Analytical Trigonometry: Reduction Formulae (180° ± θ, 90° ± θ) and General Solutions
Reduction formulae for related angles allow mathematicians to express the trigonometric value of any obtuse or reflex angle in terms of an acute angle θ in Quadrant I. Angles based on horizontal axes (180° ± θ, 360° - θ, -θ ) preserve the function name while adopting the quadrant's algebraic sign. Angles based on vertical axes (90° ± θ, 270° ± θ ) convert co-functions (sin ↔ cos, tan ↔ cot, sec ↔ csc) with the sign evaluated prior to transformation. We apply these principles to find the general solutions of equations such as sin α = cos β and tan α = cot β.
1. Reduction Rules for Related Angles
Horizontal vs Vertical Axes:
- Horizontal (180° ± θ, 360° - θ): Keeps function name:
• sin(180° - θ) = + sin θ | cos(180° - θ) = - cos θ
• sin(180° + θ) = - sin θ | tan(180° + θ) = + tan θ
• cos(-θ) = + cos θ (even function). - Vertical (90° ± θ, 270° ± θ): Changes co-function (sin ↔ cos):
• sin(90° - θ) = + cos θ | cos(90° - θ) = + sin θ
• sin(90° + θ) = + cos θ | cos(90° + θ) = - sin θ.
2. General Solutions of Equations
Periodic solution sets:
- sin α = cos β: α ± β = 90° + 360°n (or π/2 + 2πn).
- tan α = cot β: α + β = 90° + 180°n (or π/2 + πn).
- sec α = csc β: α ± β = 90° + 360°n.
Worked Examples:
Solution:
sin 150° = sin(180° - 30°) = 1/2.
cos 240° = cos(180° + 60°) = -1/2.
tan 315° = tan(360° - 45°) = -1.
Sum = (1/2) + (-1/2) - (-1) = 0 + 1 = 1.
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