Refer to Example Suppose 0 ≤ θ<2π, and rewrite the conclusion using a piecewise-defined...

Refer to Example Suppose 0 ≤ θ<2π, and rewrite the conclusion using a piecewise-defined function.

EXAMPLE Making a trigonometric substitution

Express   in terms of a trigonometric function of θ, without radicals, by making the substitution x = a sin θ for − π/2≤ θ ≤ π/2and a > 0.

SOLUTION We proceed as follows:

The last equahty is true because (1) if a > 0, then |a| = a, and (2) if − π/2≤ θ ≤ π/2, then cos θ ≥ 0 and hence |cos θ| = cos θ.

We may also use a geometric solution. If x = a sin θ, then sin θ = x/a, and the triangle in Figure illustrates the problem for 0<θ<π/2. The third side of the triangle, ,can be found by using the Pythagorean theorem. From the figure we can see that

 or, equivalently.

FIGURE