Inverse Trigonometry — Domains, Ranges and Principal Values

• Explain why inverse trig functions require restricted domains to be well-defined
• State the domain and principal value branch (range) of all six inverse trig functions
• Evaluate inverse trig at standard angles without a calculator
• Identify properties of graphs of sin⁻¹, cos⁻¹, and tan⁻¹ including symmetry and asymptotes
• Distinguish sin(sin⁻¹x) = x from sin⁻¹(sinx) = x and apply the correct reduction
• Solve problems that require reducing sin⁻¹(sinθ) for θ outside the principal branch

Level 1 — Foundations

Why restricted domains? The sine function repeats every $2\pi$; for instance $\sin(\pi/6) = \sin(5\pi/6) = 1/2$. To define a unique inverse, we restrict to the largest interval containing 0 on which sin is one-to-one: $[-\pi/2,\, \pi/2]$.

Principal value branches:

Standard values:

Level 2 — JEE Depth

Graph properties:

• $\sin^{-1}x$: odd function (symmetric about origin); increasing on $[-1,1]$; endpoints $(-1,-\pi/2)$ and $(1,\pi/2)$.
• $\cos^{-1}x$: neither even nor odd; decreasing on $[-1,1]$; endpoints $(-1,\pi)$ and $(1,0)$.
• $\tan^{-1}x$: odd function; strictly increasing on $\mathbb{R}$; horizontal asymptotes at $y = \pm\pi/2$.

Critical composition identities:

$\sin(\sin^{-1}x) = x \quad \text{for } x \in [-1,1]$ $\cos(\cos^{-1}x) = x \quad \text{for } x \in [-1,1]$ $\tan(\tan^{-1}x) = x \quad \text{for all } x \in \mathbb{R}$

But the reverse compositions are only equal to $x$ within the principal branch:

$\sin^{-1}(\sin x) = x \quad \text{only if } x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ $\cos^{-1}(\cos x) = x \quad \text{only if } x \in [0, \pi]$

Reduction for $\sin^{-1}(\sin\theta)$ outside principal branch:

Since $\sin^{-1}$ always outputs a value in $[-\pi/2, \pi/2]$, we use the identity $\sin(\pi - \theta) = \sin\theta$ to write:

• If $\theta \in (\pi/2, 3\pi/2)$: $\sin^{-1}(\sin\theta) = \pi - \theta$ (while result stays in range; check sign)
• If $\theta \in (3\pi/2, 5\pi/2)$: $\sin^{-1}(\sin\theta) = \theta - 2\pi$

More precisely, reduce $\theta$ to the equivalent angle in $[-\pi/2, \pi/2]$ using periodicity and symmetry.

Reduction for $\cos^{-1}(\cos\theta)$ outside $[0,\pi]$: Use $\cos(-\theta) = \cos\theta$ and $\cos(2\pi - \theta) = \cos\theta$ to map $\theta$ to $[0,\pi]$.

Example 1: Evaluate $\sin^{-1}\!\left(\sin\dfrac{7\pi}{6}\right)$.

Step 1 — Locate 7π/6 on the unit circle.
7π/6 is in the third quadrant (between π and 3π/2).

Step 2 — Check if 7π/6 ∈ [−π/2, π/2].
7π/6 ≈ 3.67 rad, which is outside [−π/2, π/2] ≈ [−1.57, 1.57]. So sin⁻¹(sin(7π/6)) ≠ 7π/6.

Step 3 — Find sin(7π/6).
7π/6 = π + π/6, so sin(7π/6) = −sin(π/6) = −1/2.

Step 4 — Apply sin⁻¹.
sin⁻¹(−1/2) = −π/6   (since −π/6 ∈ [−π/2, π/2] and sin(−π/6) = −1/2).

Answer: sin⁻¹(sin(7π/6)) = −π/6

Example 2: Find the principal value of $\cos^{-1}\!\left(\cos\dfrac{4\pi}{3}\right)$.

