Mathematics

Mathematics

Problem

For 0∘<θ<90∘0^{\circ}<\theta<90^{\circ}, solve sin⁡(45∘+θ)+sin⁡(45∘−θ)=1\sin(45^{\circ}+\theta)+\sin(45^{\circ}-\theta)=1. a) 60∘60^{\circ} b) 45∘45^{\circ} c) 22.5∘22.5^{\circ} d) 30∘30^{\circ}
Intuitive explanation

The key is recognizing that this sum of sines can be collapsed using the sum-to-product formula. Notice both angles are symmetric around 45°45°—this symmetry suggests using sin⁡A+sin⁡B=2sin⁡(A+B2)cos⁡(A−B2)\sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) to simplify dramatically. Once you combine them, you'll get a single trigonometric equation that's much easier to solve.

Intuitive explanation

The key is recognizing that this sum of sines can be collapsed using the sum-to-product formula. Notice both angles are symmetric around 45°45°—this symmetry suggests using sin⁡A+sin⁡B=2sin⁡(A+B2)cos⁡(A−B2)\sin A + \sin B = 2\sin\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right) to simplify dramatically. Once you combine them, you'll get a single trigonometric equation that's much easier to solve.

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