Trigonometry Examples

$\frac{\sin (\frac{5}{\sqrt{29}})}{\cos (\frac{2}{\sqrt{29}})}$

Step 1

Rewrite $\frac{\sin (\frac{5}{\sqrt{29}})}{\cos (\frac{2}{\sqrt{29}})}$ as a product.

$\sin (\frac{5}{\sqrt{29}}) \frac{1}{\cos (\frac{2}{\sqrt{29}})}$

Step 2

Write $\sin (\frac{5}{\sqrt{29}})$ as a fraction with denominator $1$ .

$\frac{\sin (\frac{5}{\sqrt{29}})}{1} \cdot \frac{1}{\cos (\frac{2}{\sqrt{29}})}$

Step 3

Simplify.

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Step 3.1

Divide $\sin (\frac{5}{\sqrt{29}})$ by $1$ .

$\sin (\frac{5}{\sqrt{29}}) \frac{1}{\cos (\frac{2}{\sqrt{29}})}$

Step 3.2

Convert from $\frac{1}{\cos (\frac{2}{\sqrt{29}})}$ to $\sec (\frac{2}{\sqrt{29}})$ .

$\sin (\frac{5}{\sqrt{29}}) \sec (\frac{2}{\sqrt{29}})$

Step 4

Multiply $\frac{5}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\sin (\frac{5}{\sqrt{29}} \cdot \frac{\sqrt{29}}{\sqrt{29}}) \sec (\frac{2}{\sqrt{29}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{5}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\sin (\frac{5 \sqrt{29}}{\sqrt{29} \sqrt{29}}) \sec (\frac{2}{\sqrt{29}})$

Step 5.2

Raise $\sqrt{29}$ to the power of $1$ .

$\sin (\frac{5 \sqrt{29}}{{\sqrt{29}}^{1} \sqrt{29}}) \sec (\frac{2}{\sqrt{29}})$

Step 5.3

Raise $\sqrt{29}$ to the power of $1$ .

$\sin (\frac{5 \sqrt{29}}{{\sqrt{29}}^{1} {\sqrt{29}}^{1}}) \sec (\frac{2}{\sqrt{29}})$

Step 5.4

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

$\sin (\frac{5 \sqrt{29}}{{\sqrt{29}}^{1 + 1}}) \sec (\frac{2}{\sqrt{29}})$

Step 5.5

Add $1$ and $1$ .

$\sin (\frac{5 \sqrt{29}}{{\sqrt{29}}^{2}}) \sec (\frac{2}{\sqrt{29}})$

Step 5.6

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$\sin (\frac{5 \sqrt{29}}{{(29^{\frac{1}{2}})}^{2}}) \sec (\frac{2}{\sqrt{29}})$

Step 5.6.2

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

$\sin (\frac{5 \sqrt{29}}{29^{\frac{1}{2} \cdot 2}}) \sec (\frac{2}{\sqrt{29}})$

Step 5.6.3

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

$\sin (\frac{5 \sqrt{29}}{29^{\frac{2}{2}}}) \sec (\frac{2}{\sqrt{29}})$

Step 5.6.4

Cancel the common factor of $2$ .

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Step 5.6.4.1

Cancel the common factor.

$\sin (\frac{5 \sqrt{29}}{29^{\frac{2}{2}}}) \sec (\frac{2}{\sqrt{29}})$

Step 5.6.4.2

Rewrite the expression.

$\sin (\frac{5 \sqrt{29}}{29^{1}}) \sec (\frac{2}{\sqrt{29}})$

Step 5.6.5

Evaluate the exponent.

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2}{\sqrt{29}})$

Step 6

Multiply $\frac{2}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2}{\sqrt{29}} \cdot \frac{\sqrt{29}}{\sqrt{29}})$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{2}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{\sqrt{29} \sqrt{29}})$

Step 7.2

Raise $\sqrt{29}$ to the power of $1$ .

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{{\sqrt{29}}^{1} \sqrt{29}})$

Step 7.3

Raise $\sqrt{29}$ to the power of $1$ .

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{{\sqrt{29}}^{1} {\sqrt{29}}^{1}})$

Step 7.4

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

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{{\sqrt{29}}^{1 + 1}})$

Step 7.5

Add $1$ and $1$ .

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{{\sqrt{29}}^{2}})$

Step 7.6

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{{(29^{\frac{1}{2}})}^{2}})$

Step 7.6.2

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

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{29^{\frac{1}{2} \cdot 2}})$

Step 7.6.3

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

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{29^{\frac{2}{2}}})$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{29^{\frac{2}{2}}})$

Step 7.6.4.2

Rewrite the expression.

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{29^{1}})$

Step 7.6.5

Evaluate the exponent.

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{29})$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sin (\frac{5 \sqrt{29}}{29}) \sec (\frac{2 \sqrt{29}}{29})$

Decimal Form:

$0.85929178 \dots$