Trigonometry Examples

$y = 6 \tan (\frac{x}{2}) - 3$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 6 \tan (\frac{x}{2}) - 3$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $6 \tan (\frac{x}{2}) - 3 = 0$ .

$6 \tan (\frac{x}{2}) - 3 = 0$

Step 1.2.2

Add $3$ to both sides of the equation.

$6 \tan (\frac{x}{2}) = 3$

Step 1.2.3

Divide each term in $6 \tan (\frac{x}{2}) = 3$ by $6$ and simplify.

Tap for more steps...

Step 1.2.3.1

Divide each term in $6 \tan (\frac{x}{2}) = 3$ by $6$ .

$\frac{6 \tan (\frac{x}{2})}{6} = \frac{3}{6}$

Step 1.2.3.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $6$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{6 \tan (\frac{x}{2})}{6} = \frac{3}{6}$

Step 1.2.3.2.1.2

Divide $\tan (\frac{x}{2})$ by $1$ .

$\tan (\frac{x}{2}) = \frac{3}{6}$

Step 1.2.3.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.3.1

Cancel the common factor of $3$ and $6$ .

Tap for more steps...

Step 1.2.3.3.1.1

Factor $3$ out of $3$ .

$\tan (\frac{x}{2}) = \frac{3 (1)}{6}$

Step 1.2.3.3.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.3.1.2.1

Factor $3$ out of $6$ .

$\tan (\frac{x}{2}) = \frac{3 \cdot 1}{3 \cdot 2}$

Step 1.2.3.3.1.2.2

Cancel the common factor.

$\tan (\frac{x}{2}) = \frac{3 \cdot 1}{3 \cdot 2}$

Step 1.2.3.3.1.2.3

Rewrite the expression.

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

Step 1.2.4

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

$\frac{x}{2} = \arctan (\frac{1}{2})$

Step 1.2.5

Simplify the right side.

Tap for more steps...

Step 1.2.5.1

Evaluate $\arctan (\frac{1}{2})$ .

$\frac{x}{2} = 0.4636476$

Step 1.2.6

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 \cdot 0.4636476$

Step 1.2.7

Simplify both sides of the equation.

Tap for more steps...

Step 1.2.7.1

Simplify the left side.

Tap for more steps...

Step 1.2.7.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.7.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 \cdot 0.4636476$

Step 1.2.7.1.1.2

Rewrite the expression.

$x = 2 \cdot 0.4636476$

Step 1.2.7.2

Simplify the right side.

Tap for more steps...

Step 1.2.7.2.1

Multiply $2$ by $0.4636476$ .

$x = 0.92729521$

Step 1.2.8

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.

$\frac{x}{2} = (3.14159265) + 0.4636476$

Step 1.2.9

Solve for $x$ .

Tap for more steps...

Step 1.2.9.1

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 ((3.14159265) + 0.4636476)$

Step 1.2.9.2

Simplify both sides of the equation.

Tap for more steps...

Step 1.2.9.2.1

Simplify the left side.

Tap for more steps...

Step 1.2.9.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.9.2.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 ((3.14159265) + 0.4636476)$

Step 1.2.9.2.1.1.2

Rewrite the expression.

$x = 2 ((3.14159265) + 0.4636476)$

Step 1.2.9.2.2

Simplify the right side.

Tap for more steps...

Step 1.2.9.2.2.1

Simplify $2 ((3.14159265) + 0.4636476)$ .

Tap for more steps...

Step 1.2.9.2.2.1.1

Add $3.14159265$ and $0.4636476$ .

$x = 2 \cdot 3.60524026$

Step 1.2.9.2.2.1.2

Multiply $2$ by $3.60524026$ .

$x = 7.21048052$

Step 1.2.10

Find the period of $\tan (\frac{x}{2})$ .

Tap for more steps...

Step 1.2.10.1

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

$\frac{π}{| b |}$

Step 1.2.10.2

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{π}{| \frac{1}{2} |}$

Step 1.2.10.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{2}}$

Step 1.2.10.4

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 2$

Step 1.2.10.5

Move $2$ to the left of $π$ .

$2 π$

Step 1.2.11

The period of the $\tan (\frac{x}{2})$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 0.92729521 + 2 π n, 7.21048052 + 2 π n$ , for any integer $n$

Step 1.2.12

Consolidate $0.92729521 + 2 π n$ and $7.21048052 + 2 π n$ to $0.92729521 + 2 π n$ .

$x = 0.92729521 + 2 π n$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0.92729521 + 2 π n, 0)$ , for any integer $n$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 6 \tan (\frac{0}{2}) - 3$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = 6 \tan (\frac{0}{2}) - 3$

Step 2.2.2

Simplify $6 \tan (\frac{0}{2}) - 3$ .

Tap for more steps...

Step 2.2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.2.1.1

Divide $0$ by $2$ .

$y = 6 \tan (0) - 3$

Step 2.2.2.1.2

The exact value of $\tan (0)$ is $0$ .

$y = 6 \cdot 0 - 3$

Step 2.2.2.1.3

Multiply $6$ by $0$ .

$y = 0 - 3$

Step 2.2.2.2

Subtract $3$ from $0$ .

$y = - 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 3)$

Step 3

List the intersections.

x-intercept(s): $(0.92729521 + 2 π n, 0)$ , for any integer $n$

y-intercept(s): $(0, - 3)$

Step 4