Algebra Examples

$\arctan (\frac{x}{3}) + arcsec (\frac{2 \sqrt{3}}{3}) = \arccos (- 1)$

Step 1

Simplify the left side.

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Step 1.1

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

$\arctan (\frac{x}{3}) + \frac{π}{6} = \arccos (- 1)$

Step 2

Simplify the right side.

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Step 2.1

The exact value of $\arccos (- 1)$ is $π$ .

$\arctan (\frac{x}{3}) + \frac{π}{6} = π$

Step 3

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

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Step 3.1

Subtract $\frac{π}{6}$ from both sides of the equation.

$\arctan (\frac{x}{3}) = π - \frac{π}{6}$

Step 3.2

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

$\arctan (\frac{x}{3}) = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 3.3

Combine $π$ and $\frac{6}{6}$ .

$\arctan (\frac{x}{3}) = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 3.4

Combine the numerators over the common denominator.

$\arctan (\frac{x}{3}) = \frac{π \cdot 6 - π}{6}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Move $6$ to the left of $π$ .

$\arctan (\frac{x}{3}) = \frac{6 \cdot π - π}{6}$

Step 3.5.2

Subtract $π$ from $6 π$ .

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

Step 4

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

$\frac{x}{3} = \tan (\frac{5 π}{6})$

Step 5

Simplify the right side.

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Step 5.1

Simplify $\tan (\frac{5 π}{6})$ .

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Step 5.1.1

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

$\frac{x}{3} = - \tan (\frac{π}{6})$

Step 5.1.2

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

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

Step 6

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = - \sqrt{3}$

Step 7

Exclude the solutions that do not make $\arctan (\frac{x}{3}) + arcsec (\frac{2 \sqrt{3}}{3}) = \arccos (- 1)$ true.

$No solution$