Enter equations of the form sin(ax+b)=c, cos(ax+b)=c, or tan(ax+b)=c — general solutions & principal values
sin equation
sin(x + ) =
cos equation
cos(x + ) =
tan equation
tan(x + ) =
Results
Step-by-Step Derivation
General Solution Formulas
sin x = a : x = (-1)ⁿ·arcsin(a) + nπ (n∈ℤ) | or x = arcsin(a) + 2nπ , x = π−arcsin(a) + 2nπ
cos x = a : x = ±arccos(a) + 2nπ (n∈ℤ)
tan x = a : x = arctan(a) + nπ (n∈ℤ) , a∈ℝ
Trigonometric equations have infinitely many solutions due to periodicity. Principal values are typically given within [0, 2π) for sin/cos or [0, π) for tan.
⚠Note: sin/cos equations require the right-hand side (c) to be within [-1,1]; otherwise no solution. tan equations require coefficient a ≠ 0.
What Are Trigonometric Equations?
Trigonometric equations involve unknown variables inside trigonometric functions. Unlike algebraic equations, they typically have infinitely many solutions because trig functions are periodic. The solution set is expressed using a general solution with integer parameter n.
Periodicity
Trigonometric functions are periodic: sin/cos have period 2π, tan has period π. Therefore, if x₀ is a solution, then x₀ plus any integer multiple of the period is also a solution.
Existence Condition
For sin x = a and cos x = a, a solution exists iff |a| ≤ 1. For tan x = a, a solution exists for all real a (but x ≠ π/2 + kπ).
General Solution Form
General solutions use integer parameter n∈ℤ. Example: sin x = 0.5 → x = π/6 + 2nπ or x = 5π/6 + 2nπ.
Substitution Method
For equations of the form sin(ax+b)=c, let u = ax+b, solve for u, then back-substitute: x = (u - b)/a.
💡 Example: Solve sin x = 0.5. Since sin(π/6) = 0.5 and sin(5π/6) = sin(π−π/6) = 0.5, within [0,2π): x₁ = π/6, x₂ = 5π/6. General solution: x = π/6 + 2kπ or x = 5π/6 + 2kπ (k∈ℤ).
How are general solutions for trig equations expressed?▼
For sin x = a: x = (-1)ⁿ·arcsin(a) + nπ (n∈ℤ). For cos x = a: x = ±arccos(a) + 2nπ (n∈ℤ). For tan x = a: x = arctan(a) + nπ (n∈ℤ).
How many solutions does sin x = 0.5 have?▼
sin x = 0.5 has infinitely many solutions: x = π/6 + 2kπ or x = 5π/6 + 2kπ (k∈ℤ). Within [0,2π), two principal solutions exist: π/6 and 5π/6.
What is the difference between algebraic and trigonometric equations?▼
Algebraic equations have finitely many solutions. Trigonometric equations have infinitely many solutions due to periodicity. Solving them requires considering periodic behavior and range restrictions.
How do I find solutions within a specific interval?▼
First find the general solution formula, then substitute integer values n to obtain solutions inside the desired interval. For example, sin x = 0.5 on [0,2π) gives x₁ = π/6, x₂ = 5π/6.
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