## anonymous one year ago Can someone help me with Problem Set 1: 1B-13

1. anonymous

The question asks us to solve for cos(Θ1−Θ2), but the answer in the supplemental packet solves for cos(Θ2−Θ1). Is this a mistake?

2. phi

cos(-x) = cos(x) so their switching theta1 and theta2 does not change the answer. (Though it would be nice if they were consistent, so as not to introduce confusion)

3. phi

Here are more details on their derivation of the cos of a difference of two angles. First, they define a unit length vector with its tail at the origin and its head lying on the unit circle. Based on this definition, we know its i and j components |dw:1442327812962:dw| we thus have two unit vectors $\vec{u_1}= < \cos \theta_1 , \sin \theta_1> \\ \vec{u_2}= < \cos \theta_2 , \sin \theta_2>$ |dw:1442328058455:dw|

4. phi

Now use the definition of a dot product $\vec{u_1} \cdot \vec{u_2} = |u_1| |u_2| \cos \phi$ where phi is the angle between the two vectors. we get (note: both vectors have unit length) $\cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2 = 1 \cdot 1 \cdot \cos \phi$ thus, where $$ϕ=θ_2−θ_1$$ we have the result $\cos \left(\theta_2 - \theta_1 \right)= \cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2$

5. anonymous

Thank you, Phi!