Trigonometry Equations

Trigonometric Equations: Solving without a Calculator

Objective

  • Solve basic trigonometric equations using general rules for trigonometric identities.

Trigonometric Equations

  • Equations involving special angles can be solved without a calculator.
  • Positive Ratios:
    • If the equation has a positive ratio (sin, cos, or tan), use the first quadrant to find the reference angle (RA) for the general solution.
  • Negative Ratios:
    • If the equation has a negative sine or tangent ratio, use the fourth quadrant to find the reference angle (RA) for the general solution.
    • If the equation has a negative cosine ratio, use the second quadrant to find the reference angle (RA) for the general solution.

Example 1: Finding the General Solution

  • Equation: tanx=−3tanx = -\sqrt{3}
  • Analysis:
    • The tangent ratio is negative.
    • Look for a solution in the second quadrant.
    • tanx=yx=−3=31tanx = \frac{y}{x} = -\sqrt{3} = \frac{\sqrt{3}}{1}
    • y=3y = \sqrt{3}
    • x=1x = 1
  • Reference Angle (RA):
    • Refer to the special angles diagram to find where, in quadrant 1, tanx=31tanx = \frac{\sqrt{3}}{1}.
    • RA=60∘RA = 60^{\circ}
  • General Solution:
    • In quadrant 2, where tan is negative:
    • x=180∘−60∘+k⋅180∘,k∈Zx = 180^{\circ} - 60^{\circ} + k \cdot 180^{\circ}, k \in Z
    • x=120∘+k⋅180∘,k∈Zx = 120^{\circ} + k \cdot 180^{\circ}, k \in Z

Example 2: Finding the General Solution

  • Equation: cosx=22cosx = \frac{\sqrt{2}}{2}
  • Analysis:
    • The cosine ratio is positive.
    • Look for solutions in the first and fourth quadrants.
    • cosx=xr=22cosx = \frac{x}{r} = \frac{\sqrt{2}}{2}
    • x=2x = \sqrt{2}
    • r=2r = 2
  • Reference Angle (RA):
    • Refer to the special angles diagram to find where, in quadrant 1, cosx=22cosx = \frac{\sqrt{2}}{2}.
    • RA=45∘RA = 45^{\circ}
  • General Solution:
    • Quadrant 1:
      • x=45∘+k⋅360∘,k∈Zx = 45^{\circ} + k \cdot 360^{\circ}, k \in Z
    • Quadrant 4:
      • x=360∘−45∘+k⋅360∘,k∈Zx = 360^{\circ} - 45^{\circ} + k \cdot 360^{\circ}, k \in Z
      • x=315∘+k⋅360∘,k∈Zx = 315^{\circ} + k \cdot 360^{\circ}, k \in Z
    • Combined General Solution:
      • x=45∘+k⋅360∘,k∈Zx = 45^{\circ} + k \cdot 360^{\circ}, k \in Z
      • x=315∘+k⋅360∘,k∈Zx = 315^{\circ} + k \cdot 360^{\circ}, k \in Z