Video Transcript
Find the solution set of tan 𝑥 plus tan seven degrees plus tan 𝑥 multiplied by tan
seven degrees equals one, where 𝑥 is greater than zero degrees and less than 360
degrees.
In this question, we want to solve a trigonometric equation involving the tangent
function, giving all the solutions between zero and 360 degrees. It may not be immediately obvious where we start here. However, it is worth recalling some of the trigonometric identities to see if this
helps.
We recall that tan of 𝐴 plus 𝐵 is equal to tan 𝐴 plus tan 𝐵 over one minus tan 𝐴
multiplied by tan 𝐵. We will therefore begin by trying to rewrite our equation in this form, where 𝐴 is
equal to 𝑥 and 𝐵 is equal to seven degrees. Subtracting tan 𝑥 tan seven degrees from both sides of the equation, we have tan 𝑥
plus tan seven degrees is equal to one minus tan 𝑥 multiplied by tan seven
degrees. We can then divide both sides of the equation by one minus tan 𝑥 tan seven degrees
as shown. The right-hand side simplifies to give us one. And the left-hand side is now written in the same form as the identity we mentioned
earlier. This means that tan of 𝑥 plus seven degrees is equal to one.
Recalling our special angles, we know that tan of 45 degrees equals one. And since tan of 𝜃 plus 180 degrees equals tan 𝜃, tan of 45 degrees plus 180
degrees equals one. And as such tan of 225 degrees equals one. We could also see this from our knowledge of the CAST diagram and the unit circle,
where the tangent function is positive in the first and third quadrants.
We therefore have two solutions for 𝑥 between zero and 360 degrees. Either 𝑥 plus seven degrees equals 45 degrees or 𝑥 plus seven degrees is equal to
225 degrees. Subtracting seven degrees from both sides of both equations, we have 𝑥 equals 38 or
218 degrees. The solution set of the equation tan 𝑥 plus tan seven degrees plus tan 𝑥 multiplied
by tan seven degrees equals one, where 𝑥 is greater than zero degrees and less than
360 degrees, is 38 degrees and 218 degrees.
