Trigonometry Examples

sin ({tan}^{-1} (u) + {sin}^{-1} (v))

$\sin (\tan^{-1} (u) + \sin^{-1} (v))$

Step 1

Apply the sum of angles identity.

sin ({tan}^{-1} (u)) cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\sin (\tan^{-1} (u)) \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Draw a triangle in the plane with vertices

(1, u)

$(1, u)$ ,

(1, 0)

$(1, 0)$ , and the origin. Then

{tan}^{-1} (u)

$\tan^{-1} (u)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through

(1, u)

$(1, u)$ . Therefore,

sin ({tan}^{-1} (u))

$\sin (\tan^{-1} (u))$ is

\frac{u}{\sqrt{1 + u^{2}}}

$\frac{u}{\sqrt{1 + u^{2}}}$ .

\frac{u}{\sqrt{1 + u^{2}}} cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\frac{u}{\sqrt{1 + u^{2}}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.2

Multiply

\frac{u}{\sqrt{1 + u^{2}}}

$\frac{u}{\sqrt{1 + u^{2}}}$ by

\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}

$\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}$ .

\frac{u}{\sqrt{1 + u^{2}}} \cdot \frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}} cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\frac{u}{\sqrt{1 + u^{2}}} \cdot \frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3

Combine and simplify the denominator.

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Step 2.1.3.1

Multiply

\frac{u}{\sqrt{1 + u^{2}}}

$\frac{u}{\sqrt{1 + u^{2}}}$ by

\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}

$\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}$ .

\frac{u \sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}} \sqrt{1 + u^{2}}} cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\frac{u \sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}} \sqrt{1 + u^{2}}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3.2

Raise

\sqrt{1 + u^{2}}

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

1

$1$ .

\frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1} \sqrt{1 + u^{2}}} cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1} \sqrt{1 + u^{2}}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3.3

Raise

\sqrt{1 + u^{2}}

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

1

$1$ .

\frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1} {\sqrt{1 + u^{2}}}^{1}} cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1} {\sqrt{1 + u^{2}}}^{1}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

\frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1 + 1}} cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{1 + 1}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3.5

Add

1

$1$ and

1

$1$ .

\frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{2}} cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\frac{u \sqrt{1 + u^{2}}}{{\sqrt{1 + u^{2}}}^{2}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3.6

Rewrite

{\sqrt{1 + u^{2}}}^{2}

${\sqrt{1 + u^{2}}}^{2}$ as

1 + u^{2}

$1 + u^{2}$ .

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Step 2.1.3.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{1 + u^{2}}

$\sqrt{1 + u^{2}}$ as

{(1 + u^{2})}^{\frac{1}{2}}

${(1 + u^{2})}^{\frac{1}{2}}$ .

\frac{u \sqrt{1 + u^{2}}}{{({(1 + u^{2})}^{\frac{1}{2}})}^{2}} cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\frac{u \sqrt{1 + u^{2}}}{{({(1 + u^{2})}^{\frac{1}{2}})}^{2}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{1}{2} \cdot 2}} cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{1}{2} \cdot 2}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{2}{2}}} cos ({sin}^{-1} (v)) + cos ({tan}^{-1} (u)) sin ({sin}^{-1} (v))

$\frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{2}{2}}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3.6.4

Cancel the common factor of

2

$2$ .

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Step 2.1.3.6.4.1

Cancel the common factor.

$\frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{\frac{2}{2}}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3.6.4.2

Rewrite the expression.

$\frac{u \sqrt{1 + u^{2}}}{{(1 + u^{2})}^{1}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.3.6.5

Simplify.

$\frac{u \sqrt{1 + u^{2}}}{1 + u^{2}} \cos (\sin^{-1} (v)) + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.4

Draw a triangle in the plane with vertices

$(\sqrt{1^{2} - v^{2}}, v)$ ,

$(\sqrt{1^{2} - v^{2}}, 0)$ , and the origin. Then

$\sin^{-1} (v)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through

$(\sqrt{1^{2} - v^{2}}, v)$ . Therefore,

$\cos (\sin^{-1} (v))$ is

$\sqrt{1 - v^{2}}$ .

$\frac{u \sqrt{1 + u^{2}}}{1 + u^{2}} \sqrt{1 - v^{2}} + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.5

Rewrite

$1$ as

$1^{2}$ .

$\frac{u \sqrt{1 + u^{2}}}{1 + u^{2}} \sqrt{1^{2} - v^{2}} + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.6

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = 1$ and

$b = v$ .

$\frac{u \sqrt{1 + u^{2}}}{1 + u^{2}} \sqrt{(1 + v) (1 - v)} + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.7

Multiply

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

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Step 2.1.7.1

Combine

$\frac{u \sqrt{1 + u^{2}}}{1 + u^{2}}$ and

$\sqrt{(1 + v) (1 - v)}$ .

$\frac{u \sqrt{1 + u^{2}} \sqrt{(1 + v) (1 - v)}}{1 + u^{2}} + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.7.2

Combine using the product rule for radicals.

$\frac{u \sqrt{(1 + v) (1 - v) (1 + u^{2})}}{1 + u^{2}} + \cos (\tan^{-1} (u)) \sin (\sin^{-1} (v))$

Step 2.1.8

Draw a triangle in the plane with vertices

$(1, u)$ ,

$(1, 0)$ , and the origin. Then

$\tan^{-1} (u)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through

$(1, u)$ . Therefore,

$\cos (\tan^{-1} (u))$ is

$\frac{1}{\sqrt{1 + u^{2}}}$ .

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

Step 2.1.9

Multiply

$\frac{1}{\sqrt{1 + u^{2}}}$ by

$\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}$ .

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

Step 2.1.10

Combine and simplify the denominator.

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Step 2.1.10.1

Multiply

$\frac{1}{\sqrt{1 + u^{2}}}$ by

$\frac{\sqrt{1 + u^{2}}}{\sqrt{1 + u^{2}}}$ .

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

Step 2.1.10.2

Raise

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

$1$ .

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

Step 2.1.10.3

Raise

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

$1$ .

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

Step 2.1.10.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.10.5

Add

$1$ and

$1$ .

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

Step 2.1.10.6

Rewrite

${\sqrt{1 + u^{2}}}^{2}$ as

$1 + u^{2}$ .

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Step 2.1.10.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{1 + u^{2}}$ as

${(1 + u^{2})}^{\frac{1}{2}}$ .

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

Step 2.1.10.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.1.10.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 2.1.10.6.4

Cancel the common factor of

$2$ .

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Step 2.1.10.6.4.1

Cancel the common factor.

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

Step 2.1.10.6.4.2

Rewrite the expression.

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

Step 2.1.10.6.5

Simplify.

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

Step 2.1.11

The functions sine and arcsine are inverses.

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

Step 2.1.12

Combine

$\frac{\sqrt{1 + u^{2}}}{1 + u^{2}}$ and

$v$ .

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

Step 2.2

Combine the numerators over the common denominator.

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