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Question Video: Solving Trigonometric Equations Involving Special Angles Mathematics • 10th Grade

Find the set of values satisfying โˆš3 cot ๐œƒ = 1 given 0ยฐ < ๐œƒ < 360ยฐ.

02:15

Video Transcript

Find the set of values satisfying root three cot ๐œƒ equals one given ๐œƒ is greater than zero and less than 360 degrees.

We can solve this problem using our knowledge of the reciprocal trigonometric functions. We recall that the cot of angle ๐œƒ is equal to one over the tan of ๐œƒ. We can rearrange the equation weโ€™re given by firstly dividing both sides by root three. The cot of ๐œƒ is therefore equal to one over root three. As tan ๐œƒ is the reciprocal of this, the tan of ๐œƒ equals root three. We can find the principal angle here using our knowledge of special angles. We know that the tan of 60 degrees is equal to root three. This means that one solution to the equation tan ๐œƒ equals root three is ๐œƒ equals 60 degrees. We are asked to find all the solutions between zero and 360 degrees. We can do this by sketching a CAST diagram.

As the tan of angle ๐œƒ is positive, there will be solutions in the first and third quadrants. We have already found that the solution in the first quadrant is equal to 60 degrees. Using the periodicity of the tangent function, we know that the tan of 180 degrees plus ๐œƒ is equal to the tan of angle ๐œƒ. Our second solution can therefore be calculated by adding 60 degrees to 180 degrees. This is equal to 240 degrees. Any further solutions found by adding or subtracting multiples of 180 degrees will be outside the required interval. So the solution set is 60 degrees and 240 degrees.

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