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Sin and Cosine rules

Naming non right-angled triangles

- CAPITALS are used for angles (usually clockwise)

- lower case letters used for sides.
The name of a side is the lower case of the letter that represents the angle opposite that side

 

The sine rule

When finding an angle, make it the numerator

To use the sine rule for anything, we need to know one complete angle & side pair such as B and b and another angle or side When finding a side, make it the numerator

Example of use

    What is the length of side x?

1) Find all the data needed 180 - (80 + 30) = 70
2) Use the rule!  

 

x/sin80 = 10/sin70

x = 10/sin70 * sin80

x = 10.5 (3s.f.)

 

3) Does the answer make sense?

Use these simple tests to make sure you haven't gone wrong...

    1- Triangle's angles add to 180!

    2- The largest side in a triangle is always opposite the largest angle

    3- No side can be greater than the sum of the other two!

 

There is also an extension to the sine rule that you should know about. sinA/a = sinB/b = sinC/c = 2r

'2r' stands for twice the radius of the circum circle that touches all three corners of the triangle

 

The cosine rule

To use the sine rule you need one complete angle & side pair such as B and b and another angle or side. Sometimes this will not be provided so you might need to use the cosine rule...

To use the cosine rule, we need to know...

all three sides to find an angle and two sides and the included angle to find a side. The 'included angle' is the angle between the known sides

What is the length of side b?

1) Find all the data needed (using the sine rule in this particular case)

 

 

 

 

 

 

sinA/20 = sin30/10

sinA = sin30/10 * 20

sin A = 1

sin-1(1) = 90

A = 90

so B = 60

 

2) Use the rule!

 

 

 

 

b2 = 102 + 202 - (2*20*10*cos60)

b2 = 300

b = √300

b = 17.3 (3s.f.)

3) Does the answer make sense?

Use these simple tests to make sure you haven't gone wrong...

    1- Triangle's angles add to 180!

    2- The largest side in a triangle is always opposite the largest angle

    3- No side can be greater than the sum of the other two!

 

Finding areas of triangles

There are two formulas for the area of a triangle

1) Find all the data needed (using the sine rule in this particular case)

 

 

cosA = (62 + 42 - 5.52) / (2*4*6)

cosA = 0.453125

A = 63.1 (3sf)

2) Use the rule!

 

1/2 * 4 * 6 * sin63.1

= 10.7 (3sf)

 

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