Inverse Circular Functions: Definitions, Domains, Ranges & Graphs

TAKEAWAYS:

• Inverse circular functions (arcsin, arccos, arctan) undo sine, cosine, and tangent functions, returning angles from ratios.
• Their domains are restricted to $[-1,1]$ (for arcsin, arccos) or all reals (for arctan) to ensure one-to-one correspondence.
• Ranges are $[-\pi /2,\pi /2]$ (arcsin), $[0,\pi ]$ (arccos), and $(-\pi /2,\pi /2)$ (arctan).
• Graphs are reflections of their parent functions across the line $y=x$.
• Key identities include $\sin (\arcsin x)=x$, $\cos (\arccos x)=x$, and $\tan (\arctan x)=x$.
• Applications include solving trigonometric equations and modeling real-world scenarios like angles in right triangles.

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1. Introduction to Inverse Circular Functions

Inverse circular functions are the reverse of the basic trigonometric functions. While sine, cosine, and tangent take an angle and return a ratio, their inverses take a ratio and return an angle.

For example:

• If $\sin \theta =\frac{1}{2}$, then $\theta =\arcsin \left(\frac{1}{2}\right)$.
• If $\tan \theta =1$, then $\theta =\arctan (1)$.

These functions are written as:

• $\arcsin x$ (or ${\sin }^{-1}x$) for inverse sine,
• $\arccos x$ (or ${\cos }^{-1}x$) for inverse cosine,
• $\arctan x$ (or ${\tan }^{-1}x$) for inverse tangent.

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2. Definitions and Restrictions

Trigonometric functions (sine, cosine, tangent) are not one-to-one over their entire domains. This means multiple angles can have the same ratio (e.g., $\sin 30^{\circ }=\sin 150^{\circ }=0.5$). To define inverses, we restrict the domain of each function to make it one-to-one.

Restricted Domains and Ranges

Why these restrictions?

• For $\arcsin x$ and $\arctan x$, the range is $[-\pi /2,\pi /2]$ because sine and tangent are one-to-one in this interval.
• For $\arccos x$, the range is $[0,\pi ]$ because cosine is one-to-one here.

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3. Graphs of Inverse Circular Functions

The graph of an inverse function is the reflection of the original function across the line $y=x$.

Graph of $y=\arcsin x$

Key Observations:

• Domain: $[-1,1]$
• Range: $[-\pi /2,\pi /2]$
• Passes through $(0,0)$, $(1,\pi /2)$, and $(-1,-\pi /2)$.

Graph of $y=\arccos x$

Key Observations:

• Domain: $[-1,1]$
• Range: $[0,\pi ]$
• Passes through $(1,0)$, $(0,\pi /2)$, and $(-1,\pi )$.

Graph of $y=\arctan x$

Key Observations:

• Domain: All real numbers ($-\infty ,\infty$)
• Range: $(-\pi /2,\pi /2)$
• Asymptotes at $y=\pi /2$ and $y=-\pi /2$ (never reaches them).

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4. Key Identities and Properties

1. Basic Identities: $$\sin (\arcsin x)=x,\cos (\arccos x)=x,\tan (\arctan x)=x$$ These identities hold only when $x$ is within the domain of the inverse function.

2. Compositions with Trigonometric Functions: $$\arcsin (\sin \theta )=\theta \text{only if}\theta \in [-\frac{\pi }{2},\frac{\pi }{2}]$$ $$\arccos (\cos \theta )=\theta \text{only if}\theta \in [0,\pi ]$$ $$\arctan (\tan \theta )=\theta \text{only if}\theta \in (-\frac{\pi }{2},\frac{\pi }{2})$$

3. Relationship Between $\arcsin x$ and $\arccos x$: $$\arcsin x+\arccos x=\frac{\pi }{2}$$ This is because $\sin \theta =\cos \left(\frac{\pi }{2}-\theta \right)$.

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5. Solved Examples

Example 1: Evaluating Inverse Functions

Find the value of: a) $\arcsin \left(\frac{\sqrt{3}}{2}\right)$ b) $\arccos \left(-\frac{1}{2}\right)$ c) $\arctan (1)$

Solution: a) $\arcsin \left(\frac{\sqrt{3}}{2}\right)=\frac{\pi }{3}$ (since $\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}$). b) $\arccos \left(-\frac{1}{2}\right)=\frac{2\pi }{3}$ (since $\cos \frac{2\pi }{3}=-\frac{1}{2}$). c) $\arctan (1)=\frac{\pi }{4}$ (since $\tan \frac{\pi }{4}=1$).

