📈 Level 2 - Topic 3: Inverse Trigonometric Functions (Arcsine, Arccosine, Arctangent) 📐🔍

1) Undoing Trigonometric Functions - Introduction to Inverses 🔍

We've spent considerable time working with trigonometric functions – sine, cosine, tangent, and their variations. These functions take an angle as input and give a ratio as output. But what if we want to go the other way? What if we know the ratio and want to find the angle? This is where inverse trigonometric functions come into play.

The Need for Inverses:

Solving for Angles: In many problems, especially in geometry, physics, and engineering, we need to find angles based on known side ratios. Inverse trigonometric functions are essential for this.
Extending Function Concepts: Just as we have inverse operations for addition (subtraction), multiplication (division), and exponentiation (logarithms, roots), we need inverses for trigonometric functions to "undo" their operation.
Mathematical Completeness: To have a complete set of mathematical tools, we need inverse functions for all fundamental trigonometric functions.

In this topic, we will introduce the three primary inverse trigonometric functions: arcsine, arccosine, and arctangent (also known as inverse sine, inverse cosine, and inverse tangent, often denoted as \( \sin^{-1}, \cos^{-1}, \tan^{-1} \)). We will explore their definitions, notations, domains, ranges, and basic properties.

2) Inverse Sine Function - Arcsine ( \( \arcsin \) or \( \sin^{-1} \) ) 📐

Let's start with the inverse sine function, also known as arcsine. We want a function that answers the question: "Given a sine ratio \( y \), what angle \( \theta \) produced this ratio?"

The Problem of Invertibility: Recall the graph of \( y = \sin(\theta) \). It's a periodic wave. This means for a given \( y \) value (ratio) between -1 and 1, there are infinitely many angles \( \theta \) that produce this sine value. For a function to have a true inverse, it must be one-to-one (each output corresponds to only one input). The sine function is not one-to-one over its entire domain.

Restricting the Domain: To define an inverse sine function, we restrict the domain of the sine function to an interval where it is one-to-one. The standard convention is to restrict the domain of sine to \( [-90^\circ, 90^\circ] \) or \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) in radians. On this restricted domain, sine is increasing and covers its full range from -1 to 1, and it becomes one-to-one.

Definition: Inverse Sine Function (Arcsine)

Let \( y = \sin(\theta) \), where \( \theta \) is restricted to the domain \( [-90^\circ, 90^\circ] \) or \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). Then the inverse sine function, denoted as \( \arcsin(y) \) or \( \sin^{-1}(y) \), is defined as the angle \( \theta \) in this restricted domain such that \( \sin(\theta) = y \).

Notation:

\( \theta = \arcsin(y) \) is equivalent to \( y = \sin(\theta) \) for \( -1 \leq y \leq 1 \) and \( -90^\circ \leq \theta \leq 90^\circ \) (or \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \)).
\( \sin^{-1}(y) \) is another common notation for arcsine, meaning the same as \( \arcsin(y) \). Note: The "-1" is NOT an exponent; it denotes the inverse function.

Domain and Range of Arcsine:

Domain of \( \arcsin(y) \) or \( \sin^{-1}(y) \): \( [-1, 1] \) (because the range of sine function is \( [-1, 1] \)).

Range of \( \arcsin(y) \) or \( \sin^{-1}(y) \): \( [-90^\circ, 90^\circ] \) or \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) (due to the restricted domain of sine).

