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TRIGO IDENTITIES - Coggle Diagram
TRIGO IDENTITIES
Trigonometric Identities
Mother Identity
From unit circle: x² + y² = 1
cos²θ + sin²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Addition and Subtraction Formulas
cos(x+y) = cosx cosy – sinx siny
cos(x–y) = cosx cosy + sinx siny
sin(x+y) = sinx cosy + cosx siny
sin(x–y) = sinx cosy – cosx siny
tan(x+y) = (tanx + tany) / (1 – tanx tany)
tan(x–y) = (tanx – tany) / (1 + tanx tany)
cot(x+y) = (cotx coty – 1) / (cotx + coty)
cot(x–y) = (cotx coty + 1) / (coty – cotx)
Double Angle Formulas
sin 2x = 2 sinx cosx
sin 2x = (2 tanx) / (1 + tan²x)
cos 2x = cos²x – sin²x
cos 2x = 2 cos²x – 1
cos 2x = 1 – 2 sin²x
cos 2x = (1 – tan²x) / (1 + tan²x)
tan 2x = (2 tanx) / (1 – tan²x)
Triple Angle Formulas
sin 3x = 3 sinx – 4 sin³x
cos 3x = 4 cos³x – 3 cosx
tan 3x = (3 tanx – tan³x) / (1 – 3 tan²x)
Half Angle Formulas
sin²(x/2) = (1 – cosx) / 2
cos²(x/2) = (1 + cosx) / 2
tan(x/2) = sinx / (1 + cosx)
tan(x/2) = (1 – cosx) / sinx
Product to Sum Formulas
sinx cosy = ½[sin(x+y) + sin(x–y)]
cosx siny = ½[sin(x+y) – sin(x–y)]
cosx cosy = ½[cos(x+y) + cos(x–y)]
sinx siny = ½[cos(x–y) – cos(x+y)]
Sum to Product Formulas
sinx + siny = 2 sin((x+y)/2) cos((x–y)/2)
sinx – siny = 2 cos((x+y)/2) sin((x–y)/2)
cosx + cosy = 2 cos((x+y)/2) cos((x–y)/2)
cosx – cosy = –2 sin((x+y)/2) sin((x–y)/2)
cos(x+y) cos(x–y) = cos²x – sin²y
sin(x+y) sin(x–y) = sin²x – sin²y
Symmetry Rules (Allied Angles)
θ → –θ : sin → –sin, cos → cos
90° – θ : sin → cos, cos → sin
90° + θ : sin → cos, cos → –sin
180° – θ : sin → sin, cos → –cos
180° + θ : sin → –sin, cos → –cos
270° – θ : sin → –cos, cos → –sin
270° + θ : sin → –cos, cos → sin
360° – θ : sin → –sin, cos → cos