Respuesta :

semsee45

Answer:

[tex]\cos \theta =\dfrac{\sqrt{21}}{5}[/tex]

Step-by-step explanation:

Given:

  • [tex]\sin \theta=\dfrac{2}{5}[/tex]
  • [tex](\sin x)^2+(\cos x)^2=1[/tex]

Substitute the given value of sin θ into the given identity and solve for cos θ:

[tex]\begin{aligned}(\sin x)^2+(\cos x)^2 & =1\\\implies (\sin \theta)^2+(\cos \theta)^2 & =1\\\left(\dfrac{2}{5}\right)^2+(\cos \theta)^2 & =1\\\left(\dfrac{2^2}{5^2}\right)+(\cos \theta)^2 & =1\\\dfrac{4}{25}+(\cos \theta)^2 & =1\\(\cos \theta)^2 & =1-\dfrac{4}{25}\\(\cos \theta)^2 & =\dfrac{21}{25}\\\cos \theta & =\sqrt{\dfrac{21}{25}}\\\cos \theta & =\dfrac{\sqrt{21}}{\sqrt{25}}\\\cos \theta & =\dfrac{\sqrt{21}}{5}\end{aligned}[/tex]

beautifuldiasaster

[tex]\large\bold{SOLUTION} \\ [/tex]

The Pythagorean Identity states that:

  • (sin x)² + (cos x)² = 1

Given:

[tex] \qquad\sf \hookrightarrow \: \boxed {\sf{\sin \theta=\dfrac{2}{5}, find \: \cos \theta}}[/tex]

Substitute the given value of sin θ

[tex]\begin{gathered}\begin{aligned}\sf\implies(\sin x)^2+(\cos x)^2 & =1\\\\\sf\implies (\sin \theta)^2+(\cos \theta)^2 & =1\\\\\sf\implies\left(\dfrac{2}{5}\right)^2+(\cos \theta)^2 & =1\\\\\sf\implies\left(\dfrac{2^2}{5^2}\right)+(\cos \theta)^2 & =1\\\\\sf\implies\dfrac{4}{25}+(\cos \theta)^2 & =1\\\\\sf\implies(\cos \theta)^2 & =1-\dfrac{4}{25}\\\\\sf\implies(\cos \theta)^2 & =\dfrac{21}{25}\\\\\sf\implies\cos \theta & =\sqrt{\dfrac{21}{25}}\\\\\sf\implies\cos \theta & =\dfrac{\sqrt{21}}{\sqrt{25}}\\\\\sf\bf\implies\cos \theta & ={\pmb{\dfrac{\sqrt{21}}{5}}}\end{aligned}\end{gathered} [/tex]

[tex] \underline{ \rule{185pt}{3pt}}[/tex]

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