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Anuta_ua [19.1K]
3 years ago
15

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 ?
Mathematics
1 answer:
slavikrds [6]3 years ago
5 0

Answer:

\text{tan}(\theta)=\frac{\sqrt{119}}{5}

Step-by-step 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 \frac{\pi}{2} and \text{cos}(\theta)=\frac{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)+(\frac{5}{12})^2=1

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

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

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

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

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

\text{sin}(\theta)=\frac{\sqrt{119}}{12}  

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

\text{tan}(\theta)=\frac{\frac{\sqrt{119}}{12}}{\frac{5}{12}}

\text{tan}(\theta)=\frac{\sqrt{119}*12}{5*12}=\frac{\sqrt{119}}{5}

Therefore, \text{tan}(\theta)=\frac{\sqrt{119}}{5}.

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