Question

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Answer 1

Answer to Question #87409 – Math – Trigonometry

Question

The following graph depicts which inverse trigonometric function?

$\varphi = \text{Arccos} \times$

$\varphi = \text{Arcsin} \times$

$\varphi = \text{Arctan} \times$

$\varphi = \text{Arcsec} \times$

Solution

1. $y = \operatorname{Arccos}(x)$

Arccosine ($y = \operatorname{Arccos}x$) is the function inverse to the cosine ($x = \cos(y)$). It has the domain $-1 \leq x \leq 1$ and the range $0 \leq y \leq \pi$.

2. $y = \operatorname{Arcsin}(x)$

Arcsine ($y = \operatorname{arcsin}x$) is the inverse function of the sine ($x = \sin(y)$). It has the domain $-1 \leq x \leq 1$ and the range $-\pi/2 \leq y \leq \pi/2$.

3. $y = \operatorname{Arctan}(x)$

The arctangent ($y = \operatorname{arctan}x$) is a function inverse to the tangent ($x = \tan(y)$), which has a domain $-\infty < x < +\infty$ and the range $-\pi/2 \leq y \leq \pi/2$.

4. $y = \operatorname{Arcsec}(x)$

$$y = \operatorname{arcsec}(x) = \arccos(1/x)$$

Arc secant is discontinuous function defined on entire real axis except the $(-1, 1)$, so its domain is $(-\infty, -1] \cup [1, +\infty)$.

Answer: y=Arcsin (x).

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