Question

Find the dimension of the subspace U = span {1,sin^2(θ), cos 2θ} of F[0, 2π]

Answer #1

**dimension** of
**Span**(U), we need to find a basis of
**Span**(U). One way to do this is to note that the
third vector is the sum of the first two vectors. Also, it's clear
that the first two vectors are linearly independent. is a basis of
**Span**(U).

Prove the following
(4 sin(θ) cos(θ))(1 − 2 sin2 (θ)) = sin(4θ)
cos(2θ) /1 + sin(2θ) = cot(θ) − 1/ cot(θ) + 1
cos(u) /1 + sin(u) + 1 + sin(u) /cos(u) = 2 sec(u)

Find f. f ''(θ) = sin(θ) + cos(θ), f(0) = 3, f '(0) = 2

Find the exact values of sin 2θ, cos 2θ, and tan 2θ for the
given value of θ.
cot θ =
4
3
; 180° < θ < 270°

Find f. f ''(θ) = sin(θ) + cos(θ), f(0) = 5, f '(0) = 4

Find f.
f ''(θ) = sin(θ) +
cos(θ), f(0) =
4, f '(0) = 3
f(θ) =

Let Θ ∼ Unif.([0, 2π]) and consider X = cos(Θ) and Y =
sin(Θ).
Can you find E[X], E[Y], and E[XY]?
clearly, x and y are not independent
I think E[X] = E[Y] = 0 but how do you find E[XY]?

If θ is in the interval [0, 2π) and cos(θ) = √ 2/2 , then θ must
be π/4 .
State true or false.

Identify the surface with parametrization x = 3 cos θ sin φ, y =
3 sin θ sin φ, z = cos φ where 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π. Hint: Find
an equation of the form F(x, y, z) = 0 for this surface by
eliminating θ and φ from the equations above. (b) Calculate a
parametrization for the tangent plane to the surface at (θ, φ) =
(π/3, π/4).

Consider the function on the interval (0, 2π).
f(x) =
sin(x)/
2 + (cos(x))2
(a) Find the open intervals on which the function is increasing
or decreasing. (Enter your answers using interval notation.)
increasing
decreasing
(b) Apply the First Derivative Test to identify the relative
extrema.
relative maximum
(x, y) =
relative minimum
(x, y) =

1. Find all angles
θ,0≤θ≤2π
(Double angle
formula, To two decimal places)
a) Tan theta
= 0.3, b) cos theta = 0.1, c) sin theta = 0.1,
d) sec theta
= 3

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