Solution of Triangles

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Solution of Triangles

• COSINE RULE

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Cosine Rule
a2 b2 c2 2bc cos A b2 c2 a2 2ca cos
B c2 a2 b2 - 2ab cos C

• ?2 sides and one included angle given.
• e.g.
• b 10cm, c 7 cm and ?A 55
• or,
• a 14cm, b 10 cm and ?C 48

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Example

• ABC is a triangle with ?A 50º, AB 9cm dan
• AC 6 cm. Find the length BC

B
Using cosine rule, a2 b2 c2 2bc cos A
62 92 2(6)(9)cos 50º 47.58 a sqrt
(47.58) 6.896 cm
9 cm
50º
C
A
6cm
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Cosine Rule
a2 b2 c2 2bc cos A b2 c2 a2 2ca cos
B c2 a2 b2 - 2ab cos C

• ?3 sides given. (solve for angles)
• hence
• Cos A (b2 c2 a2 ) 2bc
• or
• Cos B (c2 a2 b2) 2ca or
• Cos C (a2 b2 c2) ? 2ab

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EXAMPLE

• PQR is a triangle with PQ 10cm, PR 5cm, and
• QR 13 cm. Find the largest angle in the
  triangle.

Using cosine rule, p2 q2 r2 2qr cos P cos P
(52 102 132) 2(5)(10) -0.44
?P 116º 6
Q
13cm
10cm
R
P
5cm
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Solve the triangle using sine rule and cosine
rule.

• ABC is a straight line. Find ?ABD and ?BCD.

D
Using cosine rule, cos ?ABD (42 32 62)
2(3)(4) -0.45833 ?ABD
117º 17 Hence ?DBC 62?43
7cm
6cm
4cm
C
A
B
3cm
Using sine rule,
sin ?BCD 0.5079
?BCD 30?31
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Area of Triangle
h
b sin C h

• Area of triangle

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Example

• Area of triangle 0.5 ? 16.5 ? 8 ? sin 120º
• 57.16 cm2