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Question

Question: How do you solve \(2\sin x+1=0?\)...

How do you solve 2sinx+1=0?2\sin x+1=0?

Explanation

Solution

The given function in the question is the trigonometric function of the sine function. The domain and the range of the sine function is all real numbers and the range is 1θ1-1\le \theta \le 1.
The inverse of the sine function is the cosec function sinx=1cscx\sin x=\dfrac{1}{\csc x}.
In the first quadrant, the sine function increase 00to 11
In the second quadrant, the sine function decreases by 11 to 00
In the third quadrant, the sine function decreases by 00 to 1-1
In the fourth quadrant, the sine function increase by 1-1 to 00
The values of some θ\theta will help to simplify the sine function-

θ\theta 00{{0}^{0}}300{{30}^{0}}450{{45}^{0}}600{{60}^{0}}900{{90}^{0}}1200{{120}^{0}}1350{{135}^{0}}1500{{150}^{0}}1800{{180}^{0}}
Sine001/2{}^{1}/{}_{2}1/2{}^{1}/{}_{\sqrt{2}}3/2  {\sqrt{3}}/{2}\;113/2  {\sqrt{3}}/{2}\;1/2{}^{1}/{}_{\sqrt{2}}1/2{}^{1}/{}_{2}00

Complete step by step solution:
The given function is 2sinx+1=02\sin x+1=0
To simplify it we will add 1-1 in the left side and the right side of the function then it will be written as
2sinx+1+(1)=1\Rightarrow 2\sin x+1+(-1)=-1
2sinx=1\Rightarrow 2\sin x=-1
To more simplify the above term we will divide in the left side and right side of the function by 22
2sinx2=12\Rightarrow \dfrac{2\sin x}{2}=\dfrac{-1}{2}
The above term will be sinx=12\sin x=\dfrac{-1}{2}
x=sin1(12)\Rightarrow x={{\sin }^{-1}}\left( \dfrac{-1}{2} \right)
x=sin1(12)\Rightarrow x=-{{\sin }^{-1}}\left( \dfrac{1}{2} \right)
x=π6\Rightarrow x=-\dfrac{\pi }{6}
As we know that in the third and fourth quadrant it will be negative.
In the third quadrant is will be π+π6=7π6\pi +\dfrac{\pi }{6}=\dfrac{7\pi }{6}
And in the fourth quadrant, it will be 2ππ6=11π62\pi -\dfrac{\pi }{6}=\dfrac{11\pi }{6}

Hence, the answer will be π6-\dfrac{\pi }{6} or 7π6\dfrac{7\pi }{6}and 11π6\dfrac{11\pi }{6}

Note:
According to the ‘ASTC’ rule, the sine function always gets a positive value in the first quadrant and the second quadrant.
The sine function is commonly used in periodic motion, light waves, sound, etc.