Trigonometry & Inverse Trig | IISER Smart Prep

1. Core Concept

Trigonometry extends the study of right-angled triangles to the Unit Circle, allowing the definition of trigonometric functions for any angle. Inverse Trigonometric Functions (ITF) reverse this process, but their domains must be strictly restricted so they remain mathematically valid functions (one-to-one).

Graphs & Periodicity

sin x: Period = 2π, Range = [-1, 1]
cos x: Period = 2π, Range = [-1, 1]
tan x: Period = π, Range = (-∞, ∞)

Key Idea: sin and cos are bounded waves, tan has vertical asymptotes.

2. Key Formulas

Angle Measurement: π radians = 180°. Arc length l = rθ (θ in radians).

Fundamental Identities:

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

Fundamental identities (must remember)

Compound & Multiple Angles:

sin(A±B) = sinAcosB ± cosAsinB
cos(A±B) = cosAcosB ∓ sinAsinB

sin(2θ) = 2sinθcosθ = 2tanθ / (1+tan²θ)

cos(2θ) = cos²θ - sin²θ = (1 - tan²θ) / (1 + tan²θ)

General Solutions:

sin θ = sin α &implies; θ = nπ + (-1)ⁿα
cos θ = cos α &implies; θ = 2nπ ± α
tan θ = tan α &implies; θ = nπ + α

General solution represents all possible angles repeating periodically on the unit circle.

Inverse Trigonometry:

sin^-1x + cos^-1x = π/2

(for x ∈ [-1, 1])

tan^-1x + tan^-1y = tan^-1[(x+y)/(1-xy)]

(Strictly if xy < 1)

3. Principal Value Branches (ITF)

sin^-1x: [-π/2, π/2]
cos^-1x: [0, π]
tan^-1x: (-π/2, π/2)

Domain & Range of ITF

sin¹x: Domain [-1, 1], Range [-π/2, π/2]
cos¹x: Domain [-1, 1], Range [0, π]
tan¹x: Domain ℝ, Range (-π/2, π/2)

4. Conceptual Insights

Unit Circle (ASTC):

All positive (Q1), Sin positive (Q2), Tan positive (Q3), Cos positive (Q4).

Triangle Method for ITF:

To convert sin^-1(3/5) to tan^-1(), draw a right triangle where opposite=3 and hypotenuse=5. Thus, adjacent=4, so tan^-1(3/4).

5. Common Mistakes

The ITF Domain Trap: sin^-1(sin 2π/3) ≠ 2π/3. You MUST reduce it to the principal branch: sin^-1(sin(π - π/3)) = π/3.
Canceling Variables: Dividing sinθ = sin2θ by sinθ loses the roots where sinθ = 0. Always factor equations!
Squaring Equations: Squaring sides often introduces extraneous solutions. Always verify final answers in the original equation.

6. IAT Exam Focus Points

Key Exam Focus:

Intersection Points: "How many solutions for f(x) = g(x)?" Graph both sides & count intersections.
Special ITFs: Memorize values for 15° (tan 15° = 2 - √3) and 18° (sin 18° = (√5 - 1)/4).
Trig Inequalities: Solve sin x > 1/2 by identifying intervals on the sine wave graph.
Substitution: Use x = a sinθ for √(a² - x²) and x = a tanθ for √(a² + x²) to simplify complex ITFs.

7. Practice Mock Test

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Trigonometry & Inverse Trigonometry