Trigonometry Examples

$\sin (\arctan (2 x) - \arccos (x))$

Step 1

Apply the difference of angles identity.

$\sin (\arctan (2 x)) \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

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 $(1, 2 x)$ , $(1, 0)$ , and the origin. Then $\arctan (2 x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, 2 x)$ . Therefore, $\sin (\arctan (2 x))$ is $\frac{2 x}{\sqrt{1 + {(2 x)}^{2}}}$ .

$\frac{2 x}{\sqrt{1 + {(2 x)}^{2}}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.2

Simplify the denominator.

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Step 2.1.2.1

Apply the product rule to $2 x$ .

$\frac{2 x}{\sqrt{1 + 2^{2} x^{2}}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.2.2

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

$\frac{2 x}{\sqrt{1 + 4 x^{2}}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.3

Multiply $\frac{2 x}{\sqrt{1 + 4 x^{2}}}$ by $\frac{\sqrt{1 + 4 x^{2}}}{\sqrt{1 + 4 x^{2}}}$ .

$\frac{2 x}{\sqrt{1 + 4 x^{2}}} \cdot \frac{\sqrt{1 + 4 x^{2}}}{\sqrt{1 + 4 x^{2}}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4

Combine and simplify the denominator.

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Step 2.1.4.1

Multiply $\frac{2 x}{\sqrt{1 + 4 x^{2}}}$ by $\frac{\sqrt{1 + 4 x^{2}}}{\sqrt{1 + 4 x^{2}}}$ .

$\frac{2 x \sqrt{1 + 4 x^{2}}}{\sqrt{1 + 4 x^{2}} \sqrt{1 + 4 x^{2}}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4.2

Raise $\sqrt{1 + 4 x^{2}}$ to the power of $1$ .

$\frac{2 x \sqrt{1 + 4 x^{2}}}{{\sqrt{1 + 4 x^{2}}}^{1} \sqrt{1 + 4 x^{2}}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4.3

Raise $\sqrt{1 + 4 x^{2}}$ to the power of $1$ .

$\frac{2 x \sqrt{1 + 4 x^{2}}}{{\sqrt{1 + 4 x^{2}}}^{1} {\sqrt{1 + 4 x^{2}}}^{1}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4.4

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

$\frac{2 x \sqrt{1 + 4 x^{2}}}{{\sqrt{1 + 4 x^{2}}}^{1 + 1}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4.5

Add $1$ and $1$ .

$\frac{2 x \sqrt{1 + 4 x^{2}}}{{\sqrt{1 + 4 x^{2}}}^{2}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4.6

Rewrite ${\sqrt{1 + 4 x^{2}}}^{2}$ as $1 + 4 x^{2}$ .

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Step 2.1.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + 4 x^{2}}$ as ${(1 + 4 x^{2})}^{\frac{1}{2}}$ .

$\frac{2 x \sqrt{1 + 4 x^{2}}}{{({(1 + 4 x^{2})}^{\frac{1}{2}})}^{2}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4.6.2

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

$\frac{2 x \sqrt{1 + 4 x^{2}}}{{(1 + 4 x^{2})}^{\frac{1}{2} \cdot 2}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4.6.3

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

$\frac{2 x \sqrt{1 + 4 x^{2}}}{{(1 + 4 x^{2})}^{\frac{2}{2}}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4.6.4

Cancel the common factor of $2$ .

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Step 2.1.4.6.4.1

Cancel the common factor.

$\frac{2 x \sqrt{1 + 4 x^{2}}}{{(1 + 4 x^{2})}^{\frac{2}{2}}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4.6.4.2

Rewrite the expression.

$\frac{2 x \sqrt{1 + 4 x^{2}}}{{(1 + 4 x^{2})}^{1}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.4.6.5

Simplify.

$\frac{2 x \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} \cos (\arccos (x)) - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.5

The functions cosine and arccosine are inverses.

$\frac{2 x \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} x - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.6

Multiply $\frac{2 x \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} x$ .

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Step 2.1.6.1

Combine $\frac{2 x \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}}$ and $x$ .

$\frac{2 x \sqrt{1 + 4 x^{2}} x}{1 + 4 x^{2}} - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.6.2

Raise $x$ to the power of $1$ .

$\frac{2 (x^{1} x) \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.6.3

Raise $x$ to the power of $1$ .

