Question

Find the exact value of sin2Θ if tanΘ=
`(-2√(5) )/(5)` and 90 < Θ < 180.

Answer 1

`sin 2 \theta = (4 \sqrt5)/(9)`

Explanation:

Identities used :

`1. sin^2 \theta = 1 - cos^2\theta\\\\2. tan^2 \theta + 1 = sec^2 \theta\\\\3.cos \theta = (1)/(sec \theta) \\\\4. sin 2 \theta = 2 sin\theta cos\theta`

`Given \ tan \theta = -(2 \sqrt5)/(5)\\`

`we \ know \ tan ^2 \theta + 1 = sec^2\theta`

`(-(2 \sqrt 5)/(5))^2 + 1 = sec^2 \theta\\\\(4 * 5)/(25 ) + 1 = sec^2 \theta \\\\(20 + 25)/( 25) = sec^2 \theta \\\\(45)/(25) = sec^2 \theta \\\\sec \theta = \sqrt{ (45)/(25)}\\\\sec \theta = \sqrt{ (9 * 5)/(5 * 5)} \\\\sec \theta = \frac{3 \sqrt {5}}{5}\\\\`

`We \ know \ cos \theta = (1)/( sec \theta)`

`So , \cos \theta = (5)/(3 \sqrt5)`

`Next \ sin^2 \theta = 1 - cos^2 \theta`

`= 1 - ((5)/(3 \sqrt5))^2\\\\=1 - (25)/(9 * 5)\\\\=1 - (25)/(45)\\\\=(45 - 25)/(45)\\\\=(20)/(45)\\\\=(4)/(9)`

`sin \theta = \sqrt{ (4)/(9)} = (2)/(3)`

`Find \ sin 2 \theta`

`sin 2 \theta = 2 (sin \theta * cos \theta)`

`= 2 * (2)/(3) * (5)/(3\sqrt5)\\\\= (4 * \sqrt5 * \sqrt5)/(9 * \sqrt 5)\\\\=(4 \sqrt5)/(9)`