When we work out the inverse of sin cos and tan of a positive number we always find a value between and or 90^o however between 0 and or 360^o there are more values for inverse trig functions, you can see where these are by looking at a graph (below). The CAST diagram is a method of working out these other values so that we can find all the solutions of sin(x)=a,cos(y)=b,tan(z)=c for x,y,z with a,b,c constants.

There are two solutions to sin(x)=0.65 we get solution (1) when we take the inverse but we need a way of finding (2) from (1)

The four sections of the CAST are cos,all,sin and tan starting by labelling the bottom right and working round in an anti-clockwise direction.

Use the following method to work out the values for the inverse between 0 and 2pi or 360

1. find the value of the inverse of the positive between 0 and 90 or pi/2 using a normal method (eg a calculator).
   So if we want to find sin^-1(0.6) we calculate 0.6435 rads or 36.87 deg and if we want sin^-1(-0.8) we calculate sin^-1(0.8) to get 53.13 deg or 0.927 rads
2. draw in the four lines on the CAST diagram (shown in green) that represent the angle. Do this by measuring from the horizontal the angle calculated in 1
3. If the value we are finding the inverse for is negative (eg we are finding sin^-1(-0.6)) consider the quadrants that don’t include the name of the function. (so if we are finding inverse sin only consider Tan and Cos, if we are finding inverse cos only consider sin and tan and if we are finding inverse tan only consider sin and cos)
4. If the value we are finding the inverse for is positive (eg we are finding sin^-1(0.6)) only consider the quadrants with the name of the function and all. (so if we are finding inverse sin we only consider sin and all, if we are finding inverse cos only consider only consider cos and all and if we are finding inverse ta only consider tan and all)
5. Calculate the angle from the zero line anti-clockwise to the lines in the quadrant’s we are considering. The angle labels I’ve put on the axes should make this easier. These values are your solutions.

All four lines representing the angle x drawn but we will only be interested in two of them

You can check that this works by putting the values back into you calculator and if you want to check that you’ve got all of the solutions check against the graph of the function.

Example

Find all the solutions of cos(x) = 0.7 between 0 and 360^o or rads.

We can use a calculator to find the value between 0 and 90 deg or pi/2 rads
so write y = cos^-1(0.7) = 45.57 deg (we’ll work in degrees for the example to avoid repeating all the calculations)

We no draw this on the cast diagram and choose the two lines we need (in red), in this case the lines in the cos and “all” quadrants.

We then calculate the anti-clockwise angles to these lines from the horizontal:
so we get 45.57 deg for the in the “all” quadrant and 360-45.57=314.43 for the cos quadrant
so our set of solutions for x between 0 and 360 of cos(x)=0.7 are 45.57 and 314.43