Video Transcript
Consider the equation five sec 22
degrees minus 𝑥 minus 𝐴 equals three sec 22 degrees minus 𝑥. Which of the following must be
true? (A) 𝐴 equals negative two cosec 22
degrees plus 𝑥. (B) 𝐴 equals two cosec 68 degrees
plus 𝑥. (C) 𝐴 equals two cosec 68 degrees
minus 𝑥. (D) 𝐴 equals two cosec 112 degrees
plus 𝑥. Or (E) 𝐴 equals two cosec two
degrees minus 𝑥.
As we’re trying to solve for 𝐴, a
good first step would be to get 𝐴 by itself. We can do this by subtracting five
sec 22 degrees minus 𝑥 from both sides of the equation. Negative 𝐴 equals negative two sec
22 degrees minus 𝑥. Multiplying through by negative
one, we see that 𝐴 equals two times sec of 22 degrees minus 𝑥.
We have 𝐴 in terms of the secant,
and all of our options are 𝐴 written as a cosecant term. We can rewrite the secant term as a
cosecant term using the cofunction identity that the sec of 90 degrees minus 𝜃
equals the cosec of 𝜃. In this case, we want to let 22
degrees minus 𝑥 be equal to 90 degrees minus 𝜃. And we want to solve for 𝜃. Subtracting 90 degrees from both
sides of the equation, we see that negative 68 degrees minus 𝑥 equals negative
𝜃. Therefore, 𝜃 will be equal to 68
degrees plus 𝑥.
This means we’ve been able to show
that the sec of 22 degrees minus 𝑥 will be equal to the cosec of 68 degrees plus
𝑥. Two times the sec of 22 degrees
minus 𝑥 will then be equal to two times the cosec of 68 degrees plus 𝑥. Therefore, 𝐴 equals two times
cosec of 68 degrees plus 𝑥, which is option (B) here.
