Trigonometry Examples

$\sin (x) + \sqrt{3} \cos (x) < 0$

Step 1

Divide each term in the equation by $\cos (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{\sqrt{3} \cos (x)}{\cos (x)} < \frac{0}{\cos (x)}$

Step 2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \frac{\sqrt{3} \cos (x)}{\cos (x)} < \frac{0}{\cos (x)}$

Step 3

Cancel the common factor of $\cos (x)$ .

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Step 3.1

Cancel the common factor.

$\tan (x) + \frac{\sqrt{3} \cos (x)}{\cos (x)} < \frac{0}{\cos (x)}$

Step 3.2

Divide $\sqrt{3}$ by $1$ .

$\tan (x) + \sqrt{3} < \frac{0}{\cos (x)}$

Step 4

Separate fractions.

$\tan (x) + \sqrt{3} < \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) + \sqrt{3} < \frac{0}{1} \cdot \sec (x)$

Step 6

Divide $0$ by $1$ .

$\tan (x) + \sqrt{3} < 0 \sec (x)$

Step 7

Multiply $0$ by $\sec (x)$ .

$\tan (x) + \sqrt{3} < 0$

Step 8

Subtract $\sqrt{3}$ from both sides of the inequality.

$\tan (x) < - \sqrt{3}$

Step 9

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

$x < \arctan (- \sqrt{3})$

Step 10

Simplify the right side.

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Step 10.1

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

$x < - \frac{π}{3}$

Step 11

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

$x = - \frac{π}{3} - π$

Step 12

Simplify the expression to find the second solution.

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Step 12.1

Add $2 π$ to $- \frac{π}{3} - π$ .

$x = - \frac{π}{3} - π + 2 π$

Step 12.2

The resulting angle of $\frac{2 π}{3}$ is positive and coterminal with $- \frac{π}{3} - π$ .

$x = \frac{2 π}{3}$

Step 13

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

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Step 13.1

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

$\frac{π}{| b |}$

Step 13.2

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

$\frac{π}{| 1 |}$

Step 13.3

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

$\frac{π}{1}$

Step 13.4

Divide $π$ by $1$ .

$π$

Step 14

Add $π$ to every negative angle to get positive angles.

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Step 14.1

Add $π$ to $- \frac{π}{3}$ to find the positive angle.

$- \frac{π}{3} + π$

Step 14.2

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

$π \cdot \frac{3}{3} - \frac{π}{3}$

Step 14.3

Combine fractions.

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Step 14.3.1

Combine $π$ and $\frac{3}{3}$ .

$\frac{π \cdot 3}{3} - \frac{π}{3}$

Step 14.3.2

Combine the numerators over the common denominator.

$\frac{π \cdot 3 - π}{3}$

Step 14.4

Simplify the numerator.

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Step 14.4.1

Move $3$ to the left of $π$ .

$\frac{3 \cdot π - π}{3}$

Step 14.4.2

Subtract $π$ from $3 π$ .

$\frac{2 π}{3}$

Step 14.5

List the new angles.

$x = \frac{2 π}{3}$

Step 15

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

$x = \frac{2 π}{3} + π n, \frac{2 π}{3} + π n$ , for any integer $n$

Step 16

Consolidate the answers.

$x = \frac{2 π}{3} + π n$ , for any integer $n$

Step 17

Use each root to create test intervals.

Step 18

Compare the intervals to determine which ones satisfy the original inequality.

Step 19

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution