Question

If θ is a first quadrant angle in standard position with P(u,v) = (3,4) evaluate tan 2 θa) 24/7b) -8/7c) -24/7

Answer 1

We are told that

`\theta\text{ is in the first quadrant}`

and the standard position with P(u,v) = (3,4)

We want to find

`\tan 2\theta`

First, We will draw the standard position

Notice that from the diagram above, the (3,4) indicates that the horizontal distance is 3 units and the vertical distance is 4 units

From the diagram above

Using SOHCAHTOA

`\begin{gathered} \tan \theta=(opposite)/(adjacent) \\ \tan \theta=(4)/(3) \end{gathered}`

We are left with computing

`\tan 2\theta`

The addition formula for tan(A+B)

`\begin{gathered} \tan (A+B)=(\tan A+\tan B)/(1-\tan A\tan B) \\ \text{Let A = B = }\theta \\ \tan (\theta+\theta)=(\tan\theta+\tan\theta)/(1-\tan^2\theta) \\ \tan 2\theta=(2\tan\theta)/(1-\tan^2\theta) \end{gathered}`

We now imput the values of tan

`\begin{gathered} \tan 2\theta=(2\tan\theta)/(1-\tan^2\theta) \\ \tan 2\theta=(2((4)/(3)))/(1-((4)/(3))^2) \\ \tan 2\theta=((8)/(3))/(1-(16)/(9)) \\ \tan 2\theta=((8)/(3))/((9)/(9)-(16)/(9)) \\ \tan 2\theta=((8)/(3))/(-(7)/(9)) \\ \tan 2\theta=(8)/(3)*-(9)/(7) \\ \tan 2\theta=-(24)/(7) \end{gathered}`

Therefore, the correct answer is -24/7

Option C