Question

Kąt a jest ostry i sinalfa= 1/3. Oblicz 4-3tg2alfa.

Answer 1

Dane z rysunku.

[tex]\sin\alpha=\dfrac{1}{3}[/tex]

[tex]\mbox{tg }\alpha=\dfrac{1}{x}[/tex]

Obliczenie [tex]x[/tex]:

[tex]1^2+x^2=3^2\\1+x^2=9\\x^2=9-1\\x^2=8\\x=\sqrt{8}\\x=2\sqrt{2}[/tex]

Wyliczenie tangensa kąta alfa:

[tex]\mbox{tg }\alpha=\dfrac{1\cdot\sqrt{2}}{2\sqrt{2}\cdot\sqrt{2}}=\dfrac{\sqrt{2}}{2\cdot2}=\dfrac{\sqrt{2}}{4}[/tex]

Obliczenie wartości wyrażenia:

[tex]4-3\mbox{tg}^2\alpha=4-3\cdot\left(\dfrac{\sqrt{2}}{4}\right)^2=4-3\cdot\dfrac{2}{16}=4-3\cdot\dfrac{1}{8}=4-\dfrac{3}{8}=3\dfrac{5}{8}[/tex]