Video Transcript
Find the set of values satisfying
root three cot 𝜃 equals one given 𝜃 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 𝜃
is equal to one over the tan of 𝜃. We can rearrange the equation we’re
given by firstly dividing both sides by root three. The cot of 𝜃 is therefore equal to
one over root three. As tan 𝜃 is the reciprocal of
this, the tan of 𝜃 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 𝜃 equals root three is 𝜃 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 𝜃 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 𝜃 is equal to the tan of
angle 𝜃. 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.
