1.

For and :

For and :

Look at the difference!

2.

a) 0

b) π/2

c) 0

d) π/2

3.

If = /√2 + /√2, and = -sin(θ) + cos(θ), then

-sin(θ) = 1/√2 and cos(θ) = 1/√2

Based on this, possible values of θ are

sin: 5π/4, 7π/4

cos: π/4, 5π/4

There is agreement at 5π/4, and so θ = 5π/4.

4.

In polar coordinates, all we need is the magnitude of r because

r = r

We can find r by using the Pythagorean Theorem:

√(3² + 4²) = √25 = 5

Thus, the position of the peanut

r_peanut = 5

We'd also want to know the angle θ, and we can find that by knowing that

tan(θ) = y/x.  

Thus,

arctan(4/3) = θ

θ = .927