THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1013 University Mathematics II
Assignment 1
Due date: 26 Sep, 2023
Please write down your name and university number. Precise and adequate explanations should be
given to each problem.
√
1
, g(x) = tan−1 x and h(x) = x − 2. Find the natural domain and
4x − π
the corresponding range of the function f ○ (g ○ h).
1. (4 points) Let f (x) =
2. (a) (2 points) For every x ∈ [−1, 1], find a simplified formula of sin (cos−1 x) which does not
involve any trigonometric functions.
(b) (2 points) For every x, y ∈ [−1, 1], find a simplified formula of cos (cos−1 x − cos−1 y) which
does not involve any trigonometric functions.
(c) (2 points) Let a and b be positive real numbers, and let θ be any real number. Prove that
√
a sin θ + b cos θ =
a2 + b2 sin (θ + cos−1 √
a
a2 + b2
).
(d) (2 points) Solve the equation sin ϕ + 4 cos ϕ = 2.
3. Compute each of the following and express the answer in standard form. (You have to provide
some details even if some answers can be obtained directly from calculators.)
(a) (2 points) (
3 − i 101
)
2+i
√
(b) (2 points) ei(1+ 3i) (hint: for a, b ∈ R, we define ea+bi = ea ebi )
4. In each of the following, describe the geometric meaning of the set of points in the Argand
diagram.
(a) (1 point) the set of all z ∈ C such that Re(z) + Im(z) = 2.
3
(b) (1 point) the set of all nonzero z ∈ C such that z =
z
(c) (2 points) the set of all z ∈ C with z ≠ −1, i such that arg (z + 1) = arg (z − i)
1