Solving Trig Equations with Calculators, Part II : Since this document is also being prepared for viewing on the web we split this section into two parts to keep the size of the pages to a minimum.
Also, as along the last few examples in the earlier part of this section we are not going to be looking for solutions in an interval to save space. The significant part of this instance is to determine the solutions to the equation. If we'd been given an interval it would be simple enough to determine the solutions that actually fall in the interval.
In all the examples in the earlier section all the arguments, the 3t, α/7, etc., were fairly simple.
Let's take a look at an example which has a slightly more complicated argument.
Example Solve 5 cos(2 x -1) = -3 .
Solution: Note as well that the argument here is not actually all that complicated but the addition of the "-1" frequently seems to confuse people so we have to a quick example along with this kind of argument. The solution procedure is identical to all of the problems we've done to this point hence we won't be putting in much explanation. Following is the solution.
Cos( 2x -1) = - 3/5 ⇒ 2x -1 = cos-1 ( - 3/5) = 2.2143
This angle is persist in the second quadrant and hence we can use either -2.2143 or 2 ? - 2.2143 = 4.0689 for the second angle. Usually for these notes we'll employ the positive one. Thus the two angles are,
2 x -1 = 2.2143 + 2 ? n
2 x -1 = 4.0689 + 2 ? n n= 0, ±1, ±2,.......
Now, still we need to determine the actual values of x which are the solutions. These are found in the similar manner as all the problems above. First we'll add one to both sides and then divide by two. Doing this gives,
x= 1.6072 + ? n
x= 2.5345 + ? n n= 0, ±1, ±2,.......
Hence, in this example we saw an argument which was a little different from those seen beforehand, but not all that different while it comes to working the problems hence don't get too excited regarding it.