
Answers to Questions on Trigonometric Functions

 Question: Show that 1 + tan^{2}(x) =
sec^{2}(x)
Answer: tan^{2}(x) + 1 = sin^{2}(x)/cos^{2}(x)
+ 1 = [sin^{2}(x) + cos^{2}(x)]/cos^{2}(x)
= 1/cos^{2}(x) = (1/cos(x))^{2} = sec^{2}(x).
Thus tan^{2}(x) + 1 = sec^{2}(x).
 Question: Show that 1 + cot^{2}(x) =
csc^{2}(x)
Answer: Solve similarly to above or:
Divide both sides of the identity sin^{2}(x) + cos^{2}(x)
= 1 by sin^{2}(x) to get
sin^{2}(x)/sin^{2}(x) + cos^{2}(x)/sin^{2}(x)
= 1/sin^{2}(x)
Thus 1 + cot^{2}(x) = csc^{2}(x).
Return to Exercises
 Draw graphs of each of the functions. Give the period and any vertical
asymptotes. Comment on the amplitudes of these functions.
You can check these graphs with your graphing calculator. To graph
sec (x), enter it as 1/cos (x). Do not use the "cos ^{1}"
button (see notes on notation in the Field Guide Lesson.). Similarly,
enter 1/sin (x) for csc (x) and 1/tan (x) or cos (x) / sin (x) for
cot (x).
 Question: cot (x)
Answer: Cot (x) has period and vertical asymptotes at multiples of
(where sin (x) = 0).
 Question: sec (x)
Answer: Sec (x) has period 2 and vertical asymptotes at odd multiples of /2 (where cos (x) = 0).
 Question: csc (x)
Answer: Csc (x) has period 2 and vertical asymptotes at multiples of
(where sin (x) = 0).
These functions do not oscillate up and down within a finite distance
from the midline y = 0. Thus they do not have a finite amplitude.
Return to Exercises
 Give the amplitude and period for each of the following functions.
Sketch their graphs.
 Question: g(x) = 3cos 2x
Answer: Amplitude = 3, period = .
 Question: f(x) = 2.7sin 2(x  /4)
Answer: Amplitude = 2.7 (negative sign causes
a reflection in the xaxis), period = .
 Question: g(x) = sin ((1/2)(x + /3)
Answer: Amplitude = 1, period = 4 .
 Question: h(x) = 1.5cos (x  1)
Answer: Amplitude = 1.5, period = 2.
Return to Exercises
 The tidal variation in Desolation Sound on the west coast of Canada
is roughly 4 meters. That is, the difference beween water depth at high
tide and at low tide is 4 meters, with successive high tides occurring
12.5 hours apart. Suppose that at Refuge Cove in Desolation Sound, the
depth of water in meters is given by
D(t) = D_{0} + Acos B(t  t_{0})
where t is measured in hours from midnight on June 1, 1996.
 Question: What does D_{0} mean in the
context of the problem?
Answer: D_{0} is the water level midway
between high water and low water.
 Question: What is the value of A?
Answer: A = 2. It is the amplitude of oscillation,
and is half of the total tidal variation.
 Question: What is the value of B?
Answer: B = (2 )/12.5 = 0.503 (approximately).
 Question: Give the physical meaning of t_{0}
for this problem.
Answer: t_{0} is the time of the first
high tide after midnight on June 1, 1996.
Return to Exercises
