How many solutions does a trig equation have?
How many solutions does a trig equation have?
Therefore a trig equation has an infinite number of solutions if it has any. Think about an equation like sin u = 1. π/2 is a solution, but the sine function repeats its values every 2π. Therefore π/2±2π, π/2±4π, and so on are equally good solutions.
What is the general solution of a trigonometric equation?
Trigonometric Equations and its Solutions
| Trigonometrical equations | General Solutions |
|---|---|
| sin θ = sin α | θ = nπ + (-1)n α, where α ∈ [-π/2, π/2] |
| cos θ = cos α | θ = 2nπ ± α, where α ∈ (0, π] |
| tan θ = tan α | θ = nπ + α, where α ∈ (-π/2, π/2] |
| sin 2θ = sin 2α | θ = nπ ± α |
Why do many trigonometric equations have infinitely many solutions?
Because of the periodic nature of the trigonometric functions – they repeat themselves infinitely many times – the variable x can take on an infinite number of solutions.
Will there always be solutions to trigonometric equations?
Example 5: Solving a Trigonometric Equation Involving Cosecant. Solve the following equation exactly: csc θ = − 2 , 0 ≤ θ < 4 π \displaystyle \csc \theta =-2,0\le \theta <4\pi cscθ=−2,0≤θ<4π.
Do all trigonometric equations have an infinite number of solutions?
Not all trigonometric equations have an infinite number of solutions. For example, cosx=73 has no solutions. cosx=kx has a finite number of solutions (except for k=0) whose number depends on k. For example, cosx=110x has 7 solutions.
How do you solve multiple trigonometric equations?
How to Find a Solution to a Multiple-Angle Trig Equation
- Divide each side by 2; then take the square root of each side.
- Solve for 5x, which represents the angles that satisfy the equation within one rotation.
- Extend the solutions to five rotations by adding 2π to each of the original angles four times.