Find the solution set of 𝜃 that
satisfies two tan 𝜃 divided by sec 𝜃 minus one equals zero, where 𝜃 is greater
than zero degrees and less than 360 degrees.
In order to solve the equation in
this question, we will firstly need to simplify the left-hand side. We can do this by recalling some of
our trigonometric identities. Firstly, tan 𝜃 is equal to sin 𝜃
over cos 𝜃. Secondly, we recall that the secant
function is the reciprocal of the cosine function such that sec 𝜃 is equal to one
over cos 𝜃.
Rewriting two tan 𝜃 over sec 𝜃 as
two tan 𝜃 multiplied by one over sec 𝜃, we can then use our two identities to
rewrite this as two sin 𝜃 over cos 𝜃 multiplied by cos 𝜃. We can then divide the numerator
and denominator by cos 𝜃 so that this term simplifies to two sin 𝜃. The equation we are trying to solve
can therefore be written as two sin 𝜃 minus one is equal to zero. Adding one to both sides of this
equation, we have two sin 𝜃 is equal to one. And dividing through by two, we
have sin 𝜃 equals one-half.
We can now use the inverse sine
function together with the CAST diagram and our knowledge of special angles to find
the solutions to this equation. We know that if sin 𝜃 is positive
and lies between zero and one, there will be solutions in the first and second
quadrants. At this stage, we may recall that
the sine of one of our special angles, 30 degrees, is equal to one-half. Alternatively, we can take the
inverse sine of both sides of our equation such that 𝜃 is equal to the inverse sin
of one-half. Typing this into our calculator
would also give us the solution of 30 degrees.
Using the symmetry of the sine
function on our CAST diagram, we see that 𝜃 could also be equal to 180 degrees
minus 30 degrees. This is equal to 150 degrees. We can therefore conclude that the
solution set of two tan 𝜃 divided by sec 𝜃 minus one equals zero, where 𝜃 lies
between zero and 360 degrees, is 30 degrees and 150 degrees.