Example 2: Solving Equations

Solve for $x$: $${\sin }^{-1}(x)=\frac{\pi }{6}$$

Solution: Take the sine of both sides: $$x=\sin \left(\frac{\pi }{6}\right)=\frac{1}{2}$$

Example 3: Domain and Range

Determine the domain and range of $y=\arctan (2x)$.

Solution:

• The domain of $\arctan x$ is all real numbers, so the domain of $\arctan (2x)$ is also all real numbers ($-\infty ,\infty$).
• The range of $\arctan x$ is $(-\pi /2,\pi /2)$, so the range of $\arctan (2x)$ remains $(-\pi /2,\pi /2)$.

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6. Applications of Inverse Circular Functions

1. Solving Trigonometric Equations: Inverse functions help find angles when the ratio is known. For example: $$\text{If }\tan \theta =\sqrt{3},\text{ then }\theta =\arctan (\sqrt{3})=\frac{\pi }{3}.$$

2. Modeling Real-World Problems:

   • Engineering: Calculating angles in structures using inverse trigonometric functions.
   • Physics: Determining angles in projectile motion.
   • Navigation: Finding directions using bearings (e.g., $\arctan$ for compass angles).

3. Calculus: Inverse trigonometric functions appear in integration and differentiation problems, such as: $$\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}.$$

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7. Common Mistakes to Avoid

1. Forgetting Domain Restrictions:

   • $\arcsin x$ is only defined for $x\in [-1,1]$.
   • $\arctan x$ is defined for all real $x$, but its range is limited.

2. Incorrect Range Assumptions:

   • $\arccos x$ returns values in $[0,\pi ]$, not $[-\pi ,0]$.

3. Misapplying Identities:

   • $\arcsin (\sin \theta )\ne \theta$ if $\theta$ is outside $[-\pi /2,\pi /2]$.

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8. NEB-Style Questions

Short Answer Questions

1. Define $\arcsin x$. What is its domain and range?
2. If ${\tan }^{-1}(x)=\frac{\pi }{4}$, find the value of $x$.
3. Prove that $\arcsin x+\arccos x=\frac{\pi }{2}$.

Long Answer Questions

4. Sketch the graph of $y=\arcsin x$ and $y=\arccos x$ on the same axes. Label key points.
5. Solve the equation ${\sin }^{-1}(x)+{\cos }^{-1}(x)=\frac{\pi }{2}$.
6. A right triangle has sides 3, 4, and 5. Find the angle opposite the side of length 4 using inverse trigonometric functions.

Problem-Solving Questions

7. If $\theta =\arctan \left(\frac{3}{4}\right)$, find the value of $\sin \theta$ and $\cos \theta$.
8. Prove that ${\tan }^{-1}\left(\frac{1}{2}\right)+{\tan }^{-1}\left(\frac{1}{3}\right)=\frac{\pi }{4}$.

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Exam Tip

1. Memorize Ranges:

   • $\arcsin x$: $[-\pi /2,\pi /2]$
   • $\arccos x$: $[0,\pi ]$
   • $\arctan x$: $(-\pi /2,\pi /2)$

2. Practice Graphs:

   • Sketching graphs of inverse functions is a common question. Focus on key points like $(0,0)$, $(1,\pi /2)$, and $(-1,-\pi /2)$.

3. Domain and Range Questions:

   • Always check the domain of the input before applying inverse functions. For example, $\arcsin (2)$ is undefined because 2 is outside $[-1,1]$.

4. Use Identities Wisely:

   • Remember that $\arcsin x+\arccos x=\pi /2$ can simplify many problems.

5. Real-World Applications:

   • Questions may involve triangles or angles in physics/engineering. Always relate the problem to a right triangle if possible.

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Final Note: Inverse circular functions are essential for solving trigonometric equations and modeling real-world scenarios. Practice evaluating them, sketching their graphs, and applying identities to master this topic for your NEB exams!