Chapter 7. Trigonometric Functions of Real Numbers
7.1 The Unit Circle
Recall that the unit circle is the circle of radius 1 centered at the origin. Its
standard equation is
x2 + y2 = 1 .
Geometrically, the unit circle consists of all the points on the xy-plane that
are exactly 1 unit away from the origin.
We previously have seen, each point on the unit circle has coordinates in the
form (x, y) = (cos θ , sin θ ), from which we can deduce that
cos2 θ + sin2 θ = 1
Dividing both sides of the above Pythagorean relation, respectively, by
cos2 θ and sin2 θ, we get the 2 companion relations.
1 + tan2 θ = sec2 θ
1 + cot2 θ = csc2 θ
Terminal Points on the Unit Circle
Let t be a real number. Starting at the point (1, 0), corresponding to an angle
θ = 0 on the unit circle, if a particle move along the circumference of the
unit circle t units, where does it end up at? (When t > 0, the motion is
counter-clockwise. While t < 0 means a clockwise movement.) Recall that,
when given in radians, the distance traveled by the particle along the
circumference of the unit circle is exactly equal to the angle t = θ subtended
by the said circular arc. Therefore, the terminal point of this particle after
having traveled a distance t along the unit circle is precisely (cos t , sin t ).
Obviously, a completely revolution of 2π radians in either direction would
take the particle back to where it has begun. Consequently, the concept of
angles that are coterminal also applies here for any real number t in the
context of traverse along the circumference of the unit circle: numbers that
differ by 2π (radians), or its multiple, are coterminal and represent the same
point on the unit circle. For instance, t = 2π / 3, 8π / 3, 14π / 3, 20π / 3, −4π / 3,
−10π / 3 all differ by a integer multiples of 2π, and they all represent the
point (cos 2π / 3 , sin 2π / 3 ) = (−1 / 2, 3 / 2) on the xy-plane.
7.2 Trigonometric Functions of Real Numbers
Let t be any real number and let P(x, y) be the terminal point on the
unit circle determined by θ = t. Define
sin t = y
y
tan t =
x
cot t =
x
y
cos t = x
1
sec t = ,
x
csc t =
1
y,
x≠0
y≠0
Evaluating Trigonometric Functions
While the context might have changed slightly, from a right triangle to the
unit circle, the way to evaluate trigonometric functions remains the same.
Let’s recall the steps: Given a real number t, first make sure t is between 0
and 2π: add/subtract a multiple of 2π as necessary so that 0 ≤ t < 2π.
1. Find the reference angle (“reference number”) φ in quadrant I.
2. Determine the sign of the function by noting the quadrant in which
t lies.
Quadrant I,
Quadrant II,
Quadrant III,
Quadrant IV,
0 < t < π / 2:
π / 2 < t < π:
π < t < 3π / 2:
3π / 2 < t < 2π:
All positive
Sin and csc positive
Tan and cot positive
Cos and sec positive
3. Use the result of 1 and 2 to find the value of the function.
[See the separate summary sheet for the Fundamental Identities.]
Symmetries
Cosine and secant are even functions, therefore, their graphs are
symmetrical about the y-axis:
cos(−x) = cos x,
and
sec(−x) = sec(x).
Sine, tangent, cotangent, and cosecant are odd functions, therefore,
their graphs are symmetrical about the origin:
sin(−x) = −sin x,
tan(−x) = −tan(x),
csc(−x) = −csc x,
cot(−x) = −cot(x).
7.3 Trigonometric Graphs
[Graphs of the 6 basic trigonometric functions are available separately.]
[Back to using y = f (x) notation, discarding the uses of t.]
The sine and cosine functions have period 2π, therefore
sin(x + 2π) = sin x,
and
cos(x + 2π) = cos x.
Transformations of Trigonometric Graphs
Like other functions, the 6 basic trigonometric functions can be transformed
into more other related functions.
Example: Comparing y = sin x
vs
y = 5sin x
The transformation involved is simply a vertical stretching by a
factor of 5.
