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Question: How do you solve \(3\sin x+2=0\)?...

How do you solve 3sin⁡x+2=03\sin x+2=0?

Explanation

Solution

To solve the given equation first we will subtract 2 from both sides of the equation. Then we will divide the obtained equation by 3. We get the value of sin⁡x\sin x. Then by calculating the inverse we get the desired answer.

Complete step by step solution:
We have been given that 3sin⁡x+2=03\sin x+2=0.
We have to solve the given equation.
Now, to solve the given equation first we will subtract 2 from both sides of the equation. Then we will get
⇒3sin⁡x+2−2=0−2\Rightarrow 3\sin x+2-2=0-2
Now, simplifying the above obtained equation we will get
⇒3sin⁡x=−2\Rightarrow 3\sin x=-2
Now, dividing the above obtained equation by 3 we will get
⇒3sin⁡x3=−23\Rightarrow \dfrac{3\sin x}{3}=-\dfrac{2}{3}
Now, simplifying the above obtained equation we will get
⇒sin⁡x=−23\Rightarrow \sin x=-\dfrac{2}{3}
Now, taking an inverse function we will get
⇒x=sin⁡−1(−23)\Rightarrow x={{\sin }^{-1}}\left( -\dfrac{2}{3} \right)
By using the calculator we get the value as
⇒x=−0.729c\Rightarrow x={-0.729}^{c}
Hence above is the required answer.

Note: If we get the general value of sin⁡x\sin x which is given in the trigonometric ratio table, it is easy to calculate the value. We can also solve the question in alternative way as:
We get the value by simplifying the equation as
⇒sin⁡x=−23\Rightarrow \sin x=-\dfrac{2}{3}
Now, we know that there is no direct value given in the trigonometric ratio table. So when we approximate value we will get
⇒x=−41.81∘\Rightarrow x=-41.81{}^\circ
Or we can write this as
⇒x=−180+41.81∘ ⇒x=−138.19 \begin{aligned} & \Rightarrow x=-180+41.81{}^\circ \\\ & \Rightarrow x=-138.19 \\\ \end{aligned}
Hence we get the value of x.