Systems of Trigonometric Equations

How to solve graphically

When two trigonometric graphs such as sine and cosine intersect, we call that point of intersection a solution of the system of equations. This is the same meaning that solutions of systems of linear equations has.

Solutions of Systems of Sine and Cosine graphs

Problem 1

Pictured below is a system made up of the equation y = sin(x) and y = cos(x) over the interval 0≤X≤2Π

Picture of graph of system of sine and cosine equations

As you can see from the graph, this system of sine and cosine graphs intersect twice during this interval. They intercept at x = Π/4 as well as at x = 5Π/4

Problem 2

Can you figure out how many, if any, solutions there are for the following system of trigonometric equations has any solutions over the interval 0≤X≤2Π

The system of equations:

  • y = sin(x) − 1
  • y = cos(x) + 1
Problem 3

Can you figure out how many, if any, solutions there are for the following system of trigonometric equations has any solutions over the interval 0 ≤ X ≤ Π

The system of equations:

  • y = sin(2x)
  • y = cos(x)

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