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Olympiad workbook
Exam
Olympiad workbook
Subject
IOQM
Chapter
Trigonometric Equations

Olympiad workbook · IOQM · Trigonometric Equations

How many distinct triangles ABC are there, up to similarity, such...

How many distinct triangles ABC are there, up to similarity, such that the magnitudes of angles A, BB and CC in degrees are positive integers and satisfy cos⁡Acos⁡B+sin⁡Asin⁡Bsin⁡kC=1\cos A \cos B+\sin A \sin B \sin k C=1

Correct answer

6

Step-by-step solution

cos⁡Acos⁡B+sin⁡Asin⁡Bsin⁡kC=1⇒cos⁡Acos⁡B+sin⁡Asin⁡B+sin⁡Asin⁡Bsin⁡kC−sin⁡Asin⁡B=1⇒sin⁡Asin⁡B(sin⁡kC−1)=1−cos⁡(A−B)\begin{aligned} & \cos A \cos B+\sin A \sin B \sin k C=1 \\ & \Rightarrow \cos A \cos B+\sin A \sin B+\sin A \sin B \sin k C-\sin A \sin B=1 \\ & \Rightarrow \sin A \sin B(\sin k C-1)=1-\cos (A-B) \end{aligned} So (sin⁡kC−1)=0(\sin k C-1)=0 and cos⁡(A−B)=1\cos (A-B)=1 ⇒kC=90∘ and A=B\Rightarrow \mathrm{kC}=90^{\circ} \text { and } \mathrm{A}=\mathrm{B} Number of factors of 90∘90^{\circ} 90=2×32×590=2 \times 3^2 \times 5 Number of factors =2×3×2=12=2 \times 3 \times 2=12 for 6 factor A,BA, B are integers
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