Question

Find the value of cos 2 theta if theta is in the first quadrant and tan theta =15/4

Answer 1

`-(209)/(241)`

Explanation:

Given that:

`tan\theta = (15)/(4)`

Also,
`\theta` is in first quadrant.

To find:

`cos2\theta = ?`

Solution:

Let us have a look at the cosine of twice the angle in terms of tangent of the angle.

Suppose, we are given the value of
`tanA`, then the formula can be written as:

`cos2A = (1-tan^2A)/(1+tan^2A)`

In terms of
`\theta`, we can re-write the formula as:

`cos2\theta = (1-tan^2\theta)/(1+tan^2\theta)\\\Rightarrow cos2\theta = (1-((15)/(4))^2)/(1+((15)/(4))^2)\\\Rightarrow cos2\theta = (1-(225)/(16))/(1+(225)/(16))\\\Rightarrow cos2\theta = ((16-225)/(16))/((16+225)/(16))\\\Rightarrow cos2\theta = \bold{-(209)/(241)}`

`\theta` is in first quadrant, but
`2\theta` can be in the second quadrant therefore, we have a negative value of our answer i.e.
`cos2\theta`.

Therefore, the answer is:

`-(209)/(241)`