Lines.jpgThe trigonometric methods given earlier apply only to triangles containing a right angle.

For triangles without a right angle, the sine rule, the cosine rule and the area formula can be used to solve triangles and find their areas.

Sine Rule
Cosine Rule
Area Formula

New Zealand students, please note that this topic is now assessed at NCEA, Level 2 (NZ Year 12).

Sine Rule

The sine rule is concerned with triangles where pairs of angles and their opposite sides are given.

The sine rule states:

Y10_Non-right-angled_Triangles_01.gif

Y10_Non-right-angled_Triangles_02.gif

 

or, in an alternative form for use when finding angle sizes:

Y10_Non-right-angled_Triangles_03.gif

 

 

Examples
Answers

(a) Find the value of a

Y10_Non-right-angled_Triangles_04.gif

Use the formula: Y10_Non-right-angled_Triangles_05.gif

(b) Find the size of angle y

Y10_Non-right-angled_Triangles_06.gif

 

Use the formula: Y10_Non-right-angled_Triangles_07.gif

( to 1 d.p.)

 


Cosine Rule

The cosine rule is concerned with triangles where the lengths of two sides and the angle between them, or the lengths of three sides, are given.

The cosine rule states:

Y10_Non-right-angled_Triangles_08.gif

or , in alternative form for finding angles:

Y10_Non-right-angled_Triangles_09.gif

Y10_Non-right-angled_Triangles_10.gif

 

 

 

 

Examples
Answers

(a) Find the value of a

Y10_Non-right-angled_Triangles_11.gif

Use the formula: Y10_Non-right-angled_Triangles_12.gif

(b) Find the size of angle BAC

Y10_Non-right-angled_Triangles_13.gif

Use the formula: (to 1 d.p.)

Y10_Non-right-angled_Triangles_14.gif

 


Area Formula

The area of a triangle can be found given the lengths of two sides and the angle between them.

The area formula states:

 

Area of triangle = Y10_Non-right-angled_Triangles_15.gif

Y10_Non-right-angled_Triangles_16.gif

 

 

 

 

Examples
Answers

(a) Find the area of Y10_Non-right-angled_Triangles_20.gifPQR

Y10_Non-right-angled_Triangles_17.gif

Y10_Non-right-angled_Triangles_18.gif

(b) The area of the triangle is 27cm2.

Find the value of h.

Y10_Non-right-angled_Triangles_19.gif

Y10_Non-right-angled_Triangles_21.gif