Question

If tanθ= -3/4 and θ is in quadrant IV, cos2θ=

33/25
-17/25
32/25
7/25
24/25

Answer 1

Recall that

`\cos2\theta=2\cos^2\theta-1`

and

`\tan^2\theta+1=\sec^2\theta=\frac1{\cos^2\theta}`

Then

`\cos2\theta=\frac2{\tan^2\theta+1}-1\implies\cos2\theta=\boxed{\frac7{25}}`

Answer 2

`cos2\theta=(7)/(25)`

Explanation:

This is a question of Trigonometric Identities. In addition to this, In quadrant IV the cosine of the angle is naturally negative. This explains the negative value for
`tan\theta=-3/4`

The double angle formula

Let's choose a convenient identity, for the double angle
`cos2\theta`

`\\tan \theta=-3/4 \\ cos2\theta =cos^(2)\theta -sen^(2)\theta\\cos2\theta =2cos^(2)\theta-1\\\\1+tan\theta^(2) =sec^(2)\theta\\\\ 1+((-3)/(4))^(2) =(1)/(cos^2 \theta) \\\\(25)/(16)=(1)/(cos^(2)\theta)\\ cos^(2)\theta=(16)/(25)`

Finally, we can plug it in:

`cos2\theta =2cos^(2)\theta -1\\cos2\theta =2\left ( (16)/(25) \right )-1 \Rightarrow cos2\theta=(7)/(25)`