Video Transcript
Find the value of 𝑥 given tan 78
degrees equals cot 𝑥, where zero is less than 𝑥 is less than 180 degrees.
In this question, we have been
given a trigonometric equation involving the tangent and cotangent functions that we
need to solve for 𝑥. Now, our first instinct may be to
just rewrite cot 𝑥 in terms of tan 𝑥, to rearrange the equation in terms of tan
𝑥, and to apply the inverse tan function to find 𝑥. However, this requires using a
calculator when in fact this question can be solved without one if we just make use
of the right identity.
Recall that the cofunction
identities give us useful formulas for 90 degrees minus 𝜃 of each trigonometric
function. In particular, the identity which
tells us that tan of 90 minus 𝜃 equals cot 𝜃 seems like it’ll be relevant for this
question. This is because we can rewrite tan
of 78 degrees in terms of cot 𝜃 by putting it in the correct form.
Let us set tan of 78 degrees to tan
of 90 degrees minus 𝜃. This means 78 degrees equals 90
degrees minus 𝜃, which means 𝜃 is 90 minus 78 degrees, which is 12 degrees. Therefore, using the cofunction
identity, tan of 78 degrees, which is the same as tan of 90 minus 12 degrees, is
equal to cot of 12 degrees.
Now, we have been given that tan 78
degrees equals cot 𝑥. So this is equal to cot 𝑥. So, this tells us that in this case
𝑥 is equal to 12 plus any integer multiple 𝑛 of 180 degrees, because cot is a
periodic function. However, we have also been told
that 𝑥 has to be in the range from zero to 180 degrees. Therefore, the only valid value for
𝑥, given that tan of 78 degrees equals cot 𝑥, is 12 degrees.
