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Geometry in two dimensions
Right angled Trigonometry
Worked Examples

 

The Sine Rule

triangle1

 

  1. Sine Rule formula1

    or the inverse  formula2

  2. Area of triangle  formula3

 

The Cosine Rule

Given the triangle ABC as shown

triangle1

equation4

 

 

Useful rules:

  1. The largest angle is opposite the largest side, also the smallest angle is opposite the shortest side

  2. The two right angled triangles with sides in the ratio 3:4:5 and 5:12:13

 

MATH STUDIES

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Geometry and Trigonometry

Non-right angled Trigonometry: The Sine Rule and the Cosine Rule

 

The Sine Rule and the Cosine Rule : Worked Example

Given triangle ABC where AB = 5cm and BC = 7cm and angle C = 40° Find

  1. Angle A

  2. AC

  3. Area of triangle ABC

 

First sketch the triangle with the information given

(Diagram not to scale)

triangle2

 

  1. Using the sine rule:  equation5

    equation6

  2. To find AC we can use the sine or cosine rule as we now no that angle B = 180 - 40- 64.1 =75.9°

    Using the cosine rule:  AC2 = AB2 + BC2 - 2 x AB x BC cos B

    Hence   AC2 = 52 + 72 - 2 x 5 x 7 cos 75.9

    AC equation7 = 7.55 cm (correct to 3 significant figures)

    Using Sine rule

    equation8

    (there is a slight difference in values)

  3. Area triangle ABC  equation9

 

Notes:

  1. Try and use values given as much as possible to ensure greater accuracy in answers

  2. Do the calculation all in one step on the calculator to avoid rounding errors.

  3. Remember brackets on the calculator. Example sin40 should be entered as sin(40)

 

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