Trigonometry Examples

$\arcsin (\frac{10 (\sin (15))}{5})$

Step 1

Cancel the common factor of $10$ and $5$ .

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Step 1.1

Factor $5$ out of $10 (\sin (15))$ .

$\arcsin (\frac{5 (2 \sin (15))}{5})$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $5$ out of $5$ .

$\arcsin (\frac{5 (2 \sin (15))}{5 (1)})$

Step 1.2.2

Cancel the common factor.

$\arcsin (\frac{5 (2 \sin (15))}{5 \cdot 1})$

Step 1.2.3

Rewrite the expression.

$\arcsin (\frac{2 \sin (15)}{1})$

Step 1.2.4

Divide $2 \sin (15)$ by $1$ .

$\arcsin (2 \sin (15))$

Step 2

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 2.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\arcsin (2 \sin (45 - 30))$

Step 2.2

Separate negation.

$\arcsin (2 \sin (45 - (30)))$

Step 2.3

Apply the difference of angles identity.

$\arcsin (2 (\sin (45) \cos (30) - \cos (45) \sin (30)))$

Step 2.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\arcsin (2 (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)))$

Step 2.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\arcsin (2 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)))$

Step 2.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\arcsin (2 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)))$

Step 2.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\arcsin (2 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 2.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 2.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\arcsin (2 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 2.8.1.1.2

Combine using the product rule for radicals.

$\arcsin (2 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 2.8.1.1.3

Multiply $2$ by $3$ .

$\arcsin (2 (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 2.8.1.1.4

Multiply $2$ by $2$ .

$\arcsin (2 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}))$

Step 2.8.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 2.8.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\arcsin (2 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}))$

Step 2.8.1.2.2

Multiply $2$ by $2$ .

$\arcsin (2 (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}))$

Step 2.8.2

Combine the numerators over the common denominator.

$\arcsin (2 \frac{\sqrt{6} - \sqrt{2}}{4})$

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Factor $2$ out of $4$ .

$\arcsin (2 \frac{\sqrt{6} - \sqrt{2}}{2 (2)})$

Step 3.2

Cancel the common factor.

$\arcsin (2 \frac{\sqrt{6} - \sqrt{2}}{2 \cdot 2})$

Step 3.3

Rewrite the expression.

$\arcsin (\frac{\sqrt{6} - \sqrt{2}}{2})$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\arcsin (\frac{\sqrt{6} - \sqrt{2}}{2})$

Decimal Form:

$0.54408810 \dots$