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Formula Sheets / Maths

Chapter 12

Solutions of Triangles

Revise with flashcards Download this chapter (PDF) All Maths formulas (PDF)

Sine, cosine and projection rules Half angles and area Circles of a triangle m–n rule Heights and distances

Sine, cosine and projection rules

Sine rule
$\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C} = 2 R$
Cosine rule
$cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$
Projection
$a = b cos C + c cos B$
Napier's analogy
$tan \frac{B - C}{2} = \frac{b - c}{b + c} cot \frac{A}{2}$

Half angles and area

Half angles
$sin \frac{A}{2} = \frac{( s - b ) ( s - c )}{b c},$ $cos \frac{A}{2} = \frac{s ( s - a )}{b c},$ $tan \frac{A}{2} = \frac{( s - b ) ( s - c )}{s ( s - a )}$
$s = \frac{a + b + c}{2}$ .
Area
$Δ = \frac{1}{2} b c sin A = s (s - a) (s - b) (s - c) = \frac{ab c}{4 R} = r s$

Circles of a triangle

Circumradius
$R = \frac{a}{2 sin A} = \frac{ab c}{4Δ}$
Inradius
$r = \frac{Δ}{s} = (s - a) tan \frac{A}{2} = 4 R sin \frac{A}{2} sin \frac{B}{2} sin \frac{C}{2}$
Exradii
$r_{1} = \frac{Δ}{s - a} = s tan \frac{A}{2},$ $\frac{1}{r _{1}} + \frac{1}{r _{2}} + \frac{1}{r _{3}} = \frac{1}{r}$
Distances
$O I^{2} = R^{2} - 2 R r,$ distance of orthocentre from $A = 2 R cos A$
Median
$m_{a} = \frac{1}{2} 2 b^{2} + 2 c^{2} - a^{2}$

m–n rule

$(m + n) cot θ = m cot α - n cot β = n cot B - m cot C$
$D$ divides $B C$ in the ratio $) = \frac{height}{horizontal distance}$ ; $θ$ is the angle $A D C$ , $α = ∠ B A D$ , $β = ∠ D A C$ .

Heights and distances

$tan ($ angle of elevation $) = \frac{height}{horizontal distance}$
Draw the figure, mark right angles, and use the sine rule in non-right triangles.

Inverse Trigonometric Functions Sets and Relations