Trigonometry Examples

$\sin (arcsec (\frac{\sqrt{u^{2} + 25}}{u}))$

Step 1

Draw a triangle in the plane with vertices $(1, \sqrt{{(\frac{\sqrt{u^{2} + 25}}{u})}^{2} - 1^{2}})$ , $(1, 0)$ , and the origin. Then $arcsec (\frac{\sqrt{u^{2} + 25}}{u})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \sqrt{{(\frac{\sqrt{u^{2} + 25}}{u})}^{2} - 1^{2}})$ . Therefore, $\sin (arcsec (\frac{\sqrt{u^{2} + 25}}{u}))$ is $\frac{\sqrt{{(\frac{\sqrt{u^{2} + 25}}{u})}^{2} - 1}}{\frac{\sqrt{u^{2} + 25}}{u}}$ .

$\frac{\sqrt{{(\frac{\sqrt{u^{2} + 25}}{u})}^{2} - 1}}{\frac{\sqrt{u^{2} + 25}}{u}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\sqrt{{(\frac{\sqrt{u^{2} + 25}}{u})}^{2} - 1} \frac{u}{\sqrt{u^{2} + 25}}$

Step 3

Rewrite $1$ as $1^{2}$ .

$\sqrt{{(\frac{\sqrt{u^{2} + 25}}{u})}^{2} - 1^{2}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{\sqrt{u^{2} + 25}}{u}$ and $b = 1$ .

$\sqrt{(\frac{\sqrt{u^{2} + 25}}{u} + 1) (\frac{\sqrt{u^{2} + 25}}{u} - 1)} \frac{u}{\sqrt{u^{2} + 25}}$

Step 5

Simplify.

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Step 5.1

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

$\sqrt{(\frac{\sqrt{u^{2} + 25}}{u} + \frac{u}{u}) (\frac{\sqrt{u^{2} + 25}}{u} - 1)} \frac{u}{\sqrt{u^{2} + 25}}$

Step 5.2

Combine the numerators over the common denominator.

$\sqrt{\frac{\sqrt{u^{2} + 25} + u}{u} (\frac{\sqrt{u^{2} + 25}}{u} - 1)} \frac{u}{\sqrt{u^{2} + 25}}$

Step 5.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{u}{u}$ .

$\sqrt{\frac{\sqrt{u^{2} + 25} + u}{u} (\frac{\sqrt{u^{2} + 25}}{u} - 1 \cdot \frac{u}{u})} \frac{u}{\sqrt{u^{2} + 25}}$

Step 5.4

Combine $- 1$ and $\frac{u}{u}$ .

$\sqrt{\frac{\sqrt{u^{2} + 25} + u}{u} (\frac{\sqrt{u^{2} + 25}}{u} + \frac{- u}{u})} \frac{u}{\sqrt{u^{2} + 25}}$

Step 5.5

Combine the numerators over the common denominator.

$\sqrt{\frac{\sqrt{u^{2} + 25} + u}{u} \cdot \frac{\sqrt{u^{2} + 25} - u}{u}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 6

Combine fractions.

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Step 6.1

Multiply $\frac{\sqrt{u^{2} + 25} + u}{u}$ by $\frac{\sqrt{u^{2} + 25} - u}{u}$ .

$\sqrt{\frac{(\sqrt{u^{2} + 25} + u) (\sqrt{u^{2} + 25} - u)}{u \cdot u}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 6.2

Multiply $u$ by $u$ .

$\sqrt{\frac{(\sqrt{u^{2} + 25} + u) (\sqrt{u^{2} + 25} - u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 7

Expand $(\sqrt{u^{2} + 25} + u) (\sqrt{u^{2} + 25} - u)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$\sqrt{\frac{\sqrt{u^{2} + 25} (\sqrt{u^{2} + 25} - u) + u (\sqrt{u^{2} + 25} - u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 7.2

Apply the distributive property.

