Question 7.7.1: Solving an Equation for a Specified Variable Solve y = 3 cos...
Solving an Equation for a Specified Variable
Solve y = 3 \cos 2x for x, where x is restricted to the interval [ 0, \frac{π}{2} ].
We want to isolate \cos 2x on one side of the equation so that we can solve for 2x, and then for x.
y = 3 \cos 2x ← \fbox{Our goal is to isolate x.}
\frac{y}{3} = \cos 2x Divide by 3.
2x = \arccos \frac{y}{3} Definition of arccosine
x = \frac{1}{2} \arccos \frac{y}{3} Multiply by \frac{1}{2} .
An equivalent form of this answer is x = \frac{1}{2} \cos^{-1} \frac{y}{3} .
Because the function y = 3 \cos 2x is periodic, with period π, there are infinitely many domain values (x-values) that will result in a given range value ( y-value). For example, the x-values 0 and π both correspond to the y-value 3. See Figure 38. The restriction 0 ≤ x ≤ \frac{π}{2} given in the original problem ensures that this function is one-to-one, and, correspondingly, that
x =\frac{1}{2} \arccos \frac{y}{3}
has a one-to-one relationship. Thus, each y-value in [ -3, 3] substituted into this equation will lead to a single x-value.
