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Click here to lớn see ALL problems on Trigonometry-basicsQuestion 1018648: Find the exact solutions for 2sinx-1=cscx Answer by Theo(12188)
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(Show Source): You can put this solution on YOUR website! your equation is:2sin(x) - 1 = csc(x)since csc(x) = 1/sin(x), this equation becomes:2sin(x) - 1 = 1/sin(x)multiply both sides of this equation by sin(x) to lớn get:sin(x) * (2 * sin(x) - 1) = 1distribute the multiplication lớn get:2 * sin^2(x) - sin(x) = 1subtract 1 from both sides of the equation lớn get:2 * sin^2(x) - sin(x) - 1 = 0this is a quadratic equation that you can solve by using the quadratic formula.you will get sin(x) = 1 or sin(x) = -1/2.since csc(x) = 1/sin(x), then csc(x) = 1 or csc(x) = -2.when sin(x) = 1, then csc(x) - 1, và the equation of 2 * sin(x) - 1 = csc(x) becomes 2 * 1 - 1 = 1.simplify this to get 2-1 = 1 which results in 1 = 1.this confirms that sin(x) = 1 is a solution to lớn the equation.when sin(x) = -1/2, then csc(x) = -2, and the equation of 2 * sin(x) - 1 = csc(x) becomes 2 * -1/2 - 1 = -2 which becomes -1 - 1 = -2 which results in -2 = -2.this confirms that sin(x) = -1/2 is also a solution lớn the equation.your solution is therefore:sin(x) = 1 or sin(x) = -1/2.when sin(x) = 1, x = 90 degrees.when sin(x) = -1/2, your calculator will tell you that x is equal lớn -30 degrees.-30 degrees is in quadrant 4.convert that khổng lồ a positive angle by adding 360 to it and you will get x = 330 degrees.the sin is negative in quadrant 3 and quadrant 4.your reference angle is 30 degrees, which is the equivalent angle in the first quadrant.your reference angle becomes 180 + 30 = 210 degrees in quadrant 3.your solution is that:sin(x) = 1 or sin(x) = -1/2.x = 90 degrees or 210 degrees or 330 degrees.since it was not specified during what interval, then your solution is:sin(x) = 1 or sin(x) = -1/2this also implies csc(x) = 1 or csc(x) = -2.your angle will be:90 degrees plus or minus k * 360.210 degrees plus or minus k * 360.330 degrees plus or minus k * 360.k is a positive integer from 1 khổng lồ infinity.the graph of your equation will be shown below.there will be 4 graphs.1 close in view khổng lồ show you the details between 0 và 360 degrees.2 far out views khổng lồ show you the details between -720 & 720 degrees.both graphs are the same, but with different intersections shown because the intersections were too close to show them all in 1 graph.1 far far out view that goes further than -720 to lớn 720 degrees but without the details.this graph will have a horizontal đen line at y = 1 & y = -2 lớn show you where the intersection points are.this graph shows you that the possible angles for the solution go on indefinitely in both directions.here you go:close in view.....far out view 1....far out view 2....far far out view....
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