$\sqrt{\frac{\sqrt{u^{2} + 25} \sqrt{u^{2} + 25} + \sqrt{u^{2} + 25} (- u) + u (\sqrt{u^{2} + 25} - u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 7.3

Apply the distributive property.

$\sqrt{\frac{\sqrt{u^{2} + 25} \sqrt{u^{2} + 25} + \sqrt{u^{2} + 25} (- u) + u \sqrt{u^{2} + 25} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8

Simplify terms.

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Step 8.1

Combine the opposite terms in $\sqrt{u^{2} + 25} \sqrt{u^{2} + 25} + \sqrt{u^{2} + 25} (- u) + u \sqrt{u^{2} + 25} + u (- u)$ .

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Step 8.1.1

Reorder the factors in the terms $\sqrt{u^{2} + 25} (- u)$ and $u \sqrt{u^{2} + 25}$ .

$\sqrt{\frac{\sqrt{u^{2} + 25} \sqrt{u^{2} + 25} - u \sqrt{u^{2} + 25} + u \sqrt{u^{2} + 25} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.1.2

Add $- u \sqrt{u^{2} + 25}$ and $u \sqrt{u^{2} + 25}$ .

$\sqrt{\frac{\sqrt{u^{2} + 25} \sqrt{u^{2} + 25} + 0 + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.1.3

Add $\sqrt{u^{2} + 25} \sqrt{u^{2} + 25}$ and $0$ .

$\sqrt{\frac{\sqrt{u^{2} + 25} \sqrt{u^{2} + 25} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2

Simplify each term.

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Step 8.2.1

Multiply $\sqrt{u^{2} + 25} \sqrt{u^{2} + 25}$ .

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Step 8.2.1.1

Raise $\sqrt{u^{2} + 25}$ to the power of $1$ .

$\sqrt{\frac{{\sqrt{u^{2} + 25}}^{1} \sqrt{u^{2} + 25} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.1.2

Raise $\sqrt{u^{2} + 25}$ to the power of $1$ .

$\sqrt{\frac{{\sqrt{u^{2} + 25}}^{1} {\sqrt{u^{2} + 25}}^{1} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.1.3

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

$\sqrt{\frac{{\sqrt{u^{2} + 25}}^{1 + 1} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.1.4

Add $1$ and $1$ .

$\sqrt{\frac{{\sqrt{u^{2} + 25}}^{2} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.2

Rewrite ${\sqrt{u^{2} + 25}}^{2}$ as $u^{2} + 25$ .

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Step 8.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u^{2} + 25}$ as ${(u^{2} + 25)}^{\frac{1}{2}}$ .

$\sqrt{\frac{{({(u^{2} + 25)}^{\frac{1}{2}})}^{2} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.2.2

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

$\sqrt{\frac{{(u^{2} + 25)}^{\frac{1}{2} \cdot 2} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.2.3

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

$\sqrt{\frac{{(u^{2} + 25)}^{\frac{2}{2}} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.2.4

Cancel the common factor of $2$ .

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Step 8.2.2.4.1

Cancel the common factor.

$\sqrt{\frac{{(u^{2} + 25)}^{\frac{2}{2}} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.2.4.2

Rewrite the expression.

$\sqrt{\frac{{(u^{2} + 25)}^{1} + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.2.5

Simplify.

$\sqrt{\frac{u^{2} + 25 + u (- u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.3

Rewrite using the commutative property of multiplication.

$\sqrt{\frac{u^{2} + 25 - u \cdot u}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.4

Multiply $u$ by $u$ by adding the exponents.

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Step 8.2.4.1

Move $u$ .

$\sqrt{\frac{u^{2} + 25 - (u \cdot u)}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.2.4.2

Multiply $u$ by $u$ .

$\sqrt{\frac{u^{2} + 25 - u^{2}}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.3

Simplify by adding terms.

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Step 8.3.1

Combine the opposite terms in $u^{2} + 25 - u^{2}$ .

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Step 8.3.1.1

Subtract $u^{2}$ from $u^{2}$ .

