06Zsinerule

toc = The Sine Rule = (also called the Law of Sines)

Take __any__ triangle Then label
 * Label the angles ** A **, ** B **, ** C **
 * the side opposite angle ** A ** as side ** a **.
 * the side opposite angle ** B ** as side ** b **.
 * the side opposite angle ** C ** as side ** c **.

{Angle names are in capitals, side names are lower case}

This is the same labelling as is used for the Cosine Rule

For this triangle, the Sine Rule is: Use the Sine Rule when concerned with **2** sides and **2** angles where only 1 of those 4 things is unknown.

Drop off the part of the rule that is not useful
 * (in the following example, the c/sinC is not used)

Finding a Side Length


**BUT:** What do we do if the side we want is not opposite one of the known angles?

**SOLUTION:** Use the fact that the angles in a triangle add up to 180º to find the missing angle.



Finding an Angle
{It is useful to flip the Sine Rule upside down to find angles}

{Before calculating **sin** -1 **(x)** keep the full value of 0.4056 on your calculator for better accuracy}

Ambiguous Case
When finding an angle, there is one situation where __two__ answers are possible.

When you know one angle and two sides, but:
 * The known angle is __not__ between the two known sides
 * The side __opposite__ the known angle is __shorter__ than the other known side.

Find q, a < c

From solution 1, swing line ** a ** around using the top point as a pivot This creates a second solution with q being an obtuse angle

{Notice that if a > c, swinging the line a would not produce a second triangle}

Also see the Cosine Rule

For another site that explains this idea, go here: MathsIsFun

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