Final answer:
An acute angle theta is constructible with a straightedge and compass if and only if cos theta is constructible.
Step-by-step explanation:
An acute angle theta is constructible with a straightedge and compass if and only if cos theta is constructible.
To understand why this is the case, let's look at the concept of constructibility. In geometry, constructible numbers are those that can be created using a straightedge and compass. Constructible numbers include whole numbers, fractions, and surds (square roots of whole numbers). When an acute angle theta is constructible, it means that all the trigonometric functions of theta (such as sine, cosine, tangent, and cotangent) can also be constructed.
However, not all trigonometric functions need to be constructible for an angle to be constructible. In the case of an acute angle theta, cos theta is the only trigonometric function that needs to be constructible for the angle to be constructible.