Find the solution set of sin x = csc x in the interval (0°, 360°). [Answer(s) should be correct to the nearest degree.]

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justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to solve sin x = cosec x for x in the interval (0, 360).

sin x = cosec x

=> sin x = 1/sin x

=> (sin x)^2 = 1

=> sin x = 1 and sin x = -1

x = arc sin 1 and x = arc sin (-1)

=> x = 90 degrees and x = 270 degrees.

The solution of the equation in the interval (0, 360) is x = 90 degrees and x = 270 degrees.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We must find just the solutions that belong to the range (0 , 2pi).

We'll recall the identity csc x = 1/sin x and we'll substitute csc x by the equivalent ratio, into the given identity.

sin x = 1/sin x

We'll multiply by sin x both sides:

(sin x)^2 - 1 = 0

We'll recognize the difference of squares:

(sin x - 1)(sin x + 1) = 0

We'll cancel each factor:

sin x = 1 => x = pi/2 radians or 90 degrees.

sin x = -1 => x = 3pi/2 radians or 270 degrees.

The solutions of the equation in degrees, over the range (0 , 360) are: {90, 270}.

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