Cosine Rule is used:
When the length of all 3 sides of a triangle is given. OR
When 2 sides and an included angle is given. (C is an included angle of lines 43 cm and 90 cm.)
Cosine is a trigonometric function used to calculate angles of a right-angled triangle. However, the cosine rule can be used for all types of triangles including acute angled, right-angled, and obtuse-angled triangles.
Cosine Rule:
a. To calculate the length of sides.
a2 = b2 + c2 - 2bc cos A
b2 = a2 + c2 - 2ac cos B
c2 = b2 + c2 - 2ab cos C
b. To calculate the size of angles.
a2 = b2 + c2 - 2bc cos A
a2 + 2bc cos A = b2 + c2 - 2bc cos A + 2bc cos A (add 2bc cos A on both sides)
a2 - a2 + 2bc cos A = b2 + c2 - a2 (subtract a2 on both sides)
2bc cos A = b2 + c2 - a2 (divide 2bc on both sides)
cos A= (b^2 + c^2 - a^(2 ))/2bc
cos A= (b^2 + c^2 - a^(2 ))/2bc
cos B= (a^2 + c^2 - b^(2 ))/2ac
cos C= (a^2 + b^2 - c^(2 ))/2ac
1. Label the triangle
a. Label the angles A, B and C.
Any angle can be labelled as A, B or C.
b. Label the sides a, b, c.
The side opposite to angle A is labelled a, the side opposite to angle B is labelled
b, and the side opposite to angle C is labelled c.
In the triangle, a = 43 cm, b = 90 cm, c = 51 cm.
To calculate angle A:
cos A= (b2 + c2 - a2 /2bc
cos A= (902 + 512 - 432 /(2*51*90)
cos A= 0.964
A= cos-1 (0.964)
A=15.420571°
