Contents:

• Sine and Cosine Rule:
  • Example Sine Rule:
  • Example Cosine Rule:

 

Sine & Cosine Rules: These rules extend to non right angled triangles. The triangle needs to be labelled in a particular way. Each side is labelled with the same letter as the opposite angle.

The sine rule:

this can also be written the other way up.

The cosine rule: a= b+ c-2bc cos A. or rearranged as:

b= a+ c-2ac cos B. and: c= a+ b-2ac cos C.

Can also be written such: cos A = (b+ c- a)2bc. or rearranged as:

cos B = (a+ c- b)2ac. and: cos C = (a+ b- c)2ab.

These rules are applied to four basic examples. (i) Two given angles and a given side. (ii) Two given sides and a given angle not enclosed by the two sides. (iii) Two given sides and the given angle enclosed by the two sides. (iv) All sides given with no given angles.

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Examples:

Example: (i) Two angles and a side.

Enter the missing angle: 40, and letter the sides. Find the missing lengths; 'a' and 'c'.

Use Sine Rule.

Find the length of side 'a'. (asin A = bsin B). asin 80 = 8sin 60.

Rearrange for 'a': sin 80 x 8sin 60. Equals 0985 x 80866. 'a' = 91.

Similarly for length of side 'c'. (csin C = bsin B) csin 40 = 8sin 60.

Rearrange for 'c': sin 40 x 8sin 60. Equals 0643 x 80866. 'c' = 595.

(Check: using the calculated length for side 'a', side 'c' can be found. csin C = asin A. ( 0643 x 910981) = 595.)

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Example: (ii) Two sides and an unenclosed angle.

Letter the sides. Find the missing angles.

Use Sine Rule.

 

Find angle 'C': (sin Cc = sin Bb). sin C87 = sin 60 117.

Rearrange for 'C': c x sin 60 b. Equals: 87 x 0866117. 'C' = sin 0644; = 40.

(Note: The inverse of sin is used [ sin] when converting from a value that does not represent a value of degrees).

Angle 'A' is now found to be 80. The length of the side 'a' can also be calculated using the method from example (i).

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Example: (iii) Two sides and an enclosed angle.

Letter the sides. Find the missing side 'c'.

Use Cosine Rule.

 

Find the length of side 'c': (c= a+ b -2ab cos C). 675+ 45 - 2 x 675 x 45 x cos 70.

No rearranging necessary: c= 450348. Therefore 'c' = 450348. 'c' = 67.

The other angles can now be calculated using the Sine Rule method in example (ii).

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Example: (iv) Three sides and no angles.

Letter the sides. Find the missing angles.

Use Cosine Rule.

 

Find angle 'A': (a= b+ c -2bc cos A).

Rearrange for cos A: cos A = (b+ c- a)2bc.

Equals: (45 + 67 - 675)2 x 45 x 67. Equals: 0328.

'A' = cos 0328. 'A' = 7085.

(Note: The inverse of cos is used [ cos] when converting from a value that does not represent a value of degrees).

With one angle now known the other angles can be calculated using the Sine rule method in example (ii).

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