Use the half-angle formulas to simplify the expression. $$- \sqrt { \frac { 1 - \cos ( x - 3 ) } { 2 } }$$
A) $- \left| \sin \left( \frac { x - 3 } { 2 } \right) \right|$
B) $- \left| \sin ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) \right|$
C) $- | \sin ( x - 3 ) |$
D) $- \left| \sin ^ { - 1 } \left( \frac { x - 3 } { 2 } \right) \right|$
E) $- \left| \sin \left( \frac { x + 3 } { 2 } \right) \right|$

Question

Use the half-angle formulas to simplify the expression.​ $$- \sqrt { \frac { 1 - \cos ( x - 3 ) } { 2 } }$$ ​
A)​ $- \left| \sin \left( \frac { x - 3 } { 2 } \right) \right|$
B)​ $- \left| \sin ^ { - 1 } \left( \frac { x + 3 } { 2 } \right) \right|$
C)​ $- | \sin ( x - 3 ) |$
D)​ $- \left| \sin ^ { - 1 } \left( \frac { x - 3 } { 2 } \right) \right|$
E)​ $- \left| \sin \left( \frac { x + 3 } { 2 } \right) \right|$

Quizplus · Accepted Answer

The given expression $- \sqrt { \frac { 1 - \cos ( x - 3 ) } { 2 } }$ matches the half-angle formula for sine, which is $\sin \left( \frac{\theta}{2} \right) = \pm \sqrt{ \frac{1 - \cos \theta}{2} }$. Therefore, the expression simplifies to $- \left| \sin \left( \frac { x - 3 } { 2 } \right) \right|$, because the square root ensures the result is non-negative, but the negative sign outside the square root indicates the value is non-positive, hence the absolute value notation with a negative sign outside.

Use the Half-Angle Formulas to Simplify the Expression −1−cos⁡(x−3)2- \sqrt { \frac { 1 - \cos ( x - 3 ) } { 2 } }−21−cos(x−3)​​

Use the Half-Angle Formulas to Simplify the Expression $- \sqrt { \frac { 1 - \cos ( x - 3 ) } { 2 } }$