Question

In $\triangle ABC$, with usual notations, $2ac \sin\left(\frac{A-B+C}{2}\right)$ is equal to

• A. $a^{2}+b^{2}-c^{2}$
• B. $c^{2}+a^{2}-b^{2}$
• C. $b^{2}-c^{2}-a^{2}$
• D. $c^{2}-a^{2}-b^{2}$

Hint:

Trigonometry Tip: In "Properties of Triangles" problems, whenever you see an expression involving combinations of $A, B, C$ divided by 2, immediately apply $A+B+C=\pi$ to convert it into a complementary angle identity.

Answer 1

The Correct Option is B

Concept: Trigonometry - Properties of Triangles (Sine/Cosine Rules) and Angle Sum Property.

Step 1: Apply the angle sum property of a triangle. In any triangle, the sum of the internal angles is $180^\circ$ or $\pi$ radians. Therefore, $A + B + C = \pi$. We can rewrite this to group A and C: $A + C = \pi - B$.

Step 2: Substitute this into the given expression's angle. The given expression contains the angle term $\frac{A-B+C}{2}$. Rearrange the numerator to group A and C together: $\frac{(A+C)-B}{2}$. Substitute $A+C = \pi - B$ into this expression: $\frac{(\pi - B) - B}{2} = \frac{\pi - 2B}{2} = \frac{\pi}{2} - B$.

Step 3: Simplify the trigonometric function using complementary angles. Substitute the simplified angle back into the original sine function: $\sin\left(\frac{A-B+C}{2}\right) = \sin\left(\frac{\pi}{2} - B\right)$. Using the standard complementary angle identity $\sin(90^\circ - \theta) = \cos\theta$, this simplifies entirely to $\cos B$.

Step 4: Substitute this result back into the full expression. The original expression was $2ac \sin\left(\frac{A-B+C}{2}\right)$. Replacing the sine term with our result from Step 3, the expression becomes $2ac \cos B$.

Step 5: Apply the Cosine Rule to reach the final algebraic form. The Cosine Rule for an angle $B$ in a triangle states that $\cos B = \frac{a^2 + c^2 - b^2}{2ac}$. We substitute this formula directly into our current expression: $2ac \cos B = 2ac \left( \frac{a^2 + c^2 - b^2}{2ac} \right)$.
The $2ac$ terms in the numerator and denominator perfectly cancel each other out, leaving only the numerator of the Cosine Rule formula: $a^2 + c^2 - b^2$. Rearranging the terms algebraically to match the options gives $c^2 + a^2 - b^2$. $$ \therefore \text{The expression evaluates to } c^{2}+a^{2}-b^{2}. $$