General solutions of trigonometric equations: Class 11 practice
sin x = sin α ⇒ x = nπ + (−1)ⁿα; cos x = cos α ⇒ x = 2nπ ± α; tan x = tan α ⇒ x = nπ + α (n ∈ ℤ).. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Standard values
The idea behind general solutions of trigonometric equations
sin x = sin α ⇒ x = nπ + (−1)ⁿα; cos x = cos α ⇒ x = 2nπ ± α; tan x = tan α ⇒ x = nπ + α (n ∈ ℤ).
Step-by-step method
- Rewrite the right side as sin α, cos α or tan α.
- Apply the matching general-solution formula.
- List principal solutions in [0, 2π) if asked.
Worked example 1 (Basic)
Find the general solution of sin x = 1/2.
- x = nπ + (−1)ⁿ π/6
- Answer: x = nπ + (−1)ⁿ π/6
Worked example 2 (Standard)
Find the general solution of tan x = √3.
- x = nπ + π/3
- Answer: x = nπ + π/3
Worked example 3 (Challenge)
Find the solutions of cos x = −1/2 in [0, 2π).
- Principal solutions: 2π/3, 4π/3
- Answer: 2π/3, 4π/3
Common mistakes and misconceptions
- Using the sin formula for cos
- Missing the second principal solution
Practice questions
- (Basic) Find the general solution of tan x = 1.
- (Basic) Find the general solution of sin x = −1/2.
- (Standard) Find the general solution of sin x = √3/2.
- (Standard) Find the general solution of tan x = −1.
- (Challenge) Find the solutions of cos x = 0 in [0, 2π).
- (Challenge) Find the solutions of sin x = √3/2 in [0, 2π).
Answer key with full working
- Q1: x = nπ + π/4 → Answer: x = nπ + π/4
- Q2: x = nπ + (−1)ⁿ(−π/6) → Answer: x = nπ + (−1)ⁿ(−π/6)
- Q3: x = nπ + (−1)ⁿ π/3 → Answer: x = nπ + (−1)ⁿ π/3
- Q4: x = nπ − π/4 → Answer: x = nπ − π/4
- Q5: Principal solutions: π/2, 3π/2 → Answer: π/2, 3π/2
- Q6: Principal solutions: π/3, 2π/3 → Answer: π/3, 2π/3
Syllabus note
This skill is listed under "Trigonometric Functions" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is general solutions of trigonometric equations in?
"Trigonometric Functions", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.