Triangle Trigonometry
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TRIANGLE TRIGONOMETRY
In any triangle the three sides a, b, c and the three angles A, B, C are called the elements of the triangle. In a triangle ABC, the side BC, opposite to angle A, is denoted by ‘a’. The side CA, opposite to angle B, is denoted by b and the side AB by c.
THE SINE RULE
In any triangle ABC,
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a/sin A = b/sin B = c/sin C = 2R |
where R is the circumradius of the triangle.
THE COSINE RULE
In any triangle ABC,
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a2 = b2 + c2 - 2bc cos A |
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b2 = a2 + c2 -2ac cos B |
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c2 = a2 + b2 -2ab cos C |
To prove the last one :
If c is acute (Refer figure)
CD = AC cos C = b cos C
AD = AC sin C = b sin C
BC = BC - CD = a - b cos C
By Pythagoras theorem,
AB2 = BD2 + AD2
è c2 = (a - b cos C)2 + (b sin C)2
= a2 - 2ab cos C + b2 cos2 C + b2 sin2 C
= a2 + b2 - 2ab cos C
If c is obtuse
CD = AC cos (180 - C)
= - b cos C
AD = AC sin (180 - C) = b sin C
BD = BC + CD = a - b cos C
applying Pythagoras theorem,
AB2 = BD2 + AD2
è c2 = ( a - b cos c)2 + (b sin c)2
= a2- 2ab cosC + b2cos2C + b2sin2C
= a2 + b2 - 2ab cosC
So in an acute or an obtuse angled triangle, the cosine rule always holds.
Moreover, from the first figure we have
BD = AB cos B = C cos B
DC = AC cos C = b cos C
BC = BD + DC
è a = c cos B + b cos C
In the second figure when c is obtuse, we will have BD = c cos B but
DC = AC cos (180‚ - C) = - b cos C.
BC = BD - DC = c cos B - (- b cos C)
= c cos B + b cos C
So in general,
a = b cos C + c cos B
similarly, b = a cos C + c cos A
c = a cos B + b cos A