IME / ITA ⇒ (EsPCEx - 2002) Trigonometria: Relações Trigonométricas Tópico resolvido

[Moreira]

O valor de [tex3]\cos x + \text{sen} x,[/tex3] sabendo que [tex3]3\cdot \text{sen} x + 4\cdot \cos x = 5,[/tex3] é:

a) [tex3]\frac{3}{4}[/tex3]
b) [tex3]\frac{4}{5}[/tex3]
c) [tex3]1[/tex3]
d) [tex3]\frac{6}{5}[/tex3]
e) [tex3]\frac{7}{5}[/tex3]

[matbatrobin]

• [tex3]\begin{cases}\cos x=\sqrt{1-\text{sen}^2x}\\3\cdot \text{sen} x + 4\cdot \cos x = 5 \end{cases} \Longrightarrow3\cdot \text{sen} x + 4\cdot \sqrt{1-\text{sen}^2x} = 5 \Longrightarrow4\cdot \sqrt{1-\text{sen}^2x}=5-3\cdot \text{sen} x[/tex3]

• [tex3](4\cdot \sqrt{1-\text{sen}^2x})^2=(5-3\cdot \text{sen} x)^2[/tex3]

• [tex3]16\cdot (1-\text{sen}^2x)=25-30\cdot \text{sen} x+9\cdot \text{sen}^2x[/tex3]

• [tex3]25\cdot \text{sen}^2x-30\cdot \text{sen} x+9=0\Longrightarrow\text{sen}=\frac{3}{5}[/tex3]

• [tex3]\cos x=\sqrt{1-\left(\frac{3}{5}\right)^2}=\frac{4}{5}[/tex3]
  
  [tex3]\text{sen} x+\cos x=\frac{3}{5}+\frac{4}{5}=\frac{7}{5}[/tex3]

Letra (e).