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Review of Sine and Cosine Rules
We use Sine and Cosine rules to solve lengths and angles of non-right angled triangles.
We start all problems by drawing and labelling the triangle A, B, C (angles); a, b, c (side lengths); in order as shown.
Make a list of what you know & what you want to know and then choose the rule which includes all these variables.
Sine Rule
When either 2 side lengths and one of their opposing angles are known or 1 side length
and two angles are known: 𝒂
𝐬𝐢𝐧 𝑨 =
𝒃
𝐬𝐢𝐧 𝑩 =
𝒄
𝐬𝐢𝐧 𝑪
Example 1: Solve the unknown length ‘𝑥’ using the sine rule – Fill in the blanks when shown in class
1. Label the triangle
2. State variables that we know and what we want to know
3. Choose which rule to use
4. Substitute values into equation and solve
Example 2: Solve the unknown angle ‘𝑥’ using the sine rule – Fill in the blanks when shown in class
1. Label the triangle
2. State variables that we know and what we want to know
3. Choose which rule to use
4. Substitute values into equation and solve
Solve for the unknown length ‘𝒙’
Solve the unknown angle ‘𝒙’
a
B
c
a A C
c
b
Cosine Rule
When 3 side lengths of the triangle are known, solve for any angle using one of:
cos 𝐴 =𝑏2 + 𝑐2 − 𝑎2
2𝑏𝑐 cos 𝐵 =
𝑎2 + 𝑐2 − 𝑏2
2𝑎𝑐 cos 𝐶 =
𝑎2 + 𝑏2 − 𝑐2
2𝑎𝑏
When two side lengths and the angle between them are known, solve a side length using one of:
𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos 𝐴 𝑏2 = 𝑎2 + 𝑐2 − 2𝑎𝑐 cos 𝐵 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶
Example 3: Solve the unknown length ‘𝑥’ using the cosine rule – Fill in the blanks when shown in class
1. Label the triangle
2. State variables that we know and what we want to know
3. Choose which rule to use
4. Substitute values into equation and solve
Example 4: Solve the unknown angle ‘𝑥‘ using the cosine rule – Fill in the blanks when shown in class
1. Label the triangle
2. State variables that we know and what we want to know
3. Choose which rule to use
4. Substitute values into equation and solve
Solve for the unknown length ‘𝒙’
Solve the unknown angle ‘𝒙’
B
c
a
A C
c
b