Step 1 — Locate 4π/3.
4π/3 ∈ (π, 3π/2), i.e., third quadrant. Not in [0, π].

Step 2 — Find cos(4π/3).
4π/3 = π + π/3, so cos(4π/3) = −cos(π/3) = −1/2.

Step 3 — Apply cos⁻¹.
cos⁻¹(−1/2) = 2π/3   (since cos(2π/3) = −1/2 and 2π/3 ∈ [0, π]).

Answer: cos⁻¹(cos(4π/3)) = 2π/3

Verification: 2π/3 is indeed in [0, π] ✓

• Q1: Evaluate $\cos^{-1}\!\left(\cos\dfrac{5\pi}{4}\right)$.
• Q2: State the range of $\cot^{-1}x$ and explain why it cannot include 0.
• Q3: Is $\sin^{-1}x$ an even or odd function? Justify using the property $\sin^{-1}(-x)$.
• Q4: For what values of $x$ does $\tan^{-1}(\tan x) = x$ hold exactly?
• Stretch: Q5: Sketch the graph of $y = \sin^{-1}(\sin x)$ for $x\in[-2\pi, 2\pi]$ by identifying the principal branch segments and their reflections; state the period of the resulting graph.

NCERT Chapter 2 link: Section 2.2 (Basic concepts); 2.3 (Properties of inverse trig functions — Table 2.2); worked examples in 2.3 for principal value evaluation.

Exam connections: JEE Main tests principal value evaluation and composition identities (1–2 Qs/year). JEE Advanced uses them as sub-steps in equations and inequality problems; errors in principal branch selection lead to lost marks.

Study strategy: Memorise the six principal-value branches as a single table and test yourself daily for one week. For every $f^{-1}(f(x))$ problem, always ask: "Is $x$ already in the principal branch?" before cancelling. If not, use symmetry (sin is odd, cos reflects about $\pi/2$) to reduce.

• Interactive graph plotter: slide $\theta$ from $-2\pi$ to $2\pi$ and watch $\sin^{-1}(\sin\theta)$ trace its zigzag graph; colour the segment that lies in the principal branch.
• Domain-range explorer: input any $x$ value and watch which of the six inverse trig functions accept it, with a visual indicator on the unit circle.
• AI mentor reflection: "Compare the graphs of $\sin^{-1}x$ and $\cos^{-1}x$. How would you transform one graph to get the other? What identity does that transformation correspond to?"

• "Why is it necessary to restrict the domain of sine to $[-\pi/2, \pi/2]$ specifically — could we have chosen $[\pi/2, 3\pi/2]$ instead, and what would change?"
• "Without using a calculator, arrange $\sin^{-1}(0.5)$, $\cos^{-1}(0.5)$, and $\tan^{-1}(0.5)$ in increasing order. Explain your reasoning."
• Stretch: "Prove that the graph of $y = \cos^{-1}x$ can be obtained from the graph of $y = \sin^{-1}x$ by a reflection. State the exact transformation and derive the identity it represents."

• Complete the core practice + one extension activity (photo, table, short reflection, or mini-project) for base XP + topic badge.
• 5-7 day streak or family discussion note = multiplier + visible artifact in parent/principal dashboard.
• Best real-world application stories (anonymised) featured on class or national leaderboard.

• Use a servo motor and a microcontroller (Arduino): command the servo to angle $\theta$, then write code that reads the angle back and applies $\sin^{-1}(\sin(\theta))$ to confirm the principal value — observe what happens for $\theta > 90°$.
• Future Skill track: AI Mastery — In machine learning, activation functions like sigmoid and tanh are bounded outputs; connect the idea of restricted range in inverse trig to why neural networks use bounded activations to avoid exploding values.
• Coding extension: Write a Python function principal_arcsin(theta) that takes any angle (in radians) and returns $\sin^{-1}(\sin(\theta))$ by first computing $\sin(\theta)$ and then applying math.asin, with test cases for $7\pi/6$, $-5\pi/4$, and $3\pi/2$.

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