MathDB

13

Part of 2021 AMC 12/AHSME Fall

Problems(2)

Don't Trig Bash

Source: 2021 AMC 12A Fall #13

11/11/2021
The angle bisector of the acute angle formed at the origin by the graphs of the lines y=xy=x and y=3xy=3x has equation y=kxy=kx. What is kk?
<spanclass=′latex−bold′>(A)</span> 1+52<spanclass=′latex−bold′>(B)</span> 1+72<spanclass=′latex−bold′>(C)</span> 2+32<spanclass=′latex−bold′>(D)</span> 2<spanclass=′latex−bold′>(E)</span> 2+52<span class='latex-bold'>(A)</span> \: \frac{1+\sqrt{5}}{2} \qquad <span class='latex-bold'>(B)</span> \: \frac{1+\sqrt{7}}{2} \qquad <span class='latex-bold'>(C)</span> \: \frac{2+\sqrt{3}}{2} \qquad <span class='latex-bold'>(D)</span> \: 2\qquad <span class='latex-bold'>(E)</span> \: \frac{2+\sqrt{5}}{2}
AMCAMC 12AMC 12 Atrigonometry
Your sins are forgiven

Source: 12B #13

11/17/2021
Let c=2π11.c = \frac{2\pi}{11}. What is the value of
sin⁡3c⋅sin⁡6c⋅sin⁡9c⋅sin⁡12c⋅sin⁡15csin⁡c⋅sin⁡2c⋅sin⁡3c⋅sin⁡4c⋅sin⁡5c?\frac{\sin 3c \cdot \sin 6c \cdot \sin 9c \cdot \sin 12c \cdot \sin 15c}{\sin c \cdot \sin 2c \cdot \sin 3c \cdot \sin 4c \cdot \sin 5c}?
<spanclass=′latex−bold′>(A)</span> −1<spanclass=′latex−bold′>(B)</span> −115<spanclass=′latex−bold′>(C)</span> 115<spanclass=′latex−bold′>(D)</span> 1011<spanclass=′latex−bold′>(E)</span> 1<span class='latex-bold'>(A)</span>\ -1 \qquad<span class='latex-bold'>(B)</span>\ \frac{\sqrt{-11}}{5} \qquad<span class='latex-bold'>(C)</span>\ \frac{\sqrt{11}}{5} \qquad<span class='latex-bold'>(D)</span>\ \frac{10}{11} \qquad<span class='latex-bold'>(E)</span>\ 1
AMCAMC 12AMC 12 B