Trigonometry Examples

$\cos (x) > \frac{1}{2} \cdot \sin (x)$

Step 1

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

$\frac{\cos (x)}{\cos (x)} > \frac{\frac{1}{2} \cdot \sin (x)}{\cos (x)}$

Step 2

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

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Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} > \frac{\frac{1}{2} \cdot \sin (x)}{\cos (x)}$

Rewrite the expression.

$1 > \frac{\frac{1}{2} \cdot \sin (x)}{\cos (x)}$

Step 3

Separate fractions.

$1 > \frac{\frac{1}{2}}{1} \cdot \frac{\sin (x)}{\cos (x)}$

Step 4

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

$1 > \frac{\frac{1}{2}}{1} \cdot \tan (x)$

Step 5

Divide $\frac{1}{2}$ by $1$ .

$1 > \frac{1}{2} \cdot \tan (x)$

Step 6

Combine $\frac{1}{2}$ and $\tan (x)$ .

$1 > \frac{\tan (x)}{2}$

Step 7

Multiply both sides by $2$ .

$1 \cdot 2 > \frac{\tan (x)}{2} \cdot 2$

Step 8

Simplify.

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Simplify the left side.

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Multiply $2$ by $1$ .

$2 > \frac{\tan (x)}{2} \cdot 2$

Simplify the right side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

$2 > \frac{\tan (x)}{2} \cdot 2$

Rewrite the expression.

$2 > \tan (x)$

Step 9

Rewrite so $\tan (x)$ is on the left side of the inequality.

$\tan (x) < 2$

Step 10

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

$x < \arctan (2)$

Step 11

Simplify the right side.

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Evaluate $\arctan (2)$ .

$x < 1.10714871$

Step 12

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.

$x = (3.14159265) + 1.10714871$

Step 13

Solve for $x$ .

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Remove parentheses.

$x = 3.14159265 + 1.10714871$

Remove parentheses.

$x = (3.14159265) + 1.10714871$

Add $3.14159265$ and $1.10714871$ .

$x = 4.24874137$

Step 14

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

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

$\frac{π}{| b |}$

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

$\frac{π}{| 1 |}$

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

$\frac{π}{1}$

Divide $π$ by $1$ .

$π$

Step 15

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

$x = 1.10714871 + π n, 4.24874137 + π n$ , for any integer $n$

Step 16

Consolidate $1.10714871 + π n$ and $4.24874137 + π n$ to $1.10714871 + π n$ .

$x = 1.10714871 + π n$ , for any integer $n$

Step 17

Use each root to create test intervals.

$1.10714871 < x < 4.24874137$

Step 18

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Test a value on the interval $1.10714871 < x < 4.24874137$ to see if it makes the inequality true.

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Choose a value on the interval $1.10714871 < x < 4.24874137$ and see if this value makes the original inequality true.

$x = 3$

Replace $x$ with $3$ in the original inequality.

$\cos (3) > \frac{1}{2} \cdot \sin (3)$

The left side $- 0.98999249$ is not greater than the right side $0.07056$ , which means that the given statement is false.

False

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

$1.10714871 < x < 4.24874137$ False

Step 19

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

No solution