Math 1720 - Trigonometry
University of Memphis
Instructor: Gábor Mészáros
Central European University, Budapest
10.16-18.2013
Solutions
1. Use a graph drawer program (such as rechneronline.de/function-graphs/) to draw the graphs of
the following functions:
sin x
arcsin x
cos x
arccos x
tan x
arctanx
cot x
arccotx
sec x
arcsecx
csc x
arccosecx
PLEASE MAKE SURE YOU CAN USE AN APPROPRIATE GRAPH DRAWER!
2. Determine the number of intersections of the following pairs of functions on [0, π] (you may use
a graph drawer).
(a) sin x and tan x: 1
(b) sin x and sin(2x): 2
1
(c) arcsin x and arccos x:
(d) sec x and tan x: 0
(e) arcsec and arctan x:0
2
(f) arcsec and arccscx: 1
3. Evaluate the following functions at the given values:
REMEMBER: We are looking for an angle α belonging to the appropriate interval
(depending on the function, see below) such that trigfunction(α)=given value. That
is:
(a) arcsin(x) = − 21 ,
α ∈ [− π2 , π2 ], s.t sin α = − 12 . α = − π6 .
√
(b) arccos(x) = 23 ,
α ∈ [0, π], s.t cos α =
(c) arctan(x) =
α∈
(− π2 , π2 )
√
3
2
,
, s.t tan α =
. α = π3 .
√1
3
√1
3
. α = π6 .
(d) arccot(x) = −1,
α ∈ (0, π), s.t cot α = −1. α =
3π
4
.
(e) arcsec(x) = 2, α ∈ (0, π), s.t sec α = 2. α = π3 .
√
(f) arccsc(x) = 2.
√
α ∈ (− π2 , π2 ), s.t cscα = 2. α = π4 .
4. Prove that tan−1 (x) 6=
sin−1 (x)
cos−1 (x)
by showing that the two functions have dierent domains.
The domain of tan (x) is R while the right side has a restricted domain:
−1
• sin−1 (x) is only dened on [−1, 1],
3
• cos−1 (x) is only dened on [−1, 1],
• cos−1 (x) 6= 0, that is, x 6= 0.
In total, x ∈ [−1, 1]\{0}.
5. Find the appropriate domain and image of the function sec−1 (x). Verify your answer by drawing
the graph.
Domain: (−∞, −1] ∪ [1, ∞).
Range: (− π2 , π2 )\{0}.
4