Question

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 ?

Answer 1

`\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)`.