It is easiest to verify the formulas:
The set of all solutions to y = cos(x) is x = 2kp ± cos-1(y) and the set of all solutions to y = tan(x) is x = kp + tan-1(y), where k can be any integer.
algebraically, using the sum and difference formulas for cos and the sum formula for tan (which is similar to the difference formula for tan that we derived in the Exercises, and which can be derived in a similar manner):
cos( 2kp + cos-1(y))
= cos( 2kp)cos( cos-1(y)) -
sin( 2kp)sin( cos-1(y)) =
1·y - 0·sin( cos-1(y)) = y
cos( 2kp - cos-1(y)) = cos( 2kp)cos( cos-1(y))
+ sin( 2kp)sin( cos-1(y)) =
1·y + 0·sin( cos-1(y)) = y
We can see that these formulas are correct by looking at how each of their graphs intersect a horizontal line. For example, the graph of cos looks like:
We can see the two "smallest" solutions at ±cos-1(y); all the other solutions are translations by (multiples of) 2p in either direction.
Similarly, the graph of tan looks like:
This time there is only one "smallest" solution, tan-1(y); all the other solutions are translations by (multiples of) p in either direction, since tan has period p.
Go back to Solving Trigonometric Equations
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