Example: Graph of y = 2sin(x) + 3
Obtained by vertically stretch the graph of y = sin x by a
factor of 2, then vertically shift up 3 units.
Example: Graph of y = cos x
vs
y = 3cos(x − π) + 1
The latter function can be obtained from the graph of y = cos x by a
right horizontal shift of π, followed by vertically stretch of 3 times
magnification, and lastly a vertical shift of 1 unit.
Horizontal scaling (stretching or shrinking) changes period, and therefore,
frequency. Stretching increases period and decreases frequency. Shrinking
decreases period and increase frequency.
Examples: From Chapter 3:
Comparing the graphs of y = sin x vs. y = sin (x /2)
sin x has frequency of 1 and period 2π, the horizontal stretching by a factor 2
halves the frequency and double the period to 4π.
And compare both against the graph of y = sin 2x (freq = 2, period = π).
Amplitude, Frequency, Period, and Phase Shift
Phase shift = horizontal shift in the context of a sinusoidal function.
Due to the periodic nature of trigonometric functions, horizontal shift by a
multiple of whole period result in an identically function/graph. Therefore,
meaningful phase shifts are limited to less than one period in magnitude.
Amplitude = Half of the range of the given sinusoidal function.
General Sine and Cosine Curves
The sine and cosine curves (for k > 0)
y = a sin k(x − b)
and
y = a cos k(x − b)
have amplitude | a |, frequency k, period 2π / k, and phase shift b.
Example: Graph of y = 5cos(2x − π / 3) + 2
Example: The sum of a sine and a cosine curves
The sum of a sine and a cosine curves (of the same frequency) is
another sinusoidal curve of the original frequency. The result is, therefore,
just a transformed version of sine/cosine function.
Graph of y = cos x − sin x
Verify that the graph is also that of y =
2 cos( x + π / 4) , a cosine
function of amplitude 2 , period 2π, and a phase shift of π / 4
leftward. Hence, y =
2 cos( x + π / 4) = cos x − sin x.
Graph of f (x) = sin(x) / x
The function is undefined for x = 0, its domain consists of all other
real numbers. The graph is continuous except when x = 0. There is
not a vertical asymptote; instead there is a missing point on the curve
at exactly the point (0, 1). The fact that the curve is converging to y =
1 near x = 0 gives rise to the familiar approximation
When the angle x is very small,
sin x ≈ x.
It is of great importance for calculus. The fact underpins the entire
calculus treatment of trigonometric functions.
The function also has a horizontal asymptote y = 0. It is of some
interests to students of algebra, because the graph crosses the
horizontal asymptote infinitely often (except when x = 0, exactly once
every π). Hence it is another counter-example to disprove the
common misconception that a graph could never cross an asymptote –
while true for vertical asymptotes, it is definitely FALSE for a
horizontal asymptote.
7.4 More Trigonometric Graphs
[The graphs of tan x, cot x, and sec x are available separately.]
The graph of y = csc x
All 4 functions’ graphs have infinitely many vertical asymptotes π units
apart between each pair. They are located at the zeros of their respective
denominator: For tan x and sec x, when cos x = 0, i.e., whenever x = ±π / 2,
±3π / 2, ±5π / 2, ±7π / 2, … (odd integer multiples of ±π / 2). For cot x and csc
x, when sin x = 0, i.e., whenever x = ±π , ±2π , ±3π, ±4π, … (all integer
multiples of ±π ).
Periodic Properties:
The tangent and cotangent functions have period π, therefore
tan(x + π) = tan x,
and
cot(x + π) = cot x.
The secant and cosecant functions have period 2π, therefore
sec(x + 2π) = sec x,
and
csc(x + 2π) = csc x.
When combined with horizontal scaling, the period is also stretched /
shrunk. Thus,
y = a tan kx
and
y = a cot kx
have frequency a (radians / unit time), and period π / k. As well,
y = a sec kx
and
y = a csc kx
have frequency a (radians / unit time), and period 2π / k.
The graph of y = −sec(2x) + 1
The graph of y = tan(x + π / 4) + 5