Trigonometry Examples

$\cos (a) < \sin (a)$

Step 1

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

$\frac{\cos (a)}{\cos (a)} < \frac{\sin (a)}{\cos (a)}$

Step 2

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

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Step 2.1

Cancel the common factor.

$\frac{\cos (a)}{\cos (a)} < \frac{\sin (a)}{\cos (a)}$

Step 2.2

Rewrite the expression.

$1 < \frac{\sin (a)}{\cos (a)}$

Step 3

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

$1 < \tan (a)$

Step 4

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

$\tan (a) > 1$

Step 5

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

$a > \arctan (1)$

Step 6

Simplify the right side.

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Step 6.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$a > \frac{π}{4}$

Step 7

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.

$a = π + \frac{π}{4}$

Step 8

Simplify $π + \frac{π}{4}$ .

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Step 8.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$a = π \cdot \frac{4}{4} + \frac{π}{4}$

Step 8.2

Combine fractions.

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Step 8.2.1

Combine $π$ and $\frac{4}{4}$ .

$a = \frac{π \cdot 4}{4} + \frac{π}{4}$

Step 8.2.2

Combine the numerators over the common denominator.

$a = \frac{π \cdot 4 + π}{4}$

Step 8.3

Simplify the numerator.

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Step 8.3.1

Move $4$ to the left of $π$ .

$a = \frac{4 \cdot π + π}{4}$

Step 8.3.2

Add $4 π$ and $π$ .

$a = \frac{5 π}{4}$

Step 9

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

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Step 9.1

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

$\frac{π}{| b |}$

Step 9.2

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

$\frac{π}{| 1 |}$

Step 9.3

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

$\frac{π}{1}$

Step 9.4

Divide $π$ by $1$ .

$π$

Step 10

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

$a = \frac{π}{4} + π n, \frac{5 π}{4} + π n$ , for any integer $n$

Step 11

Consolidate the answers.

$a = \frac{π}{4} + π n$ , for any integer $n$

Step 12

Use each root to create test intervals.

$\frac{π}{4} < a < \frac{5 π}{4}$

Step 13

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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Step 13.1

Test a value on the interval $\frac{π}{4} < a < \frac{5 π}{4}$ to see if it makes the inequality true.

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Step 13.1.1

Choose a value on the interval $\frac{π}{4} < a < \frac{5 π}{4}$ and see if this value makes the original inequality true.

$a = 2$

Step 13.1.2

Replace $a$ with $2$ in the original inequality.

$\cos (2) < \sin (2)$

Step 13.1.3

The left side $- 0.41614683$ is less than the right side $0.90929742$ , which means that the given statement is always true.

True

Step 13.2

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

$\frac{π}{4} < a < \frac{5 π}{4}$ True

Step 14

The solution consists of all of the true intervals.

$\frac{π}{4} + π n < a < \frac{5 π}{4} + π n$ , for any integer $n$

Step 15