Problem 104 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When using the half-angle formulas for trigonometric functions of \(\frac{\alpha}{2},\) I determine the sign based on the quadrant in which \(\alpha\) lies.

Step-by-Step Solution

Answer

The statement does not make sense because the sign for the half-angle formulas is determined by the quadrant in which α/2 lies, not α.

Step 1: Break down Half-Angle Formulas

The Half-Angle Formulas are given by:- \(sin(\frac{\alpha}{2}) = ±\sqrt{\frac{1 - cos(\alpha)}{2}}\)- \(cos(\frac{\alpha}{2}) = ±\sqrt{\frac{1 + cos(\alpha)}{2}}\)- \(tan(\frac{\alpha}{2}) = ±\sqrt{\frac{1 - cos(\alpha)}{1 + cos(\alpha)}}\)The sign (positive or negative) of the trigonometric function depends not on the quadrant in which α lies, but on the quadrant in which α/2 lies. This is because it is not the angle α being considered, but half of that angle

Step 2: Analyze the Statement

To determine the sign of the result one would have to know in which quadrant the half angle, not the original angle, lies. Given this, it's clear that the reasoning in the statement is flawed.

Step 3: Final Assessment

The statement does not make sense because the sign of the result of trigonometric functions using half-angle formulas are based on the quadrant of alpha/2, not on the quadrant of α itself.