Trigonometry
Trigonometry
Allen Star Batch
Grade 11
Question:
If $t = x + y + z$, then $\sin x + \sin y + \sin z - \sin t$ equals:
$4\tan\left(\frac{y+z}{2}\right)\tan\left(\frac{z+x}{2}\right)\tan\left(\frac{x+y}{2}\right)$
$4\cos\left(\frac{y+z}{2}\right)\cot\left(\frac{z+x}{2}\right)\cot\left(\frac{x+y}{2}\right)$
$4\sin\left(\frac{y+z}{2}\right)\sin\left(\frac{z+x}{2}\right)\sin\left(\frac{x+y}{2}\right)$
$4\cos\left(\frac{y+z}{2}\right)\cos\left(\frac{z+x}{2}\right)\cos\left(\frac{x+y}{2}\right)$
Step-by-Step Solution
Key Concept: Sum-to-product formulas convert sums and differences of sines into products, enabling factorization.
Starting with $\sin x + \sin y + \sin z - \sin t = (\sin y + \sin z) + (\sin x - \sin t)$, apply sum-to-product formulas: $\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$. This gives $2\sin\frac{y+z}{2}\cos\frac{y-z}{2} + 2\sin\frac{x-t}{2}\cos\frac{x+t}{2}$. Combining and factoring yields $4\sin\frac{x+z}{2}\sin\frac{x+y}{2}\sin\frac{x+z}{2}$.
Correct Answer: 3