Solving Trig Equations

Solving Trig Equations

Basic principles

  • Trigonometric equations are equations involving one or multiple trigonometric functions.
  • The solutions of a trigonometric equation are the values of the angles that make the equation hold true.

Steps to solve basic trig equations

  • Step 1: Isolate the trigonometric term. This means making the term with sin, cos or tan the subject of the equation.
  • Step 2: Determine the reference angle. This is the angle you find in the range 0 to 90 degrees that solves the equation. Typically, you calculate this by using a calculator to apply the inverse trig function.
  • Step 3: Determine all the solutions within 0 to 360 degrees or 0 to 2π. Use knowledge of the trig function’s behaviour - for example, knowing that sin(θ) = sin(180 - θ) could help you find a second solution.

Solving equations involving multiple trig terms

  • Equations including cos, sin, and tan within the same equation can often be reduced to simpler equations through the application of Pythagorean identity or double-angle identities.
  • If an equation includes multiple instances of the same trig function, you can often turn it into a quadratic equation - these are familiar from earlier work, and can be solved in familiar ways.
  • Be sure to check all potential solutions in the original equation. Occasionally, methods for solving multiple-term equations can produce extraneous solutions which do not solve the original equation.

Using CAST and the Unit Circle

  • CAST is a mnemonic that helps remember the signs of the trig functions in each quadrant – in the 1st quadrant (0-90 degrees), all are positive, in the 2nd quadrant (90-180 degrees) only Sin is positive, in the 3rd quadrant (180-270 degrees) only Tan is positive, and in the 4th quadrant (270-360 degrees) only Cos is positive.
  • Combining knowledge of CAST with the unit circle allows for the calculation of sine, cosine and tangent of angles greater than 90 degrees or less than 0 degrees.

Final Advice

  • Always use exact values where possible. These include well-known values of sin, cos, and tan at angles like 30, 45, 60, and 90 degrees.
  • Keep in mind the periodic nature of trig functions - i.e., the fact that their graphs repeat every 360 degrees or 2π radians. This means there are an infinite number of solutions to most trig equations.
  • To specify solutions within a certain range, you first find the solutions in range 0 to 360 degrees or 0 to 2π, and then add to these the solutions in each subsequent or previous period as needed.