Can someone help me with fully illustrated steps for the following please?:
Solve the following trigonometric equation. Give the exact value(s).
2 sin^2 (x) - 3 cos (x) = 3 x ∈ [0, π]
Any help would be greatly appreciated!
Thanks in advance!
Can someone help me with fully illustrated steps for the following please?:
Solve the following trigonometric equation. Give the exact value(s).
2 sin^2 (x) - 3 cos (x) = 3 x ∈ [0, π]
Any help would be greatly appreciated!
Thanks in advance!
Hello, s3a!
Solve the following trigonometric equation. Give the exact value(s).
. . $\displaystyle 2\sin^2\!x - 3\cos x \:=\:3 \qquad x \in [0, \pi]$
$\displaystyle \text{We have: }\;2\underbrace{\sin^2\!x} - 3\cos x \;=\;3$
. . . . . $\displaystyle 2\overbrace{(1-\cos^2\!x)} - 3\cos x \;=\;3$
. . which simplifies to: .$\displaystyle 2\cos^2\!x + 3\cos x + 1 \:=\:0 $
. . which factors: .$\displaystyle (\cos x + 1)(2\cos x + 1) \:=\:0$
And we have two equations to solve:
. . $\displaystyle \cos x +1\:=\:0 \quad\Rightarrow\quad \cos x \:=\:-1 \quad\Rightarrow\quad\boxed{ x \:=\:\pi}$
. . $\displaystyle 2\cos x + 1 \:=\:0 \quad\Rightarrow\quad \cos x \:=\:-\tfrac{1}{2} \quad\Rightarrow\quad\boxed{ x \:=\:\tfrac{2\pi}{3}}$