Question

If x = 6 tan θ, write the expression θ 2 − sin 2θ 4 in terms of just x. (Simplify the double angle using a double-angle identity.)

Answer 1

Given that


`x=6\tan\theta \\ \\ \Rightarrow\tan\theta= (x)/(6)`

Consider a right triangle with angle θ, the side opposite the angle θ is x and that side adjacent angle x is 6.

Then the hypothenus is given by
`√(x^2+36)`


`\sin{ (2\theta)/(4)} =\sin{2\left( (\theta)/(2) \right)} \\ \\ =2\sin{(\theta)/(2)}\cos{(\theta)/(2)}=2\cdot \sqrt{(1-cos(\theta))/(2)} \cdot \sqrt{(1+cos(\theta))/(2)} \\ \\ =√(1-\cos^2\theta)`

Recall that
`\cos\theta= (adjacent)/(hypothenuse) = (6)/(√(x^2+36))`

Therefore,


`(\theta)/(2) -\sin (2\theta)/(4) = (1)/(2) \tan^(-1) (x)/(6) -√(1-\cos^2\theta) \\ \\ =(1)/(2) \tan^(-1) (x)/(6)-\sqrt{1-\left((6)/(√(x^2+36))\right)^2}=(1)/(2) \tan^(-1) (x)/(6)-\sqrt{1-(36)/(x^2+36)} \\ \\ =(1)/(2) \tan^(-1) (x)/(6)-\sqrt{(x^2)/(x^2+36)}=\bold{(1)/(2) \tan^(-1) (x)/(6)-(x)/(√(x^2+36))}`