Qualitative Reasoning about Double Angle Signs or Inequalities

Determine the sign, quadrant, or relative magnitude of a double angle expression (e.g., sin 2α > 0?) or compare trigonometric expressions using identities without computing exact values.

gaokao 2020 Q2 5 marks View

If $\alpha$ is an angle in the fourth quadrant, then
A． $\cos 2 \alpha > 0$
B． $\cos 2 \alpha \leqslant 0$
C． $\sin 2 \alpha > 0$
D． $\sin 2 \alpha < 0$

isi-entrance 2026 QB10 View

Let $\frac { \tan 3 \theta } { \tan \theta } = k$. Then
(A) $k \in ( 1 / 3,3 )$
(B) $k \notin ( 1 / 3,3 )$
(C) $\frac { \sin 3 \theta } { \sin \theta } = \frac { 2 k } { k - 1 }$.
(D) $\frac { \sin 3 \theta } { \sin \theta } > \frac { 2 k } { k - 1 }$