Problem:
What is the degree measure of the acute angle formed by lines with slopes 2 and 1/3?
Answer Choices:
A. 30
B. 37.5
C. 45
D. 52.5
E. 60
Solution:
Consider lines from the origin O to points P(3,1) and Q(−1,3). The lines are perpendicular because their slopes are negative reciprocals of each other. Points P and Q are equidistant from O, so their midpoint lies on the bisector of ∠POQ. The midpoint is
M=(23+(−1),21+3)=(1,2)
and the line OM has slope 2. Angle ∠MOP is formed by lines with slopes 2 and 31 and is half of a right angle, so its measure is (C)45∘.
OR
Define points O(0,0),M(1,2), and P(3,1), so that lines OM and OP have the required slopes. Then OM=12+22=5,MP=(3−1)2+(2−1)2=5, and OP=32+12=10.
Furthermore, OP=2⋅OM. It follows that △OMP is an isosceles right triangle with hypotenuse OP, so ∠MOP=(C)45∘.
OR
The specified angle is the difference between angles α and β whose tangents are 2 and 31, respectively. The Tangent Difference Formula gives
tan(α−β)=1+tanα⋅tanβtanα−tanβ=1+322−31=1
The acute angle whose tangent is 1 is (C)45∘.
\section*{OR}
Consider vectors a=⟨3,1⟩ and b=⟨1,2⟩. The cosine of the angle between them is
