TRIGONOMETRY IN THE UNIT CIRCLE
PART I
I) IN THE UNIT CIRCLE
Any real number θ may be interpreted as the radian measure of an angle as follows: If θ ≥ 0 , think of wrapping a
of string around the standard unit circle C in the plane, with initial point P(1,0), and proceeding
counterclockwise around the circle; do the same if θ < 0 , but wrap the string clockwise around the circle.
length
θ
As a matter of common practice and convenience, it is useful to measure angles in degrees, which are defined by
partitioning one whole revolution into 360 equal parts, each of which is then called one degree. In this way, one whole
revolution around the unit circle measures 2π radians and also 360 degrees, that is: 360° = 2π radians. Each degree may be
further subdivided into 60 parts, called minutes, and each minute may be subdivided into another 60 parts, called
seconds.
7π
Infinitely many angles can have the same initial and terminal sides. For example, angles of radian whose measures are
3 or −17π 3 have the same initial and terminal sides as the angle of π 3 . More generally, an angle of measure t rad has
the same initial and terminal sides as the angles of radian measure t + 2kπ for any integer k (positive or negative).
Conversely, any two or more angles with the same initial and terminal sides must have radian measures that differ by an
integer multiple of 2π . Such angles are called coterminal.
Refering to the left diagram, we now define the trigonometric functions as follows: the sine of θ (denoted as sin θ )
is the y-coordinate of P, and the cosine of θ (denoted as cos θ ) is the x-coordinate of
P.
Now we are able to compute trigonometric functions of arbitrary angles from
our knowledge of theses trigonometric functions of the associated angles between 0 and
π/2. We can easily find these equalities:
sin(π - θ) =
cos(π - θ) =
sin(π + θ) =
cos(π + θ) =
sin(π/2 - θ) =
cos(π/2 - θ) =
II ) IN THE RIGHT TRIANGLE
This section reviews the basic trigonometric functions. The trigonometric functions are important because
they are periodic, or repeating, and therefore model many naturally occurring periodic processes.
You are probably familiar with defining the trigonometric functions of an acute angle in terms of the sides of a right
triangle.
cos x =
adjacent
hypotenuse
sin x =
opposite
hypotenuse
tan( x) =
sin( x)
cos( x)
and
cos ²( x) + sin ²( x) = 1
Tan has period π , and sin and cos have period 2π . Periodic functions are important
because much behaviour studied in science is approximately periodic.
III) EXERCISES
sin(π / 3) = 3 / 2 and cos(π / 4) = 2 / 2
b) Check that f ( x ) = cos(6 x ) has π / 3 for period
a) Using a right triangle, explain why
cos( x) = 1/ 2 and sin( x) = − 2 / 2 on the interval [0; 2π [ then on ℝ
d) Solve cos ²( x ) − 1/ 2 < 0 on ] − π ; π ]
e) Thanks to your calculator, find the period of the function f ( x ) = cos ²( x ) − sin ²( x) . Check it algebraically.
c) Solve the equations