Trigonometric Equations

• The difference between identities (always true) and equations (true for specific angles).
• To solve basic equations sin x = k, cos x = k, tan x = k using the unit circle and general solutions.
• General solution formulas for sin x = sin α, cos x = cos α, tan x = tan α — essential for JEE.
• To solve equations reducible to quadratic form in sin x, cos x, or tan x.

Level 1 — Principal and general solutions

Verbal: Because trig functions are periodic, equations have infinitely many solutions. The general solution gives all angles in terms of integer n.

Symbolic:

• sin x = 0 → x = nπ, n ∈ ℤ
• cos x = 0 → x = (2n+1)π/2
• tan x = 0 → x = nπ
• sin x = 1 → x = (4n+1)π/2 ... wait, sin x = 1 → x = π/2 + 2nπ

For sin x = sin α: x = nπ + (−1)ⁿ α, n ∈ ℤ

For cos x = cos α: x = 2nπ ± α, n ∈ ℤ

For tan x = tan α: x = nπ + α, n ∈ ℤ (α ≠ π/2 + kπ)

Domain restriction: |sin x| ≤ 1 and |cos x| ≤ 1. Equations sin x = 2 have no real solution.

Level 2 — Quadratic and factorisation methods

Type: a sin²x + b sin x + c = 0. Substitute t = sin x (|t| ≤ 1), solve quadratic, back-substitute.

Type: sin x + cos x = 1. Square both sides cautiously (may introduce extraneous roots) OR divide by √2 to use compound angles.

Type: 2 sin x cos x = sin x → sin x (2 cos x − 1) = 0 → sin x = 0 or cos x = 1/2.

NCERT range: Often solutions required in [0, 2π) unless general solution asked.

NCERT spotlight — General solutions and extraneous roots

Principal values of inverse trig functions restrict ranges: sin inverse in [-pi/2, pi/2], cos inverse in [0, pi]. When solving sin 2x = sin x, factor to sin x (2 cos x - 1) = 0 rather than dividing by sin x and losing roots.

Squaring hazard: Squaring sin x + cos x = 1 introduces extraneous solutions such as x = pi. Always substitute back into the original equation.

Graphical counting: The number of solutions of sin x = x/10 equals intersection points of y = sin x and y = x/10 — a common JEE graphical question.

Solve 2 sin²x − 3 sin x + 1 = 0 for x ∈ [0, 2π).

Step 1 — Let t = sin x. Equation: 2t² − 3t + 1 = 0.
Step 2 — Factor: (2t − 1)(t − 1) = 0 → t = 1/2 or t = 1.
Step 3 — sin x = 1/2 → x = π/6 or 5π/6 in [0, 2π).
Step 4 — sin x = 1 → x = π/2.
Step 5 — Solutions: x = π/6, 5π/6, π/2.

Applications — physics oscillations

Simple harmonic motion x = A sin(omega t + phi) requires solving for time when x = A/2. sin(omega t + phi) = 1/2 gives omega t + phi = pi/6 + 2n pi or 5pi/6 + 2n pi. General solution captures all repeat times — essential for phase problems in waves chapter.

Review and practice drill

Review checklist: (1) General solution templates for sin, cos, tan. (2) Factor do not divide — may lose roots. (3) Verify after squaring. (4) Principal value ranges for inverse trig. Practice: Solve 2 cos x = sqrt(3) in [0, 2 pi). cos x = sqrt(3)/2 gives x = pi/6 and 11 pi/6.

• Solve cos x = −√3/2 for x ∈ [0, 2π).
• Write the general solution of tan x = −1.
• Solve sin x + cos x = 0 in [0, 2π).

Open the Practice tab for graded questions on Equations.

• Use the platform-native live simulation or PhET-style tool for this topic (number line, Venn, physics playground, molecule builder, sensor dashboard, etc.).
• Mirror / body / home activity: physically do the concept (count objects, measure, role-play) and photograph or describe for portfolio.
• Voice or text reflection with AI Mentor: explain the concept to a younger student or family member.

• "Explain this concept to a Class 6 student using one real example from an Indian home, school, market, or festival."
• "What is one common mistake students make here, and how would you catch yourself making it?"
• Stretch: "How does this connect to coding, robotics, money, health, environment, or a future career?"

• 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.

• One hands-on project or measurement using the Drishti kit or household items that makes the concept physical.
• Direct link to at least one Future Skill track (Money Management, Green Tech, Cyber Defenders, Micro-Entrepreneurship, AI Mastery, Sustainable Living, Personality Development).
• Coding extension where relevant (simple script, simulation, or data logging).

This material emphasises experiential "learning by doing", competency (apply/create/analyse), vocational exposure, critical thinking, and multidisciplinary connections. Designed to feed live worlds, AI Mentor (with memory), gamification, robotics, parent analytics, and future skills — not just exam prep.

Portfolio Evidence Idea: Your photo/table/reflection/project + one sentence on "How this helps me in real life or a possible future path."

Open the Practice tab for aligned questions (easy/medium/hard + case-based) with full AI scaffolding.

See curriculum for cross-links and the full future-skills/robotics chapters.