Question

Use the given information to find each of the following.sin A/2, given cos A/2 = - 3, 90° \u003c A \u003c 180°

Accepted Answer

First, recognize that the problem gives you $$ \cos \frac{A}{2} = -3 $$ and the angle $$ A  is $$between 90° and 180°. Since $$ \cos \frac{A}{2} $$ must be between -1 and 1 for real angles, check if the given value is valid or if there might be a typo or misunderstanding in the problem statement. Assuming the value is valid or corrected, recall the Pythagorean identity for sine and cosine: $$ \sin^2 	heta + \cos^2 	heta = 1 . $$Here, $$ 	heta = \frac{A}{2} . $$Use this to express $$ \sin \frac{A}{2}  in $$terms of $$ \cos \frac{A}{2} $$: $$ \sin \frac{A}{2} = \pm \sqrt{1 - \cos^2 \frac{A}{2}} $$ Determine the correct sign (positive or negative) for $$ \sin \frac{A}{2}  by $$considering the quadrant in which $$ \frac{A}{2} $$ lies. Since $$ 90^\circ \u003c A \u003c 180^\circ $$, then $$ 45^\circ \u003c \frac{A}{2} \u003c 90^\circ $$, which places $$ \frac{A}{2}  in $$the first quadrant where sine is positive. Finally, substitute the value of $$ \cos \frac{A}{2} $$ into the formula and simplify under the square root to find $$ \sin \frac{A}{2} . $$Remember to choose the positive root based on the quadrant analysis.

Use the given information to find each of the following.
sin A/2, given cos A/2 = - 3, 90° < A < 180°

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Key Concepts

Half-Angle Identities

Sign Determination in Quadrants

Pythagorean Identity

Use the given information to find each of the following.sin A/2, given cos A/2 = - 3, 90° < A < 180°