Property 18
Half Angle Formulas
\(\blacksquare\quad \sin(2a)=\sin(a+a)=\sin a\cdot\cos a+\cos a\cdot\sin a\)
\(\blacksquare\quad \sin(2a)=2\cdot\sin a\cdot\cos a\)
\(\blacksquare\quad \cos(2a)=\cos(a+a)=\cos a\cdot\cos a-\sin a\cdot\sin a\)
\(\blacksquare\quad \cos(2a)=\cos^2a-\sin^2a\)
\(\blacksquare\quad \cos(2a)=1-2\sin^2a\)
\(\blacksquare\quad \cos(2a)=2\cos^2a-1\)
\(\blacksquare\quad \tan(2a)=\tan(a+a)=\samefrac{\tan a+\tan a}{1-\tan a\cdot\tan a}\)
\(\blacksquare\quad \tan(2a)=\samefrac{2\tan a}{1-\tan^2a}\)
\(\blacksquare\quad \cot(2a)=\samefrac{\cot^2a-1}{2\cot a}\)
1.
\(2\sin x\cos x=?\)
2.
\(x\in\left(0,\samefrac\pi2\right),\ \cos x=\samefrac35\)
\(\sin2x=?\)
3.
\(x\in\left(\pi,\samefrac{3\pi}{2}\right),\ \tan x=\samefrac34\)
\(\cos2x=?\)
4.
\(2\sin75^\circ\cos75^\circ=?\)
5.
\(\cos^215^\circ-\sin^215^\circ=?\)
6.
\(\sin20^\circ=k\)
\(\cos50^\circ=?\)
7.
\(\cos25^\circ=t\)
\(\cos50^\circ=?\)
8.
\(\sin x-\cos x=\samefrac12\)
\(\sin2x=?\)
9.
\(\sin x+\cos x=\samefrac13\)
\(\sin2x=?\)
10.
\(\samefrac{\sin2x}{2\cos^2(x/2)-1}=?\)
11.
\(\sin10^\circ\cos10^\circ\cos20^\circ\cos40^\circ=x\sin80^\circ\)
\(x=?\)
12.
\(\sin x=\samefrac45\)
\(\sin\left(\samefrac x2\right)=?\)
13.
\(\cos2x=\samefrac35\)
\(\tan x=?\)
14.
\(\samefrac{\sin60^\circ}{\cos15^\circ}-\samefrac{\cos60^\circ}{\sin15^\circ}=?\)
15.
\(0<x<\samefrac\pi2\)
\(\sqrt{1+\sin2x}=?\)
16.
\(\samefrac\pi2<x<\pi\)
\(\cot x=-\samefrac43\)
\(\Rightarrow\tan(2x)=?\)
17.
\(0<x<\samefrac\pi4\)
\(\tan x=\samefrac12\)
\(\Rightarrow\tan(2x)=?\)
18.
\(\tan22.5^\circ=?\)