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Symbol Defintion
Example : Sqr(x) = square root of x
Q01. Diagram : Proof of sine law
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Q02. Coine Law
Cosine law.
- Defintion : Trianle ABC
- a^2 = b^2 + c^2 - 2*b*c*cos(A).
- b^2 = c^2 + a^2 - 2*c*a*cos(B).
- c^2 = a^2 + b^2 - 2*b*c*cos(C).
- Application : Solve triangle if SAS are given
- Solve triangle if b, A, c are given.
- Solve triangle if c, B, a are given.
- Solve triangle if a, C, b are given.
- Construction
- Draw triangle ABC.
- Draw AH perpendicular to AB and let AH = h.
- Let AH = x and BH = c - x
- Proof of a^2 = b^2 + b^2 - 2*b*b*cos(A).
- Triangle BCH
- a^2 = h^2 + (c - x)^2.
- a^2 = h^2 + c^2 + x^2 - 2*c*x.
- Triangle AHC : x = b*cos(A) and h = b*sin(A)
- a^2 = (b^2)*sin(A)^2 + b^2)*cos(A)^2 + c^2 - 2*b*c*cos(A).
- Since sin(A)^2 + cos(A)^2 = 1
- Hence a^2 = b^2 + c^2 - 2*b*c*cos(A)
- Triangle BCH
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Use cosine law to prove
- sin(A/2) = Sqr((s-b)*(s-c)/(b*c))
- cos(A/2) = Sqr(s*(s-a)/(b*c))
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Q03. Area triangle = b*c*Sin(A)/2
Proof
- Proof sin(A) = b*c*sin(A)/2
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Q04. Sin(A) = 2*Sqr(s*(s-a)*(s-b)*(s-c))/(b*c)
Proof
- Proof sin(A) = 2*Sqr(s*(s-a)(s-b)*(s-c))/(b*c).
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Q05. Area of triangle = Sqr(s(s-a)*(s-b)*(s-c)/(b*c))
Proof
- Proof Area of triangle = Sqr(s*(s-a)(s-b)*(s-c))
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Q06. Solve triangle if SAS are given
Example : if a = 2 and b = 3 and angle C = 60 degrees, find other side c.
- c^2 = 2^2 + 3^2 - 2*2*3*cos(60).
- c^2 = 4 + 9 - 12*(1/2).
- c^2 = 7.0
- Hence c = 2.64575.
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Q07. Application : Direction of sum of two vectors
Application : Sum of two vectors
- From A walks 2 km to east at B. Then walks 1 km to N.E. at C.
- Find the distance and direction between A and C.
- Solution
- let AC = b, AB = c = 2 km, BC = a = 1 km.
- Angle ABC = B = 180 - 45 degrees = 135 degrees.
- Hence b^2 = a^2 + c^2 - 2*a*c*cos(135)
- Hence b^2 = 1^2 + 2^2 + 4*cos(45) = 5 + 2*Sqr(2).
- Hence b = Sqr(5 + 2*Sqr(2)) = Sqr(7.828427) = 2.79793 km.
- Find direction by sine law in triangle ABC.
- a/sin(A) = b/sin(B)
- Hence sin(A) = (a*sin(B))/b = sin(135)/2.79793 = 0.50545.
- Hence A = arcsin(0.50545)30.3612 degrees.
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