Trigonometry Examples

$1 = \tan (x) - \sqrt{3}$

Step 1

Rewrite the equation as $\tan (x) - \sqrt{3} = 1$ .

$\tan (x) - \sqrt{3} = 1$

Step 2

Add $\sqrt{3}$ to both sides of the equation.

$\tan (x) = 1 + \sqrt{3}$

Step 3

Convert the right side of the equation to its decimal equivalent.

$\tan (x) = 2.7320508$

Step 4

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

$x = \arctan (2.7320508)$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arctan (2.7320508)$ .

$x = 1.21991691$

Step 6

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = (3.14159265) + 1.21991691$

Step 7

Solve for $x$ .

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Step 7.1

Remove parentheses.

$x = 3.14159265 + 1.21991691$

Step 7.2

Remove parentheses.

$x = (3.14159265) + 1.21991691$

Step 7.3

Add $3.14159265$ and $1.21991691$ .

$x = 4.36150956$

Step 8

Find the period of $\tan (x)$ .

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Step 8.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 8.2

Replace $b$ with $1$ in the formula for period.

$\frac{π}{| 1 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{π}{1}$

Step 8.4

Divide $π$ by $1$ .

$π$

Step 9

The period of the $\tan (x)$ function is $π$ so values will repeat every $π$ radians in both directions.

$x = 1.21991691 + π n, 4.36150956 + π n$ , for any integer $n$

Step 10

Consolidate $1.21991691 + π n$ and $4.36150956 + π n$ to $1.21991691 + π n$ .

$x = 1.21991691 + π n$ , for any integer $n$