$\sqrt{\frac{0 + 25}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.3.1.2

Add $0$ and $25$ .

$\sqrt{\frac{25}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 8.3.2

Rewrite $25$ as $5^{2}$ .

$\sqrt{\frac{5^{2}}{u^{2}}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 9

Rewrite $\frac{5^{2}}{u^{2}}$ as ${(\frac{5}{u})}^{2}$ .

$\sqrt{{(\frac{5}{u})}^{2}} \frac{u}{\sqrt{u^{2} + 25}}$

Step 10

Pull terms out from under the radical, assuming positive real numbers.

$\frac{5}{u} \cdot \frac{u}{\sqrt{u^{2} + 25}}$

Step 11

Combine.

$\frac{5 u}{u \sqrt{u^{2} + 25}}$

Step 12

Cancel the common factor of $u$ .

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Step 12.1

Cancel the common factor.

$\frac{5 u}{u \sqrt{u^{2} + 25}}$

Step 12.2

Rewrite the expression.

$\frac{5}{\sqrt{u^{2} + 25}}$

Step 13

Multiply $\frac{5}{\sqrt{u^{2} + 25}}$ by $\frac{\sqrt{u^{2} + 25}}{\sqrt{u^{2} + 25}}$ .

$\frac{5}{\sqrt{u^{2} + 25}} \cdot \frac{\sqrt{u^{2} + 25}}{\sqrt{u^{2} + 25}}$

Step 14

Combine and simplify the denominator.

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Step 14.1

Multiply $\frac{5}{\sqrt{u^{2} + 25}}$ by $\frac{\sqrt{u^{2} + 25}}{\sqrt{u^{2} + 25}}$ .

$\frac{5 \sqrt{u^{2} + 25}}{\sqrt{u^{2} + 25} \sqrt{u^{2} + 25}}$

Step 14.2

Raise $\sqrt{u^{2} + 25}$ to the power of $1$ .

$\frac{5 \sqrt{u^{2} + 25}}{{\sqrt{u^{2} + 25}}^{1} \sqrt{u^{2} + 25}}$

Step 14.3

Raise $\sqrt{u^{2} + 25}$ to the power of $1$ .

$\frac{5 \sqrt{u^{2} + 25}}{{\sqrt{u^{2} + 25}}^{1} {\sqrt{u^{2} + 25}}^{1}}$

Step 14.4

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

$\frac{5 \sqrt{u^{2} + 25}}{{\sqrt{u^{2} + 25}}^{1 + 1}}$

Step 14.5

Add $1$ and $1$ .

$\frac{5 \sqrt{u^{2} + 25}}{{\sqrt{u^{2} + 25}}^{2}}$

Step 14.6

Rewrite ${\sqrt{u^{2} + 25}}^{2}$ as $u^{2} + 25$ .

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Step 14.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u^{2} + 25}$ as ${(u^{2} + 25)}^{\frac{1}{2}}$ .

$\frac{5 \sqrt{u^{2} + 25}}{{({(u^{2} + 25)}^{\frac{1}{2}})}^{2}}$

Step 14.6.2

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

$\frac{5 \sqrt{u^{2} + 25}}{{(u^{2} + 25)}^{\frac{1}{2} \cdot 2}}$

Step 14.6.3

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

$\frac{5 \sqrt{u^{2} + 25}}{{(u^{2} + 25)}^{\frac{2}{2}}}$

Step 14.6.4

Cancel the common factor of $2$ .

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Step 14.6.4.1

Cancel the common factor.

$\frac{5 \sqrt{u^{2} + 25}}{{(u^{2} + 25)}^{\frac{2}{2}}}$

Step 14.6.4.2

Rewrite the expression.

$\frac{5 \sqrt{u^{2} + 25}}{{(u^{2} + 25)}^{1}}$

Step 14.6.5

Simplify.

$\frac{5 \sqrt{u^{2} + 25}}{u^{2} + 25}$