Trigonometry Examples

$\arcsin (x) - \arctan (1) = - \frac{π}{6}$

Step 1

Simplify the left side.

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Step 1.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$\arcsin (x) - \frac{π}{4} = - \frac{π}{6}$

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Add $\frac{π}{4}$ to both sides of the equation.

$\arcsin (x) = - \frac{π}{6} + \frac{π}{4}$

Step 2.2

To write $- \frac{π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\arcsin (x) = - \frac{π}{6} \cdot \frac{2}{2} + \frac{π}{4}$

Step 2.3

To write $\frac{π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\arcsin (x) = - \frac{π}{6} \cdot \frac{2}{2} + \frac{π}{4} \cdot \frac{3}{3}$

Step 2.4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{π}{6}$ by $\frac{2}{2}$ .

$\arcsin (x) = - \frac{π \cdot 2}{6 \cdot 2} + \frac{π}{4} \cdot \frac{3}{3}$

Step 2.4.2

Multiply $6$ by $2$ .

$\arcsin (x) = - \frac{π \cdot 2}{12} + \frac{π}{4} \cdot \frac{3}{3}$

Step 2.4.3

Multiply $\frac{π}{4}$ by $\frac{3}{3}$ .

$\arcsin (x) = - \frac{π \cdot 2}{12} + \frac{π \cdot 3}{4 \cdot 3}$

Step 2.4.4

Multiply $4$ by $3$ .

$\arcsin (x) = - \frac{π \cdot 2}{12} + \frac{π \cdot 3}{12}$

Step 2.5

Combine the numerators over the common denominator.

$\arcsin (x) = \frac{- π \cdot 2 + π \cdot 3}{12}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $2$ by $- 1$ .

$\arcsin (x) = \frac{- 2 π + π \cdot 3}{12}$

Step 2.6.2

Move $3$ to the left of $π$ .

$\arcsin (x) = \frac{- 2 π + 3 \cdot π}{12}$

Step 2.6.3

Add $- 2 π$ and $3 π$ .

$\arcsin (x) = \frac{π}{12}$

Step 3

Take the inverse arcsine of both sides of the equation to extract $x$ from inside the arcsine.

$x = \sin (\frac{π}{12})$

Step 4

Simplify the right side.

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Step 4.1

The exact value of $\sin (\frac{π}{12})$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 4.1.1

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$x = \sin (\frac{π}{4} - \frac{π}{6})$

Step 4.1.2

Apply the difference of angles identity.

$x = \sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})$

Step 4.1.3

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

$x = \frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6})$

Step 4.1.4

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$x = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6})$

Step 4.1.5

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$x = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6})$

Step 4.1.6

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$x = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 4.1.7

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

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Step 4.1.7.1

Simplify each term.

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Step 4.1.7.1.1

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

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Step 4.1.7.1.1.1

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

$x = \frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 4.1.7.1.1.2

Combine using the product rule for radicals.

$x = \frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 4.1.7.1.1.3

Multiply $2$ by $3$ .

$x = \frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 4.1.7.1.1.4

Multiply $2$ by $2$ .

$x = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 4.1.7.1.2

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

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Step 4.1.7.1.2.1

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

$x = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}$

Step 4.1.7.1.2.2

Multiply $2$ by $2$ .

$x = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$

Step 4.1.7.2

Combine the numerators over the common denominator.

$x = \frac{\sqrt{6} - \sqrt{2}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{6} - \sqrt{2}}{4}$

Decimal Form:

$x = 0.25881904 \dots$