Trigonometry Examples

$\sin (\arcsin (\frac{1}{6}) + \arctan (- 5))$

Step 1

Use the sum formula for sine to simplify the expression. The formula states that $\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$ .

$\sin (\arcsin (\frac{1}{6})) \cos (\arctan (- 5)) + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 2

Remove parentheses.

$\sin (\arcsin (\frac{1}{6})) \cos (\arctan (- 5)) + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3

Simplify each term.

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Step 3.1

The functions sine and arcsine are inverses.

$\frac{1}{6} \cos (\arctan (- 5)) + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2

Draw a triangle in the plane with vertices $(1, - 5)$ , $(1, 0)$ , and the origin. Then $\arctan (- 5)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, - 5)$ . Therefore, $\cos (\arctan (- 5))$ is $\frac{\sqrt{26}}{26}$ .

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Step 3.2.1

Multiply $\frac{1}{\sqrt{26}}$ by $\frac{\sqrt{26}}{\sqrt{26}}$ .

$\frac{1}{6} \cdot \frac{\sqrt{26}}{\sqrt{26} \sqrt{26}} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2.2

Raise $\sqrt{26}$ to the power of $1$ .

$\frac{1}{6} \cdot \frac{\sqrt{26}}{{\sqrt{26}}^{1} \sqrt{26}} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2.3

Raise $\sqrt{26}$ to the power of $1$ .

$\frac{1}{6} \cdot \frac{\sqrt{26}}{{\sqrt{26}}^{1} {\sqrt{26}}^{1}} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2.4

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

$\frac{1}{6} \cdot \frac{\sqrt{26}}{{\sqrt{26}}^{1 + 1}} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2.5

Add $1$ and $1$ .

$\frac{1}{6} \cdot \frac{\sqrt{26}}{{\sqrt{26}}^{2}} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2.6

Rewrite ${\sqrt{26}}^{2}$ as $26$ .

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Step 3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{26}$ as $26^{\frac{1}{2}}$ .

$\frac{1}{6} \cdot \frac{\sqrt{26}}{{(26^{\frac{1}{2}})}^{2}} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2.6.2

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

$\frac{1}{6} \cdot \frac{\sqrt{26}}{26^{\frac{1}{2} \cdot 2}} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2.6.3

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

$\frac{1}{6} \cdot \frac{\sqrt{26}}{26^{\frac{2}{2}}} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2.6.4

Cancel the common factor of $2$ .

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Step 3.2.6.4.1

Cancel the common factor.

$\frac{1}{6} \cdot \frac{\sqrt{26}}{26^{\frac{2}{2}}} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2.6.4.2

Rewrite the expression.

$\frac{1}{6} \cdot \frac{\sqrt{26}}{26^{1}} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.2.6.5

Evaluate the exponent.

$\frac{1}{6} \cdot \frac{\sqrt{26}}{26} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.3

Multiply $\frac{1}{6} \cdot \frac{\sqrt{26}}{26}$ .

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Step 3.3.1

Multiply $\frac{1}{6}$ by $\frac{\sqrt{26}}{26}$ .

$\frac{\sqrt{26}}{6 \cdot 26} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.3.2

Multiply $6$ by $26$ .

$\frac{\sqrt{26}}{156} + \cos (\arcsin (\frac{1}{6})) \sin (\arctan (- 5))$

Step 3.4

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{1}{6})}^{2}}, \frac{1}{6})$ , $(\sqrt{1^{2} - {(\frac{1}{6})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{1}{6})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{1}{6})}^{2}}, \frac{1}{6})$ . Therefore, $\cos (\arcsin (\frac{1}{6}))$ is $\sqrt{\frac{35}{36}}$ .

$\frac{\sqrt{26}}{156} + \sqrt{\frac{35}{36}} \sin (\arctan (- 5))$

Step 3.5

Rewrite $\sqrt{\frac{35}{36}}$ as $\frac{\sqrt{35}}{\sqrt{36}}$ .

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{\sqrt{36}} \sin (\arctan (- 5))$

Step 3.6

Simplify the denominator.

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Step 3.6.1

Rewrite $36$ as $6^{2}$ .

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{\sqrt{6^{2}}} \sin (\arctan (- 5))$

Step 3.6.2

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

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} \sin (\arctan (- 5))$

Step 3.7

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Step 3.7.1

Multiply $\frac{5}{\sqrt{26}}$ by $\frac{\sqrt{26}}{\sqrt{26}}$ .

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{\sqrt{26} \sqrt{26}})$

Step 3.7.2

Raise $\sqrt{26}$ to the power of $1$ .

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{{\sqrt{26}}^{1} \sqrt{26}})$

Step 3.7.3

Raise $\sqrt{26}$ to the power of $1$ .

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{{\sqrt{26}}^{1} {\sqrt{26}}^{1}})$

Step 3.7.4

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

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{{\sqrt{26}}^{1 + 1}})$

Step 3.7.5

Add $1$ and $1$ .

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{{\sqrt{26}}^{2}})$

Step 3.7.6

Rewrite ${\sqrt{26}}^{2}$ as $26$ .

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Step 3.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{26}$ as $26^{\frac{1}{2}}$ .

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{{(26^{\frac{1}{2}})}^{2}})$

Step 3.7.6.2

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

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{26^{\frac{1}{2} \cdot 2}})$

Step 3.7.6.3

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

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{26^{\frac{2}{2}}})$

Step 3.7.6.4

Cancel the common factor of $2$ .

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Step 3.7.6.4.1

Cancel the common factor.

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{26^{\frac{2}{2}}})$

Step 3.7.6.4.2

Rewrite the expression.

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{26^{1}})$

Step 3.7.6.5

Evaluate the exponent.

$\frac{\sqrt{26}}{156} + \frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{26})$

Step 3.8

Multiply $\frac{\sqrt{35}}{6} (- \frac{5 \sqrt{26}}{26})$ .

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Step 3.8.1

Multiply $\frac{\sqrt{35}}{6}$ by $\frac{5 \sqrt{26}}{26}$ .

$\frac{\sqrt{26}}{156} - \frac{\sqrt{35} (5 \sqrt{26})}{6 \cdot 26}$

Step 3.8.2

Combine using the product rule for radicals.

$\frac{\sqrt{26}}{156} - \frac{5 \sqrt{35 \cdot 26}}{6 \cdot 26}$

Step 3.8.3

Multiply $35$ by $26$ .

$\frac{\sqrt{26}}{156} - \frac{5 \sqrt{910}}{6 \cdot 26}$

Step 3.8.4

Multiply $6$ by $26$ .

$\frac{\sqrt{26}}{156} - \frac{5 \sqrt{910}}{156}$

Step 4

Combine the numerators over the common denominator.

$\frac{\sqrt{26} - 5 \sqrt{910}}{156}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{26} - 5 \sqrt{910}}{156}$

Decimal Form:

$- 0.93417956 \dots$