Example 1: Evaluating Arcsine

Find \( \arcsin(1/2) \). We need to find an angle \( \theta \) in \( [-90^\circ, 90^\circ] \) such that \( \sin(\theta) = 1/2 \). We know \( \sin(30^\circ) = 1/2 \), and \( 30^\circ \) is in the range \( [-90^\circ, 90^\circ] \). So, \( \arcsin(1/2) = 30^\circ \) or \( \frac{\pi}{6} \) radians.
Find \( \sin^{-1}(-\frac{\sqrt{2}}{2}) \). We need to find \( \theta \in [-90^\circ, 90^\circ] \) with \( \sin(\theta) = -\frac{\sqrt{2}}{2} \). We know \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \), and sine is negative in the 4th quadrant. So, \( \sin(-45^\circ) = -\frac{\sqrt{2}}{2} \). Since \( -45^\circ \) is in \( [-90^\circ, 90^\circ] \), \( \sin^{-1}(-\frac{\sqrt{2}}{2}) = -45^\circ \) or \( -\frac{\pi}{4} \) radians.
Find \( \arcsin(2) \). Since the domain of arcsine is \( [-1, 1] \), \( \arcsin(2) \) is undefined (or does not exist as a real number). There is no real angle whose sine is 2.

3) Inverse Cosine Function - Arccosine ( \( \arccos \) or \( \cos^{-1} \) ) 📐

Next, let's consider the inverse cosine function, or arccosine. Similar to sine, the cosine function is also periodic and not one-to-one over its entire domain.

Restricting Domain for Cosine: To define arccosine, we need to restrict the domain of the cosine function to make it one-to-one. The standard restricted domain for cosine is \( [0^\circ, 180^\circ] \) or \( [0, \pi] \) in radians. On this domain, cosine is decreasing and covers its full range from -1 to 1, becoming one-to-one.

Definition: Inverse Cosine Function (Arccosine)

Let \( y = \cos(\theta) \), where \( \theta \) is restricted to the domain \( [0^\circ, 180^\circ] \) or \( [0, \pi] \). Then the inverse cosine function, denoted as \( \arccos(y) \) or \( \cos^{-1}(y) \), is defined as the angle \( \theta \) in this restricted domain such that \( \cos(\theta) = y \).

Notation:

\( \theta = \arccos(y) \) is equivalent to \( y = \cos(\theta) \) for \( -1 \leq y \leq 1 \) and \( 0^\circ \leq \theta \leq 180^\circ \) (or \( 0 \leq \theta \leq \pi \)).
\( \cos^{-1}(y) \) is another common notation for arccosine, same meaning as \( \arccos(y) \).

Domain and Range of Arccosine:

Domain of \( \arccos(y) \) or \( \cos^{-1}(y) \): \( [-1, 1] \) (because the range of cosine function is \( [-1, 1] \)).

Range of \( \arccos(y) \) or \( \cos^{-1}(y) \): \( [0^\circ, 180^\circ] \) or \( [0, \pi] \) (due to the restricted domain of cosine).

Example 2: Evaluating Arccosine

Find \( \arccos(\frac{\sqrt{3}}{2}) \). We need to find an angle \( \theta \) in \( [0^\circ, 180^\circ] \) such that \( \cos(\theta) = \frac{\sqrt{3}}{2} \). We know \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), and \( 30^\circ \) is in the range \( [0^\circ, 180^\circ] \). So, \( \arccos(\frac{\sqrt{3}}{2}) = 30^\circ \) or \( \frac{\pi}{6} \) radians.
Find \( \cos^{-1}(-\frac{1}{2}) \). We need to find \( \theta \in [0^\circ, 180^\circ] \) with \( \cos(\theta) = -\frac{1}{2} \). We know \( \cos(60^\circ) = \frac{1}{2} \), and cosine is negative in the 2nd quadrant. So, \( \cos(180^\circ - 60^\circ) = \cos(120^\circ) = -\frac{1}{2} \). Since \( 120^\circ \) is in \( [0^\circ, 180^\circ] \), \( \cos^{-1}(-\frac{1}{2}) = 120^\circ \) or \( \frac{2\pi}{3} \) radians.
Find \( \arccos(-1.5) \). Since the domain of arccosine is \( [-1, 1] \), \( \arccos(-1.5) \) is undefined.

4) Inverse Tangent Function - Arctangent ( \( \arctan \) or \( \tan^{-1} \) ) 📐

Finally, let's consider the inverse tangent function, or arctangent. Like sine and cosine, the tangent function is periodic and needs a restricted domain to be invertible.

