Law of Sines versus Law of Cosines

Plugging in the values correctly and using the calculator correctly are both difficult.  But, with practice, those issues become routine.  The trickiest part about the Laws of Sines and Cosines is knowing which you can use, and when.

Another pitfall that traps students, frequently, is that they fail to use the simplest methods available.  It is as though they feel obligated to use the most recent tools they’ve developed.  If you’re dealing with a right triangle, there is absolutely no need or reason to use the sine rule, the cosine rule of the sine formula for the area of a triangle.  Everything can be found with sine, cosine and tangent, the Pythagorean Theorem, or the sum of angles of a triangle is 180 degrees.

This table is really not much help because the moment you understand it you no longer need it.  The way you come to understand it through experience and practice. But, it’s a good reference to give you some confidence moving forward. 

Let’s see an example.

So, the missing side’s length is required.  We cannot find it using the law of sines because we do not know the value of the angle k

However, we can find the unmarked angle, as it is opposite of 8 and we know the angle that’s 82 degrees and its opposite side.

The missing angle is 41.3°.  Now, we can find the measurement of angle k, by subtracting 82 and 41.3 from 180.  That gives us k = 56.7°.

Now we can find the missing side with either the sine or the cosine rule.  If given the choice, the sine rule is simpler on the calculator, so it is probably best.

The big idea here is that you must understand each formula.  Be strict with the angle and its opposite side.  That is where mistakes are made.  If you choose the wrong formula, but are strict with the relationships between opposite angles and sides you’ll end up with an equation that you cannot solve.  That means you need to try something else!  That is it…no harm, no foul, just move on and learn from it.

Here’s an example problem for you to try.  Find the values of x and y

Did you realize you could use the sine or the cosine rule for x?

x = 12.3, y = 25.9°

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