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Ambiguous Case of the Sine Rule
Non Right-Angled Triangle Trigonometry
By the end of this lesson you will be able to explain/calculate the
following:
1. The Ambiguous Case of the Sine Rule
• On your calculator, investigate the values for
each of these pairs of sine ratios:
▫ sin 30° and sin 150°
▫ sin 110° and sin 70°.
• Similarly, sin 60° and sin 120° give an identical
value of 0.8660.
• Find the inverse sine of these values
▫ sin−1 (0.8660) is 60°.
▫ The obtuse (greater than 90°) angle is not given by
the calculator.
• When using the inverse sine function on your
calculator, the calculator will give only the
acute angle.
b  sin A  a  sin B
(2 x  5)  12  ( x  3) 
2
2
2x  5
2 ( x  3)

2
2
2 x  5  2 ( x  3)
2x  5  2x  3 2
2x  2x  5  3 2
x(2  2 )  5  3 2
53 2
x
2 2
53 2
x
2 2
53 2 2 2
x

2 2 2 2
(5  3 2 )( 2  2 )
x
(2  2 )( 2  2 )
10  5 2  6 2  6
x
22  ( 2 ) 2
16  11 2
x
2
x  8  112 2