TRIGONOMETRY: COSINE RULE
The purpose of this worksheet is to discover a formula called Cosine Rule.
Section A: Exploration
1.
Draw any triangle in the space below. It does not matter whether it is right-angled, acuteangled, obtuse-angled, equilateral, isosceles or scalene. But try to draw as big as possible.
2.
Label the vertices of the triangle A, B and C (the order does not matter).
3.
Label the side of the triangle opposite the vertex A as a (the order is important).
Label the side of the triangle opposite the vertex B as b.
Label the side of the triangle opposite the vertex C as c.
4.
Using a protractor and a ruler, complete the table below. Your measurements will be
different from your teacher or your classmates.
Angle
5.
Angle in Degrees
Length of Side
A
a
B
b
C
c
Length in cm
Using a calculator, evaluate the expressions in the table below.
Expression
Value (to 2 s.f.)
Expression
a2
b 2  c 2  2bc cos A 
b2
a 2  c 2  2ac cos B 
c2
a 2  b 2  2ab cos C 
© Joseph Yeo, 1998-2010: This worksheet is FREE for anyone to use
Value (to 2 s.f.)
Page 1
Section B: Findings
6.
What do you notice about the second and fourth columns in the table in Q5?
7.
Check the value of a 2 with a classmate. Does it have the same value?
8.
Check the values of all the 6 expressions with your classmate. What do you notice?
9.
Does your finding in Q6 and Q8 depend on which triangle you or your classmate draw?
Section C: Conclusion
10.
The relationship that you have discovered in Q6 and Q8 is called the Cosine Rule. State the
Cosine Rule in the box below.
11.
Make the angle the subject of the formula for the Cosine Rule.
© Joseph Yeo, 1998-2010: This worksheet is FREE for anyone to use
Page 2
Answer Keys
6.
7.
8.
9.
10.
a 2  b 2  c 2  2bc cos A ; b 2  a 2  c 2  2ac cos B ; c 2  a 2  b 2  2ab cos C
No
a 2  b 2  c 2  2bc cos A ; b 2  a 2  c 2  2ac cos B ; c 2  a 2  b 2  2ab cos C
No
a 2  b 2  c 2  2bc cos A ; b 2  a 2  c 2  2ac cos B ; c 2  a 2  b 2  2ab cos C
11.
cos A 
b2  c2  a2
a2  c2  b2
a2  b2  c2
; cos B 
; cos C 
2bc
2ac
2ab
© Joseph Yeo, 1998-2010: This worksheet is FREE for anyone to use
Page 3