Mathematics: Solve 2sin2(x) – 5sin(x) – 3 = 0. let u = sin(x). which of the following is equivalent to the given equation?

Solve 2sin2(x) – 5sin(x) – 3 = 0. let u = sin(x).

Ответ:

caeyanij

10.03.2022

Mathematics

$x=(-1)^{k+1}\cdot \dfrac{\pi}{6}+\pi k,\ k\in Z$

Step-by-step explanation:

Consider the equation

$2\sin^2x-5\sin x-3=0$

Substitute

$u=\sin x$

into the equation:

2u^2-5u-3=0

This is quadratic equation, where

$D=(-5)^2-4\cdot 2\cdot (-3)=25+24=49\\ \\\sqrt{D}=7$

Hence,

$u_1=\dfrac{-(-5)-\sqrt{D}}{2\cdot 2}=\dfrac{5-7}{4}=-\dfrac{1}{2}\\ \\u_2=\dfrac{-(-5)+\sqrt{D}}{2\cdot 2}=\dfrac{5+7}{4}=3$

Now, use the substitution again:

1. When $u_1=-\dfrac{1}{2},$ then

$\sin x=-\dfrac{1}{2}\\ \\x=(-1)^k\sin^{-1}\left(-\dfrac{1}{2}\right)+\pi k,\ k\in Z\\ \\x=(-1)^k\cdot \left(-\dfrac{\pi}{6}\right)+\pi k,\ k\in Z\\ \\x=(-1)^{k+1}\cdot \dfrac{\pi}{6}+\pi k,\ k\in Z$

2. When u_2=3, then

$\sin x=3$

This equation has no solutions, because the range of $\sin x$ is [-1,1] and 3>1.

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Solve 2sin2(x) – 5sin(x) – 3 = 0. let u = sin(x). which of the following is equivalent to the given equation?

Ответ:

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