Graphs of Inverse Trigonometric Functions

Graphs of Inverse Trigonometric Functions

The inverse trigonometric functions are the inverses of the basic trigonometric functions (sine, cosine, tangent, etc.). They allow you to determine the angle that corresponds to a specific trigonometric value. These inverse functions are typically written as:

• Inverse sine: $\sin^{-1}(x)$ or $\arcsin(x)$
• Inverse cosine: $\cos^{-1}(x)$ or $\arccos(x)$
• Inverse tangent: $\tan^{-1}(x)$ or $\arctan(x)$
• Inverse secant: $\sec^{-1}(x)$ or $\arcsec(x)$
• Inverse cosecant: $\csc^{-1}(x)$ or $\arccsc(x)$
• Inverse cotangent: $\cot^{-1}(x)$ or $\arccot(x)$

These functions are used when you know the value of a trigonometric ratio and need to find the corresponding angle. Let’s explore the properties and graphs of these functions.

---

1. Inverse Sine Function ($y = \sin^{-1}(x)$ or $y = \arcsin(x)$)

• Domain: The domain of the inverse sine function is restricted to $x \in [-1, 1]$. This is because the sine function only takes values between -1 and 1.
• Range: The range of $\sin^{-1}(x)$ is $y \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$, meaning the output is an angle between $-90^\circ$ and $90^\circ$.
• Key Points:
  • $\sin^{-1}(0) = 0$
  • $\sin^{-1}(1) = \frac{\pi}{2}$
  • $\sin^{-1}(-1) = -\frac{\pi}{2}$

Graph Characteristics:

• The graph of $y = \arcsin(x)$ is increasing.
• It starts at $(-1, -\frac{\pi}{2})$ and ends at $(1, \frac{\pi}{2})$.
• It is a smooth curve that passes through the origin $(0, 0)$.

---

2. Inverse Cosine Function ($y = \cos^{-1}(x)$ or $y = \arccos(x)$)

• Domain: The domain of the inverse cosine function is $x \in [-1, 1]$, since cosine values range between -1 and 1.
• Range: The range of $\cos^{-1}(x)$ is $y \in [0, \pi]$, meaning the output is an angle between 0 and $180^\circ$.
• Key Points:
  • $\cos^{-1}(0) = \frac{\pi}{2}$
  • $\cos^{-1}(1) = 0$
  • $\cos^{-1}(-1) = \pi$

Graph Characteristics:

• The graph of $y = \arccos(x)$ is decreasing.
• It starts at $(-1, \pi)$ and ends at $(1, 0)$.
• The curve is smooth and passes through the point $(0, \frac{\pi}{2})$.

---

3. Inverse Tangent Function ($y = \tan^{-1}(x)$ or $y = \arctan(x)$)

• Domain: The domain of the inverse tangent function is all real numbers $x \in (-\infty, \infty)$.
• Range: The range of $\tan^{-1}(x)$ is $y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$, meaning the output is an angle between $-90^\circ$ and $90^\circ$.
• Key Points:
  • $\tan^{-1}(0) = 0$
  • $\tan^{-1}(1) = \frac{\pi}{4}$
  • $\tan^{-1}(-1) = -\frac{\pi}{4}$

Graph Characteristics:

• The graph of $y = \arctan(x)$ is an increasing curve.
• As $x$ approaches $\infty$, $y$ approaches $\frac{\pi}{2}$, and as $x$ approaches $-\infty$, $y$ approaches $-\frac{\pi}{2}$.
• The curve passes through the origin $(0, 0)$.

---

4. Inverse Secant Function ($y = \sec^{-1}(x)$ or $y = \arcsec(x)$)

• Domain: The domain of the inverse secant function is $x \in (-\infty, -1] \cup [1, \infty)$, because secant values must be either less than or equal to -1, or greater than or equal to 1.
• Range: The range of $\sec^{-1}(x)$ is $y \in [0, \pi]$ except $\frac{\pi}{2}$, meaning the output angle is between 0 and $\pi$, but not equal to $\frac{\pi}{2}$.
• Key Points:
  • $\sec^{-1}(1) = 0$
  • $\sec^{-1}(-1) = \pi$

Graph Characteristics:

• The graph of $y = \arcsec(x)$ is composed of two separate branches.
• One branch is in the interval $[1, \infty)$ and the other branch is in the interval $(-\infty, -1]$.
• The curve approaches $0$ as $x \to 1^+$, and it approaches $\pi$ as $x \to -1^-$.

---

5. Inverse Cosecant Function ($y = \csc^{-1}(x)$ or $y = \arccsc(x)$)

• Domain: The domain of the inverse cosecant function is $x \in (-\infty, -1] \cup [1, \infty)$, because cosecant values must be either less than or equal to -1, or greater than or equal to 1.
• Range: The range of $\csc^{-1}(x)$ is $y \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ except $0$, meaning the output angle is between $-90^\circ$ and $90^\circ$, but not equal to 0.
• Key Points:
  • $\csc^{-1}(1) = \frac{\pi}{2}$
  • $\csc^{-1}(-1) = -\frac{\pi}{2}$

Graph Characteristics:

• The graph of $y = \arccsc(x)$ is similar to the graph of $\arcsec(x)$ and consists of two separate branches.
• One branch is in the interval $[1, \infty)$ and the other branch is in the interval $(-\infty, -1]$.
• The curve approaches $\frac{\pi}{2}$ as $x \to 1^+$, and approaches $-\frac{\pi}{2}$ as $x \to -1^-$.

---

6. Inverse Cotangent Function ($y = \cot^{-1}(x)$ or $y = \arccot(x)$)

• Domain: The domain of the inverse cotangent function is all real numbers $x \in (-\infty, \infty)$.
• Range: The range of $\cot^{-1}(x)$ is $y \in (0, \pi)$, meaning the output angle is between 0 and $\pi$.
• Key Points:
  • $\cot^{-1}(0) = \frac{\pi}{2}$
  • $\cot^{-1}(1) = \frac{\pi}{4}$
  • $\cot^{-1}(-1) = \frac{3\pi}{4}$

Graph Characteristics:

• The graph of $y = \arccot(x)$ is a decreasing curve.
• As $x$ approaches $\infty$, $y$ approaches $0$, and as $x$ approaches $-\infty$, $y$ approaches $\pi$.
• The curve passes through the point $(0, \frac{\pi}{2})$.

---

Summary of Graph Characteristics of Inverse Trigonometric Functions:

Function	Domain	Range	Key Points
$\arcsin(x)$	$[-1, 1]$	$\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$	$\arcsin(0) = 0$, $\arcsin(1) = \frac{\pi}{2}$, $\arcsin(-1) = -\frac{\pi}{2}$
$\arccos(x)$	$[-1, 1]$	$[0, \pi]$	$\arccos(0) = \frac{\pi}{2}$, $\arccos(1) = 0$, $\arccos(-1) = \pi$
$\arctan(x)$	$(-\infty, \infty)$	$\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$	$\arctan(0) = 0$, $\arctan(1) = \frac{\pi}{4}$, $\arctan(-1) = -\frac{\pi}{4}$
$\arcsec(x)$	$(-\infty, -1] \cup [1, \infty)$	$$ [0