General Solutions of Trigonometric Equations

Today in class we learned about General Solutions of Trigonometric Equations.
What we learned was that General Solutions of Trigonometric Equations are:

• Not bounded by an interval
• Have unlimited coterminal answers
  

We also learned that

• For each solution of sinθ or cosθ in degrees, add 360°k or 2kπ for radians
• For each solution of tanθ in degrees, add 180°k or kπ for radians

For each answer, we include:

• Where kЄI (where k is an element of integers)
  

k accounts for all positive and negative coterminal integers

EXAMPLE of this type of equation




First step: Create your special triangle on unit circle

• Since tan is positive in quadrants 1 and 3, that's where we create our triangles

Second step: Figure out the angles for each quadrant they are in, making sure to add 360°or 180° or 2kπ or kπ


Q1 -> Ref. angle = Rel. angle
Q2 -> 180° - Ref. angle = Rel. angle
Q3 -> 180° + Ref. angle = Rel. angle
Q4 -> 360° - Ref. angle = Rel. angle

therefore
(in degrees)
Q1 = 30°+180°k, where kЄI
Q2 = 180°+30° = 210°+180°k, where kЄI

(in radians)
Q1 = π/6 + 2kπ
Q2 = 7π/6 + 2kπ


θ = π/6+2kπ , 7π/6+2kπ
where kЄI