(PAS UEM 2021) Trigonometria

[latex]\mathrm{Item\ 01)\ Do\ enunciado: 2sin(2x)-cos(4x)=3}[/latex]

[latex]\mathrm{cos(2x+2x)=cos(2x)cos(2x)-sin(2x)sin(2x)=cos^2(2x)-sin^2(2x)}[/latex]

[latex]\mathrm{Mas,cos^2(2x)=1-sin^2(2x),logo,cos(4x)=1-2sin^2(2x)}[/latex]

[latex]\mathrm{Logo:2sin(2x)-cos(4x)=3\leftrightarrow 2sin^2(2x)+2sin(2x)-4=0}[/latex]

[latex]\mathrm{Sendo\ sin(2x)=t,vem:2t^2+2t-4=0\ \therefore\ (t_1,t_2)=(-2,1)}[/latex]

[latex]\mathrm{Sendo\ sin(2x)\in [-1,1], logo,sin(2x)=t_1=-2\notin \mathbb{R}\ (N\tilde{a}o\ conv\acute{e}m!)}[/latex]

[latex]\mathrm{Para\ sin(2x)=t_2=1\to sin(2x)=sin\left ( \frac{\pi }{2} \right )\to x=\frac{\pi }{4}+k\pi ,k\in\mathbb{Z}}[/latex]

[latex]\mathrm{Para\ x\in [0,2\pi]\ e\ k=\left \{ 0,1 \right \},logo,S=\left \{ \frac{\pi }{4},\frac{5\pi }{4} \right \}.}[/latex]

[latex]\mathrm{Item\ 02)\ Do\ enunciado:sin(x+\pi )cos\left ( x-\frac{\pi }{2} \right )\leq 0}[/latex]

[latex]\mathrm{[sin(x)cos(\pi )+sin(\pi )cos(x)]\left [ cos(x)cos\left ( \frac{\pi }{2} \right )+sin(x)sin\left ( \frac{\pi }{2} \right ) \right ]\leq 0}[/latex]

[latex]\mathrm{Dos\ c\acute{a}lculos,vem:-sin^2(x)\leq 0}[/latex]

[latex]\mathrm{O\ termo\ sin^2(x)\ \acute{e}\ maior\ ou\ igual\ a\ zero\ para\ todo\ x\in \mathbb{R}.}[/latex]

[latex]\mathrm{Logo, o\ sinal\ negativo\ de\ f(x)=-sin^2(x)\ implica\ f(x)\leq 0,\forall\ x\in \mathbb{R}.}[/latex]

[latex]\mathrm{Item\ 04)\ Seja\ f(x)=a+bsin(cx+d)}[/latex]

[latex]\mathrm{Para\ f(x)=sin\left ( x+\frac{\pi }{2} \right ),vem:(a,b,c,d)=\left ( 0,1,1,\frac{\pi }{2} \right )}[/latex]

[latex]\mathrm{P=\frac{2\pi }{|c|}\to P=\frac{2\pi }{|1|}\to P=2\pi }[/latex]

[latex]\mathrm{Item\ 08)}[/latex]



[latex]\mathrm{Item\ 16)\ 3sin^2(x)+cos^2(x)=4}[/latex]

[latex]\mathrm{3sin^2(x)+1-sin^2(x)=4\to sin^2(x)=\frac{3}{2}}[/latex]

[latex]\mathrm{Mas,f(x)=sin^2(x)\in [0,1],logo,sin^2(x)=\frac{3}{2}\notin\mathbb{R}}[/latex]