Mathematics: Find the exact value of sin2Θ if tanΘ= and 90 Θ 180.

i wont stopStep-by-step explanation:...

Find the exact value of sin2Θ if tanΘ= and 90 Θ

Ответ:

rozlynnv22

11.07.2021

Mathematics

$sin 2 \theta = \frac{4 \sqrt5}{9}$

Step-by-step explanation:

Identities used :

$1. sin^2 \theta = 1 - cos^2\theta\\\\2. tan^2 \theta + 1 = sec^2 \theta\\\\3.cos \theta = \frac{1}{sec \theta} \\\\4. sin 2 \theta = 2 sin\theta cos\theta$

$Given \ tan \theta = -\frac{2 \sqrt5}{5}\\$

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

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