Find The Principal And General Solutions Of The Following Equations: Cosec X = -2

Given: Cosec x = -2

\( \Rightarrow \frac{1}{sin x} = -2 \\\Rightarrow sin x = -\frac{1}{2}\)

we know that sin 30 = \( -\frac{1}{2}\)

But we need the negative value of sin x, sin is negative in the III & IV quadrant.

Value in III quadrant = 180° + 30° = 210°

Value in IV quadrant = 360° – 30° = 330°

so principal solutions are

x = 210° = \( \frac{7π}{6}\)

x = 330°

\( \Rightarrow 330° * \frac{π}{180} = \frac{11π}{6}\)

General solution:

Let sinx = sin y—[1]

sin x = \( -\frac{1}{2}\)—[2]

From the above equation [1] and [2], we get;

sin y = \( -\frac{1}{2} \\sin y = sin \frac{7π}{6}\\\Rightarrow y = \frac{7π}{6}\)

Therefore, sin x = sin y

General solution is

\( x = n\Pi + (-1)^{n} y, n \in 1 \\ x = n\Pi + (-1)^{n} \frac{7\Pi}{6}, n \in 1\)

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