 1

Simplify the numerator.

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Step 1.1

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

$K = \frac{(\frac{1}{2} + \sec (60 °)) (\tan (45 °) + \sec^{2} (45 °))}{5 \cos (37 °)}$

Step 1.2

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

$K = \frac{(\frac{1}{2} + 2) (\tan (45 °) + \sec^{2} (45 °))}{5 \cos (37 °)}$

Step 1.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$K = \frac{(\frac{1}{2} + 2 \cdot \frac{2}{2}) (\tan (45 °) + \sec^{2} (45 °))}{5 \cos (37 °)}$

Step 1.4

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

$K = \frac{(\frac{1}{2} + \frac{2 \cdot 2}{2}) (\tan (45 °) + \sec^{2} (45 °))}{5 \cos (37 °)}$

Step 1.5

Combine the numerators over the common denominator.

$K = \frac{\frac{1 + 2 \cdot 2}{2} (\tan (45 °) + \sec^{2} (45 °))}{5 \cos (37 °)}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $2$ by $2$ .

$K = \frac{\frac{1 + 4}{2} (\tan (45 °) + \sec^{2} (45 °))}{5 \cos (37 °)}$

Step 1.6.2

Add $1$ and $4$ .

$K = \frac{\frac{5}{2} (\tan (45 °) + \sec^{2} (45 °))}{5 \cos (37 °)}$

Step 1.7

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

$K = \frac{\frac{5}{2} (1 + \sec^{2} (45 °))}{5 \cos (37 °)}$

Step 1.8

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

$K = \frac{\frac{5}{2} (1 + {(\frac{2}{\sqrt{2}})}^{2})}{5 \cos (37 °)}$

Step 1.9

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

$K = \frac{\frac{5}{2} (1 + {(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2})}{5 \cos (37 °)}$

Step 1.10

Combine and simplify the denominator.

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Step 1.10.1

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

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2})}{5 \cos (37 °)}$

Step 1.10.2

Raise $\sqrt{2}$ to the power of $1$ .

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2})}{5 \cos (37 °)}$

Step 1.10.3

Raise $\sqrt{2}$ to the power of $1$ .

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2})}{5 \cos (37 °)}$

Step 1.10.4

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

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2})}{5 \cos (37 °)}$

Step 1.10.5

Add $1$ and $1$ .

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2})}{5 \cos (37 °)}$

Step 1.10.6

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

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Step 1.10.6.1

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

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2})}{5 \cos (37 °)}$

Step 1.10.6.2

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

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2})}{5 \cos (37 °)}$

Step 1.10.6.3

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

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2})}{5 \cos (37 °)}$

Step 1.10.6.4

Cancel the common factor of $2$ .

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Step 1.10.6.4.1

Cancel the common factor.

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2})}{5 \cos (37 °)}$

Step 1.10.6.4.2

Rewrite the expression.

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{2^{1}})}^{2})}{5 \cos (37 °)}$

Step 1.10.6.5

Evaluate the exponent.

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{2})}^{2})}{5 \cos (37 °)}$

Step 1.11

Cancel the common factor of $2$ .

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Step 1.11.1

Cancel the common factor.

$K = \frac{\frac{5}{2} (1 + {(\frac{2 \sqrt{2}}{2})}^{2})}{5 \cos (37 °)}$

Step 1.11.2

Divide $\sqrt{2}$ by $1$ .

$K = \frac{\frac{5}{2} (1 + {\sqrt{2}}^{2})}{5 \cos (37 °)}$

Step 1.12

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

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Step 1.12.1

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

$K = \frac{\frac{5}{2} (1 + {(2^{\frac{1}{2}})}^{2})}{5 \cos (37 °)}$

Step 1.12.2

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

$K = \frac{\frac{5}{2} (1 + 2^{\frac{1}{2} \cdot 2})}{5 \cos (37 °)}$

Step 1.12.3

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

$K = \frac{\frac{5}{2} (1 + 2^{\frac{2}{2}})}{5 \cos (37 °)}$

Step 1.12.4

Cancel the common factor of $2$ .

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Step 1.12.4.1

Cancel the common factor.

$K = \frac{\frac{5}{2} (1 + 2^{\frac{2}{2}})}{5 \cos (37 °)}$

Step 1.12.4.2

Rewrite the expression.

$K = \frac{\frac{5}{2} (1 + 2^{1})}{5 \cos (37 °)}$

Step 1.12.5

Evaluate the exponent.

$K = \frac{\frac{5}{2} (1 + 2)}{5 \cos (37 °)}$

Step 1.13

Add $1$ and $2$ .

$K = \frac{\frac{5}{2} \cdot 3}{5 \cos (37 °)}$

Step 2

Evaluate $\cos (37 °)$ .

$K = \frac{\frac{5}{2} \cdot 3}{5 \cdot 0.79863551}$

Step 3

Combine $\frac{5}{2}$ and $3$ .

$K = \frac{\frac{5 \cdot 3}{2}}{5 \cdot 0.79863551}$

Step 4

Multiply.

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Step 4.1

Multiply $5$ by $0.79863551$ .

$K = \frac{\frac{5 \cdot 3}{2}}{3.99317755}$

Step 4.2

Multiply $5$ by $3$ .

$K = \frac{\frac{15}{2}}{3.99317755}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$K = \frac{15}{2} \cdot \frac{1}{3.99317755}$

Step 6

Divide $1$ by $3.99317755$ .

$K = \frac{15}{2} \cdot 0.25042713$

Step 7

Multiply $\frac{15}{2} \cdot 0.25042713$ .

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Step 7.1

Combine $\frac{15}{2}$ and $0.25042713$ .

$K = \frac{15 \cdot 0.25042713}{2}$

Step 7.2

Multiply $15$ by $0.25042713$ .

$K = \frac{3.75640697}{2}$

Step 8

Divide $3.75640697$ by $2$ .

$K = 1.87820348$