Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
If in a $\triangle ABC$, $a = 5$, $b = 4$ and $\cos(A - B) = \frac{31}{32}$, then the third side $c$ is equal to
Step-by-Step Solution
Key Concept: Apply half-angle identities and the relationship between sides and angles to solve for trigonometric values.
Given $\tan\left(\frac{A-B}{2}\right) = \frac{a-b}{a+b}\cot\left(\frac{C}{2}\right)$, we use the half-angle formula $\cot\left(\frac{C}{2}\right) = \frac{1 + \tan^2\left(\frac{A-B}{2}\right)}{1 + \frac{1}{81}\cot^2\left(\frac{C}{2}\right)}$. Through algebraic manipulation, we find $\cot^2\left(\frac{C}{2}\right) = \frac{9}{7}$ and ultimately $\cos(C) = \frac{1}{8}$ with $c = 6$.
Correct Answer: 6