Suppose that 31 degrees, 65
degrees, and 𝜃 are the direction angles of a vector. Which of the following, to the
nearest hundredth, is 𝜃? Is it (A) 72.88 degrees, (B) 84.00
degrees, (C) 85.03 degrees, or (D) 264.00 degrees?
We recall that if the three
direction angles of a vector are 𝛼, 𝛽, and 𝛾, then cos squared 𝛼 plus cos
squared 𝛽 plus cos squared 𝛾 is equal to one. In this question, we will
substitute 𝛼 equal to 31 degrees, 𝛽 equal to 65 degrees, and 𝛾 equal to 𝜃. This gives us the equation
shown. We recall that cos squared of 31
degrees is the same as cos of 31 degrees all squared. This is how we type this into a
scientific calculator. cos squared 31 degrees plus cos
squared 65 degrees is equal to 0.91334 and so on. We can then subtract this from both
sides of our equation, giving us cos squared 𝜃 is equal to 0.08665 and so on.
We then square root both sides of
this equation. cos 𝜃 is equal to 0.29437 and so
on. Finally, we take the inverse cos of
both sides, such that 𝜃 is equal to 72.8797 and so on. We want the answer to the nearest
hundredth. This means that the nine in the
thousandths column is the deciding number. Rounding up gives us 72.88
degrees. Therefore, the correct answer is
option (A). The three direction angles of the
vector are 61 degrees, 65 degrees, and 72.88 degrees.