Restricting Domain for Tangent: The standard restricted domain for tangent to make it one-to-one is \( (-90^\circ, 90^\circ) \) or \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) in radians. Note that we use open intervals because tangent is undefined at \( \pm 90^\circ \). On this domain, tangent is increasing and covers its full range from \( -\infty \) to \( \infty \), and it becomes one-to-one.

Definition: Inverse Tangent Function (Arctangent)

Let \( y = \tan(\theta) \), where \( \theta \) is restricted to the domain \( (-90^\circ, 90^\circ) \) or \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). Then the inverse tangent function, denoted as \( \arctan(y) \) or \( \tan^{-1}(y) \), is defined as the angle \( \theta \) in this restricted domain such that \( \tan(\theta) = y \).

Notation:

\( \theta = \arctan(y) \) is equivalent to \( y = \tan(\theta) \) for \( y \in \mathbb{R} \) (any real number) and \( -90^\circ < \theta < 90^\circ \) (or \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \)).
\( \tan^{-1}(y) \) is another common notation for arctangent, same meaning as \( \arctan(y) \).

Domain and Range of Arctangent:

Domain of \( \arctan(y) \) or \( \tan^{-1}(y) \): \( (-\infty, \infty) \) or all real numbers (because the range of tangent function is \( (-\infty, \infty) \)).

Range of \( \arctan(y) \) or \( \tan^{-1}(y) \): \( (-90^\circ, 90^\circ) \) or \( (-\frac{\pi}{2}, \frac{\pi}{2}) \) (due to the restricted domain of tangent).

Example 3: Evaluating Arctangent

Find \( \arctan(1) \). We need to find an angle \( \theta \) in \( (-90^\circ, 90^\circ) \) such that \( \tan(\theta) = 1 \). We know \( \tan(45^\circ) = 1 \), and \( 45^\circ \) is in the range \( (-90^\circ, 90^\circ) \). So, \( \arctan(1) = 45^\circ \) or \( \frac{\pi}{4} \) radians.
Find \( \tan^{-1}(-\sqrt{3}) \). We need to find \( \theta \in (-90^\circ, 90^\circ) \) with \( \tan(\theta) = -\sqrt{3} \). We know \( \tan(60^\circ) = \sqrt{3} \), and tangent is negative in the 4th quadrant. So, \( \tan(-60^\circ) = -\sqrt{3} \). Since \( -60^\circ \) is in \( (-90^\circ, 90^\circ) \), \( \tan^{-1}(-\sqrt{3}) = -60^\circ \) or \( -\frac{\pi}{3} \) radians.
Find \( \arctan(0) \). We need to find \( \theta \in (-90^\circ, 90^\circ) \) with \( \tan(\theta) = 0 \). We know \( \tan(0^\circ) = 0 \), and \( 0^\circ \) is in \( (-90^\circ, 90^\circ) \). So, \( \arctan(0) = 0^\circ \) or \( 0 \) radians.

5) Summary of Inverse Trigonometric Functions - Domains and Ranges 📝

Let's summarize the key properties of the inverse trigonometric functions we've learned, focusing on their domains and ranges:

Summary Table: Inverse Trigonometric Functions

Inverse Function	Notation	Domain (Input Values)	Range (Output Angles)
Arcsine	\( \arcsin(y) \) or \( \sin^{-1}(y) \)	\( [-1, 1] \)	\( [-90^\circ, 90^\circ] \) or \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)
Arccosine	\( \arccos(y) \) or \( \cos^{-1}(y) \)	\( [-1, 1] \)	\( [0^\circ, 180^\circ] \) or \( [0, \pi] \)
Arctangent	\( \arctan(y) \) or \( \tan^{-1}(y) \)	\( (-\infty, \infty) \)	\( (-90^\circ, 90^\circ) \) or \( (-\frac{\pi}{2}, \frac{\pi}{2}) \)

