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Sandra knows the Pythagorean identity sin 2 ⁡ θ + cos 2 ⁡ θ = 1 . If she is told that 0 ≤ θ ≤ π 2 and cos ⁡ θ = 5 12, what will she get when she correctly calculates tan ⁡ θ ?

User Jacek
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1 Answer

3 votes

Answer:


\text{tan}(\theta)=(√(119))/(5)

Explanation:

We have been given that Sandra knows the Pythagorean identity
\text{sin}^2(\theta)+\text{cos}^2(\theta)=1. She is told that
0\leq \theta\leq (\pi)/(2) and
\text{cos}(\theta)=(5)/(12).

First of all, we will find value of sine theta using the given identity.


\text{sin}^2(\theta)+\text{cos}^2(\theta)=1


\text{sin}^2(\theta)+((5)/(12))^2=1


\text{sin}^2(\theta)+(25)/(144)=1


\text{sin}^2(\theta)+(25)/(144)-(25)/(144)=1-(25)/(144)


\text{sin}^2(\theta)=(144-25)/(144)


\text{sin}^2(\theta)=(119)/(144)


\text{sin}(\theta)=\sqrt{(119)/(144)}


\text{sin}(\theta)=(√(119))/(12)


\text{tan}(\theta)=\frac{\text{sin}(\theta)}{\text{cos}(\theta)}


\text{tan}(\theta)=((√(119))/(12))/((5)/(12))


\text{tan}(\theta)=(√(119)*12)/(5*12)=(√(119))/(5)

Therefore,
\text{tan}(\theta)=(√(119))/(5).

User Giridharan
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4.1k points