Question

2sinsquaretheta-4=5costheta

Answer 1

Note:

let theta=a

Given

$\sf 2sin^2a-4=5cosa$

To find

The value of a

Explanation:

consider

$= > 2 {sin}^{2} a - 4 = 5cosa$

$\boxed{\bf sin^2a+cos^2a=1}$

$\boxed{\bf sin^2a=1-cos^2a}$

$= > 2(1 - {cos}^{2} a) - 4 = 5cosa$

$= > 2 - 2 {cos}^{2} a - 4 = 5cosa$

$= > 2 {cos}^{2} a + 5cosa + 2 = 0$

Splitting the middle term

$= > 2 {cos}^{2} a + 4cosa + cosa + 2 = 0$

$= > 2cosa(cosa + 2) + 1(cosa + 2) = 0$

$= > (2cosa + 1)(cosa + 2) = 0$

$= > 2cosa + 1 = 0 \\ = > cosa = \dfrac{ - 1}{2}$

$= > cosa + 2 = 0 \\ = > cosa = - 2 \: (negligible)$

As cosa range is [-1,1]

$= > cosa = \dfrac{ - 1}{2}$

$\boxed{\sf a=120\degree}$

Important formulas

$= > {cos}^{2} a + {sin}^{2} a = 1 \\ \\ = > {sec}^{2} a - {tan}^{2} a = 1 \\ \\ = > {cosec}^{2} a - {cot}^{2} a = 1$