6) Practice Questions 🎯

6.1 Fundamental – Evaluating Inverse Trigonometric Functions

1. Evaluate \( \arcsin(\frac{\sqrt{2}}{2}) \) in degrees and radians.

2. Evaluate \( \cos^{-1}(0) \) in degrees and radians.

3. Evaluate \( \arctan(\sqrt{3}) \) in degrees and radians.

4. Find the value of \( \sin^{-1}(-1) \) in degrees and radians.

5. Find the value of \( \arccos(-\frac{\sqrt{3}}{2}) \) in degrees and radians.

6. What is the domain of the function \( f(x) = \arcsin(x) \)?

7. What is the range of the function \( g(x) = \arccos(x) \) in degrees?

8. Is \( \arcsin(1.5) \) defined? Explain why or why not.

9. In what quadrant(s) does the angle \( \arcsin(y) \) lie, for any valid input \( y \)?

10. In what quadrant(s) does the angle \( \arccos(y) \) lie, for any valid input \( y \)?

6.2 Challenging – Conceptual and Application Questions 💪🚀

1. Explain why it is necessary to restrict the domains of sine, cosine, and tangent functions to define their inverse functions. What would happen if we didn't restrict the domains?

2. Why is the range of \( \arcsin(x) \) chosen to be \( [-\frac{\pi}{2}, \frac{\pi}{2}] \) and not, for example, \( [\frac{\pi}{2}, \frac{3\pi}{2}] \)? Is there a specific reason for this choice, or is it arbitrary?

3. Suppose you know that \( \sin(\theta) = 0.6 \) and \( \theta \) is in the second quadrant \( (90^\circ < \theta < 180^\circ) \). Is \( \theta = \arcsin(0.6) \)? If not, how can you find the value of \( \theta \) using \( \arcsin \) or other inverse trigonometric functions?

4. Consider a right-angled triangle where the opposite side to angle \( \theta \) is 5 units and the hypotenuse is 7 units. Express \( \theta \) using an inverse trigonometric function. Calculate the approximate value of \( \theta \) in degrees.

5. (Conceptual) How are the graphs of \( y = \sin(x) \) (with restricted domain) and \( y = \arcsin(x) \) related geometrically? (Hint: Think about reflection across a line).

7) Summary - Unveiling Inverse Trigonometric Functions 🎉

Inverse Trig Functions: Arcsine (\( \arcsin \) or \( \sin^{-1} \)), Arccosine (\( \arccos \) or \( \cos^{-1} \)), Arctangent (\( \arctan \) or \( \tan^{-1} \)) are inverses of sine, cosine, and tangent respectively.
Need for Domain Restriction: Sine, Cosine, and Tangent are not one-to-one over their entire domains, so domains are restricted to define inverses.
Arcsine: Inverse of sine with domain restricted to \( [-\frac{\pi}{2}, \frac{\pi}{2}] \). Domain of \( \arcsin \) is \( [-1, 1] \), Range is \( [-\frac{\pi}{2}, \frac{\pi}{2}] \).
Arccosine: Inverse of cosine with domain restricted to \( [0, \pi] \). Domain of \( \arccos \) is \( [-1, 1] \), Range is \( [0, \pi] \).
Arctangent: Inverse of tangent with domain restricted to \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). Domain of \( \arctan \) is \( (-\infty, \infty) \), Range is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).
Domains and Ranges are Key: Understanding the specific domains and ranges of inverse trigonometric functions is crucial for correct usage.
Applications: Inverse trigonometric functions are essential for solving for angles in trigonometry and its applications.

Excellent! You've now been introduced to the inverse trigonometric functions – arcsine, arccosine, and arctangent. Understanding these functions, especially their domains and ranges, is vital for further work in trigonometry and related fields. In the following topics, we'll apply these functions and explore more trigonometric relationships and identities! 📐🔍🌟