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Question Video: Solving a Trigonometric Equation Involving the Cosecant Function Mathematics • 10th Grade

Find the set of values satisfying โˆš(3) csc (๐œƒ) โˆ’ 2 = 0 given that 0ยฐ โ‰ค ๐œƒ < 360ยฐ.

02:37

Video Transcript

Find the set of values satisfying root three csc ๐œƒ minus two equals zero, given that ๐œƒ is greater than or equal to zero degrees and less than 360 degrees.

In order to solve the trigonometric equation in this question, we begin by adding two to both sides. This gives us root three csc ๐œƒ is equal to two. Next, we divide through by root three, giving us csc ๐œƒ is equal to two over root three. We recall that the cosecant function is the reciprocal of the sine function such that csc ๐œƒ is equal to one over sin ๐œƒ. This means that one over sin ๐œƒ is equal to two over root three. And sin ๐œƒ must therefore be equal to root three over two.

We are now in a position to solve the equation using the inverse trigonometric functions together with our knowledge of the CAST diagram and special angles. Since sin ๐œƒ is equal to a positive value, we know that if there are any solutions, they will lie in the first and second quadrants. We may recall that sin of 60 degrees is equal to root three over two. Alternatively, we could take the inverse sine of both sides of our equation, giving us ๐œƒ is equal to the inverse sin of root three over two. Ensuring that our calculator is in degree mode, typing in the right-hand side, we see that ๐œƒ is equal to 60 degrees. This is the solution in the first quadrant. And using the symmetry of the sine function in the CAST diagram, we have a second solution equal to 180 degrees minus 60 degrees, which is equal to 120 degrees.

The set of values satisfying root three csc ๐œƒ minus two equals zero, given that ๐œƒ is greater than or equal to zero and less than 360, are 60 degrees and 120 degrees.

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