This calculator helps you solve trigonometric equations of the form A * func(B*x + C) = D for x within a specified range.
sin
cos
tan
Understanding Trigonometric Equations
Trigonometric equations are equations that involve trigonometric functions of an unknown angle. Solving them means finding the values of the angle that satisfy the equation. These equations often have multiple solutions due to the periodic nature of trigonometric functions.
The General Form: A * func(B*x + C) = D
This calculator focuses on solving equations in the form A * func(B*x + C) = D, where:
A is a coefficient that scales the amplitude of the function.
func is a trigonometric function (sine, cosine, or tangent).
B is a coefficient that affects the period of the function.
x is the unknown variable we are solving for.
C is a phase shift, moving the graph horizontally.
D is a constant value.
To solve such an equation, the first step is usually to isolate the trigonometric function:
func(B*x + C) = D / A
Steps to Solve:
Isolate the Trig Function: Divide both sides by A to get func(B*x + C) = D/A. Let Y = D/A.
Find the Principal Value: Use the inverse trigonometric function (arcsin, arccos, arctan) to find the principal value of the angle. For example, if sin(θ) = Y, then θ = arcsin(Y).
Determine General Solutions: Due to periodicity, there are infinitely many solutions.
For sin(θ) = Y: General solutions are θ = arcsin(Y) + 2nπ and θ = (π - arcsin(Y)) + 2nπ, where n is an integer.
For cos(θ) = Y: General solutions are θ = arccos(Y) + 2nπ and θ = -arccos(Y) + 2nπ, where n is an integer.
For tan(θ) = Y: General solutions are θ = arctan(Y) + nπ, where n is an integer.
Substitute and Solve for x: Replace θ with B*x + C and solve for x. For example, if B*x + C = arcsin(Y) + 2nπ, then x = (arcsin(Y) + 2nπ - C) / B.
Filter Solutions within a Range: Iterate through integer values of n to find all solutions for x that fall within the specified Range Start and Range End.
Important Considerations:
Domain and Range: For sine and cosine functions, the value of D/A must be between -1 and 1 (inclusive). If it falls outside this range, there are no real solutions.
Units: Angles are typically measured in radians when using inverse trigonometric functions in mathematics and programming. Ensure your range is also in radians.
Periodicity: The B coefficient affects the period. A larger B means a shorter period and potentially more solutions within a given range.
Examples:
Example 1: Simple Sine Equation
Solve 2 * sin(x) = 1 for x in the range [0, 2π].
A = 2, func = sin, B = 1, C = 0, D = 1.
Isolate: sin(x) = 1 / 2 = 0.5.
Principal value: x = arcsin(0.5) = π/6 (approx 0.5236 radians).
General solutions for x:
x = π/6 + 2nπ
x = (π - π/6) + 2nπ = 5π/6 + 2nπ
Within [0, 2π] (approx [0, 6.283]):
For n=0: x = π/6 and x = 5π/6.
Using the calculator: Set A=2, func=sin, B=1, C=0, D=1, Range Start=0, Range End=6.283185307.
Expected Solutions: 0.5236, 2.6180.
Example 2: Cosine Equation with Period Change
Solve cos(2x) = 0.5 for x in the range [0, 2π].
A = 1, func = cos, B = 2, C = 0, D = 0.5.
Isolate: cos(2x) = 0.5.
Principal value: 2x = arccos(0.5) = π/3 (approx 1.0472 radians).
General solutions for 2x:
2x = π/3 + 2nπ
2x = -π/3 + 2nπ
Solve for x:
x = π/6 + nπ
x = -π/6 + nπ
Within [0, 2π]:
For x = π/6 + nπ:
n=0: x = π/6 (approx 0.5236)
n=1: x = π/6 + π = 7π/6 (approx 3.6652)
For x = -π/6 + nπ:
n=1: x = -π/6 + π = 5π/6 (approx 2.6180)
n=2: x = -π/6 + 2π = 11π/6 (approx 5.7596)
Using the calculator: Set A=1, func=cos, B=2, C=0, D=0.5, Range Start=0, Range End=6.283185307.