Video Transcript
Find the solution set of 𝜃 that
satisfies sec 𝜃 is equal to negative root two given that 𝜃 is greater than or
equal to zero degrees and less than 360 degrees.
In order to answer this question,
we begin by recalling that the secant function is the reciprocal of the cosine
function, such that sec 𝜃 is equal to one over cos 𝜃. This means that we can rewrite the
equation as one over cos is equal to negative root two. And in turn, cos 𝜃 is equal to
negative one over root two. We can solve this equation using
inverse trigonometric functions together with our knowledge of the CAST diagram and
special angles.
Since the value of cos 𝜃 is
negative and lies between zero and negative one, we know we will have solutions in
the second and third quadrants. At this stage, we may recall that
the cos of 45 degrees is equal to one over root two. We can use this fact together with
the symmetry of the cosine function in the CAST diagram to find the solutions of our
equation in the second and third quadrants.
The values of 𝜃 that satisfy the
equation cos 𝜃 is equal to negative one over root two are 𝜃 is equal to 180
degrees minus 45 degrees and 𝜃 is equal to 180 degrees plus 45 degrees. This gives us our two solutions of
135 degrees and 225 degrees.
It is worth noting we could also
have solved the equation by taking the inverse cosine of both sides, such that 𝜃 is
equal to the inverse cos of negative one over root two. Ensuring that our calculator is in
degree mode, typing in the right-hand side would have given us our first solution of
135 degrees. We could then have found the second
solution by subtracting this value from 360 degrees, giving us our second solution
of 225 degrees.
Either way, the solution set of 𝜃
that satisfies sec 𝜃 is equal to negative root two where 𝜃 is greater than or
equal to zero and less than 360 degrees is 135 degrees and 225 degrees.
