Trigonometry Examples

$6 \arccos (3 x) = 5 π$

Step 1

Divide each term in $6 \arccos (3 x) = 5 π$ by $6$ and simplify.

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Step 1.1

Divide each term in $6 \arccos (3 x) = 5 π$ by $6$ .

$\frac{6 \arccos (3 x)}{6} = \frac{5 π}{6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $6$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{6 \arccos (3 x)}{6} = \frac{5 π}{6}$

Step 1.2.1.2

Divide $\arccos (3 x)$ by $1$ .

$\arccos (3 x) = \frac{5 π}{6}$

Step 2

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

$3 x = \cos (\frac{5 π}{6})$

Step 3

Simplify the right side.

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Step 3.1

Simplify $\cos (\frac{5 π}{6})$ .

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Step 3.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$3 x = - \cos (\frac{π}{6})$

Step 3.1.2

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

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

Step 4

Divide each term in $3 x = - \frac{\sqrt{3}}{2}$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 x = - \frac{\sqrt{3}}{2}$ by $3$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $x$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.2

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

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Step 4.3.2.1

Multiply $\frac{1}{3}$ by $\frac{\sqrt{3}}{2}$ .

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

Step 4.3.2.2

Multiply $3$ by $2$ .

$x = - \frac{\sqrt{3}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\sqrt{3}}{6}$

Decimal Form:

$x = - 0.28867513 \dots$