$\frac{2 (x^{1} x^{1}) \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.6.4

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

$\frac{2 x^{1 + 1} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.6.5

Add $1$ and $1$ .

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \cos (\arctan (2 x)) \sin (\arccos (x))$

Step 2.1.7

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

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{1}{\sqrt{1 + {(2 x)}^{2}}} \sin (\arccos (x))$

Step 2.1.8

Simplify the denominator.

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Step 2.1.8.1

Apply the product rule to $2 x$ .

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{1}{\sqrt{1 + 2^{2} x^{2}}} \sin (\arccos (x))$

Step 2.1.8.2

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

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{1}{\sqrt{1 + 4 x^{2}}} \sin (\arccos (x))$

Step 2.1.9

Multiply $\frac{1}{\sqrt{1 + 4 x^{2}}}$ by $\frac{\sqrt{1 + 4 x^{2}}}{\sqrt{1 + 4 x^{2}}}$ .

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - (\frac{1}{\sqrt{1 + 4 x^{2}}} \cdot \frac{\sqrt{1 + 4 x^{2}}}{\sqrt{1 + 4 x^{2}}}) \sin (\arccos (x))$

Step 2.1.10

Combine and simplify the denominator.

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Step 2.1.10.1

Multiply $\frac{1}{\sqrt{1 + 4 x^{2}}}$ by $\frac{\sqrt{1 + 4 x^{2}}}{\sqrt{1 + 4 x^{2}}}$ .

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{\sqrt{1 + 4 x^{2}} \sqrt{1 + 4 x^{2}}} \sin (\arccos (x))$

Step 2.1.10.2

Raise $\sqrt{1 + 4 x^{2}}$ to the power of $1$ .

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{{\sqrt{1 + 4 x^{2}}}^{1} \sqrt{1 + 4 x^{2}}} \sin (\arccos (x))$

Step 2.1.10.3

Raise $\sqrt{1 + 4 x^{2}}$ to the power of $1$ .

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{{\sqrt{1 + 4 x^{2}}}^{1} {\sqrt{1 + 4 x^{2}}}^{1}} \sin (\arccos (x))$

Step 2.1.10.4

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

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{{\sqrt{1 + 4 x^{2}}}^{1 + 1}} \sin (\arccos (x))$

Step 2.1.10.5

Add $1$ and $1$ .

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{{\sqrt{1 + 4 x^{2}}}^{2}} \sin (\arccos (x))$

Step 2.1.10.6

Rewrite ${\sqrt{1 + 4 x^{2}}}^{2}$ as $1 + 4 x^{2}$ .

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Step 2.1.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + 4 x^{2}}$ as ${(1 + 4 x^{2})}^{\frac{1}{2}}$ .

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{{({(1 + 4 x^{2})}^{\frac{1}{2}})}^{2}} \sin (\arccos (x))$

Step 2.1.10.6.2

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

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{{(1 + 4 x^{2})}^{\frac{1}{2} \cdot 2}} \sin (\arccos (x))$

Step 2.1.10.6.3

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

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{{(1 + 4 x^{2})}^{\frac{2}{2}}} \sin (\arccos (x))$

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.

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{{(1 + 4 x^{2})}^{\frac{2}{2}}} \sin (\arccos (x))$

Step 2.1.10.6.4.2

Rewrite the expression.

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{{(1 + 4 x^{2})}^{1}} \sin (\arccos (x))$

Step 2.1.10.6.5

Simplify.

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} \sin (\arccos (x))$

Step 2.1.11

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

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} \sqrt{1 - x^{2}}$

Step 2.1.12

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

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} \sqrt{1^{2} - x^{2}}$

Step 2.1.13

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 $b = x$ .

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} \sqrt{(1 + x) (1 - x)}$

Step 2.1.14

Multiply $- \frac{\sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} \sqrt{(1 + x) (1 - x)}$ .

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Step 2.1.14.1

Combine $\sqrt{(1 + x) (1 - x)}$ and $\frac{\sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}}$ .

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{(1 + x) (1 - x)} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}}$

Step 2.1.14.2

Combine using the product rule for radicals.

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}}}{1 + 4 x^{2}} - \frac{\sqrt{(1 + x) (1 - x) (1 + 4 x^{2})}}{1 + 4 x^{2}}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{2 x^{2} \sqrt{1 + 4 x^{2}} - \sqrt{(1 + x) (1 - x) (1 + 4 x^{2})}}{1 + 4 x^{2}}$