Find the x and y Intercepts y=tan(x-pi/2)

$y = \tan (x - \frac{π}{2})$

Find the x-intercepts.

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

$0 = \tan (x - \frac{π}{2})$

Solve the equation.

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

$\tan (x - \frac{π}{2}) = 0$

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

$x - \frac{π}{2} = \arctan (0)$

Simplify the right side.

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

$x - \frac{π}{2} = 0$

Add $\frac{π}{2}$ to both sides of the equation.

$x = \frac{π}{2}$

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.

$x - \frac{π}{2} = π + 0$

Solve for $x$ .

Add $π$ and $0$ .

$x - \frac{π}{2} = π$

Move all terms not containing $x$ to the right side of the equation.

Add $\frac{π}{2}$ to both sides of the equation.

$x = π + \frac{π}{2}$

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = π \cdot \frac{2}{2} + \frac{π}{2}$

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} + \frac{π}{2}$

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 + π}{2}$

Simplify the numerator.

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π + π}{2}$

Add $2 π$ and $π$ .

$x = \frac{3 π}{2}$

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

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 $1$ .

$π$

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

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

Consolidate the answers.

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

x-intercept(s) in point form.

x-intercept(s): $(\frac{π}{2} + π n, 0)$ , for any integer $n$

Find the y-intercepts.

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

$y = \tan ((0) - \frac{π}{2})$

Solve the equation.

Remove parentheses.

$y = \tan (0 - \frac{π}{2})$

Remove parentheses.

$y = \tan ((0) - \frac{π}{2})$

Simplify the right side.

Simplify $\tan ((0) - \frac{π}{2})$ .

Rewrite $\tan ((0) - \frac{π}{2})$ in terms of sines and cosines.

$y = \frac{\sin (0 - \frac{π}{2})}{\cos (0 - \frac{π}{2})}$

Subtract $\frac{π}{2}$ from $0$ .

$y = \frac{\sin (0 - \frac{π}{2})}{\cos (- \frac{π}{2})}$

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$y = \frac{\sin (0 - \frac{π}{2})}{\cos (\frac{3 π}{2})}$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$y = \frac{\sin (0 - \frac{π}{2})}{\cos (\frac{π}{2})}$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$y = \frac{\sin (0 - \frac{π}{2})}{0}$

The equation cannot be solved because it is undefined.

$Undefined$

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

y-intercept(s): $None$

List the intersections.

x-intercept(s): $(\frac{π}{2} + π n, 0)$ , for any integer $n$

y-intercept(s): $None$

Find the x and y Intercepts y=tan(x-pi/2)

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