Solving trigonometric equations using the t-formulae

Solving trigonometric equations using the t-formulae

Basics of Trigonometric Equations Using the t-Formulae

  • The t-formulae are used to simplify trigonometric equations.
  • The t-formulae let us represent sin θ and cos θ in terms of a variable t, which converts trigonometric equations into algebraic equations.
  • The formulae are as follows: sin θ = 2t / (1 + t^2), and cos θ = (1 - t^2) / (1 + t^2), where t = tan(θ / 2).
  • These equations can be derived from the double angle formulae.
  • The variable t in the t-formulae represents tan(θ / 2) and so will range from -∞ to ∞.
  • Remember that θ ≠ nπ ± π / 2 since tan(θ / 2) is undefined at these points.

Solving Trigonometric Equations Using t-Formulae

  • Begin using t-formulae to solve trigonometric equations by making θ the subject of the equations.
  • Then substitute the expressions for sin θ and cos θ.
  • You will now have an algebraic equation that is easier to solve in terms of t.
  • Remember to put your final solutions back in terms of θ to get your solutions for the original problem.
  • However, be careful with the signs of your solutions, making sure to account for plus or minus in your final answers.
  • As an added step, you can always verify your solutions by plugging them back into the original equation.

Key Considerations

  • Be careful with quadrants. Always remember to consider the quadrant of θ in calculating your final solutions.
  • Remember the range of your solutions. See if the question asks for solutions in a specific range and adjust accordingly.

Common Pitfalls

  • One common pitfall when solving trigonometric equations using t-formulae is forgetting to convert the solutions back into terms of θ.
  • Ensure that your algebra is correct and you have properly simplified and solved the equations.
  • Make sure not to forget about multiple solutions to the equations.

Remember, successfully applying the t-formulae requires a good understanding of basic trigonometry and algebra. Be sure to practice not just the theory, but also solving a variety of trigonometric equations using these techniques to be fully prepared.