Tangent

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ACT Math › Tangent

Questions 1 - 10

1

Find the domain of $y=tan(x-\frac{3\pi}{4})$ . Assume is for all real numbers.

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \pi k$

$x=\frac{5\pi}{4}\pm \pi k$

$\textup{Everywhere except: }x=\frac{3\pi}{4}\pm \pi k$

$x=\frac{3\pi}{4}\pm \pi k$

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \frac{\pi}{2} k$

Explanation

The domain of does not exist at $\frac{\pi}{2}\pm \pi k$ , for is an integer.

The ends of every period approaches to either positive or negative infinity. Notice that for this problem, the entire graph shifts to the right $\frac{3\pi}{4}$ units. This means that the asymptotes would also shift right by the same distance.

The asymptotes will exist at:

$x=\frac{\pi}{2}\pm \pi k + \frac{3\pi}{4} = \frac{5\pi}{4}\pm \pi k$

Therefore, the domain of $y=tan(x-\frac{3\pi}{4})$ will exist anywhere EXCEPT:

$\frac{5\pi}{4}\pm \pi k$

2

Find the domain of $y=tan(x-\frac{3\pi}{4})$ . Assume is for all real numbers.

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \pi k$

$x=\frac{5\pi}{4}\pm \pi k$

$\textup{Everywhere except: }x=\frac{3\pi}{4}\pm \pi k$

$x=\frac{3\pi}{4}\pm \pi k$

$\textup{Everywhere except: }x=\frac{5\pi}{4}\pm \frac{\pi}{2} k$

Explanation

The domain of does not exist at $\frac{\pi}{2}\pm \pi k$ , for is an integer.

The ends of every period approaches to either positive or negative infinity. Notice that for this problem, the entire graph shifts to the right $\frac{3\pi}{4}$ units. This means that the asymptotes would also shift right by the same distance.

The asymptotes will exist at:

$x=\frac{\pi}{2}\pm \pi k + \frac{3\pi}{4} = \frac{5\pi}{4}\pm \pi k$

Therefore, the domain of $y=tan(x-\frac{3\pi}{4})$ will exist anywhere EXCEPT:

$\frac{5\pi}{4}\pm \pi k$

3

What is the range of the given trigonometric function:

$n(t) = 4\cot(2t)$

$(-\infty, \infty)$

Explanation

The range of a function is every value that the funciton's results take. For tangent and cotangent, the function spans from $(-\infty, \infty)$ and so the range is:

$(-\infty, \infty)$

4

What is the range of the given trigonometric function:

$n(t) = 4\cot(2t)$

$(-\infty, \infty)$

Explanation

The range of a function is every value that the funciton's results take. For tangent and cotangent, the function spans from $(-\infty, \infty)$ and so the range is:

$(-\infty, \infty)$

5

What is the period of the following trigonometric equation:
$h(t) = 4\tan(-3t)$

$\frac{\pi}{3}$

$-\frac{\pi}{3}$

$\frac{\pi}{2}$

$4\pi$

$2\pi$

Explanation

For tangent and cotangent the period is given by the formula:

$\omega = \frac{\pi}{\left | b\right |}$ where comes from $h(t)= a\tan(bt)$ .

Thus we see from our equation and so
$\omega = \frac{\pi}{\left | -3\right |}=\frac{\pi}{3}$ .

6

What is the period of the following trigonometric equation:
$h(t) = 4\tan(-3t)$

$\frac{\pi}{3}$

$-\frac{\pi}{3}$

$\frac{\pi}{2}$

$4\pi$

$2\pi$

Explanation

For tangent and cotangent the period is given by the formula:

$\omega = \frac{\pi}{\left | b\right |}$ where comes from $h(t)= a\tan(bt)$ .

Thus we see from our equation and so
$\omega = \frac{\pi}{\left | -3\right |}=\frac{\pi}{3}$ .

7

What is the period of the following trigonometric function:

$m(t) = -3\cot(-2t)$

$\omega = \frac{\pi}{2}$

$\omega = \pi$

$\omega = -\pi$

$\omega = -\frac{\pi}{2}$

$\omega = \frac{\pi}{3}$

Explanation

To find the period of a tangent or cotangent function use the following formula:

$\omega = \frac{\pi}{\left| b\right|}$

from the general tirogonometric formula:

$m(t) = a\cot(bt)$

Since we have,

$m(t) = -3\cot(-2t)$

we have

.

Thus we get that

$\omega = \frac{\pi}{2}$

8

What is the range of the trigonometric fuction defined by:
$f(t) = -2 \tan(3t)$ ?

$\mathbb{R}$

$\mathbb{N}$

$\mathbb{Z}$

Explanation

For tangent and cotangent functions, the range is always all real numbers.

9

What is the period of the following trigonometric function:

$m(t) = -3\cot(-2t)$

$\omega = \frac{\pi}{2}$

$\omega = \pi$

$\omega = -\pi$

$\omega = -\frac{\pi}{2}$

$\omega = \frac{\pi}{3}$

Explanation

To find the period of a tangent or cotangent function use the following formula:

$\omega = \frac{\pi}{\left| b\right|}$

from the general tirogonometric formula:

$m(t) = a\cot(bt)$

Since we have,

$m(t) = -3\cot(-2t)$

we have

.

Thus we get that

$\omega = \frac{\pi}{2}$

10

What is the range of the trigonometric fuction defined by:
$f(t) = -2 \tan(3t)$ ?

$\mathbb{R}$

$\mathbb{N}$

$\mathbb{Z}$

Explanation

For tangent and cotangent functions, the range is always all real numbers.

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