Up until now we have found the trigonometric functional value of some angle in standard
position given some point (x, y) on the terminal ray. Now we will find the trigonometric functional values for some special angles.We will begin with the quadrantal
|Definition A quadrantal angle is an angle in standard position whose terminal ray lies along
one of the axes.
Examples of quadrantal angles include, 0, π/2 , π , and 3π/ 2.
coterminal with these angles are, of course, also quadrantal. We are interested
in finding the six trigonometric functional values of these special angles, and we will begin
with θ = 0. Since any point (x, y) on the terminal ray of an angle with measure
0 has y coordinate equal to 0, we know that r = |x|, and we have,
Using a similar approach, we can find the six trigonometric functional values for
θ = π/2, θ = π, and θ = 3π/2 as,
The trigonometric functional values of angles coterminal with 0,
π/2 , π, and 3π/2
are the same
as those above, and the trigonometric functional values repeat themselves (e.g., π and 3π are coterminal and sin (π) = sin (π + 2π) = sin (3π) = 0). This illustrates the
fact that the trigonometric functions are periodic. We will discuss this in greater detail in
the next section.
To find the trigonometric functional values of other special angles, we will redefine the six
trigonometric functions relative to an angle of a right triangle. Consider the following
We can find the trigonometric functional values using the following definitions:
where adj, opp, and hyp are short for side adjacent to θ, side opposite θ , and
hypotenuse, respectively. You should be convinced that these new definitions are consistent
with the previous ones. Now consider the following 30° - 60° - 90° triangle:
It may be helpful to recall the Pythagorean Theorem,
a2 + b2 = c2,
where a and b are the legs of a right triangle, and c is the hypotenuse. Using the
above definitions of the trigonometric functions, we can compute the trigonometric functional values
of θ = π/
6 (30° ) and θ = π/
3 (60°) as,
We will now find the trigonometric functional values of another special angle, the 45°-45°-90° triangle, as depicted
in the following figure:
Using the above triangle we find the six trigonometric functional values of θ = π/
Other Special Angles
We can also find the exact trigonometric functional values of a few other angles that relate
to the angles θ = π/6, θ = π/4, and θ = π/3 in certain ways. To look at these other
angles, we introduce the concept of a reference angle, denotated as θ'. To compute
the reference angle for an angle that has negative measure, find the first positive
coterminal angle. To find the reference angle for an angle that is larger than 360° ,
find a coterminal angle that lies between 0° and 360°. Once you have an angle
θ with 0° ≤ θ ≤ 360°, you can find its reference angle as follows:
1. If the terminal ray of θ lies in quadrant I, then θ ' = θ.
2. If the terminal ray of θ lies in quadrant II, then θ ' = 180° − θ.
3. If the terminal ray of θ lies in quadrant III, then θ ' = θ − 180° .
4. If the terminal ray of θ lies in quadrant IV, then θ ' = 360°− θ.
The reference angle is used to find the exact trigonometric functional values of angles that
terminate outside of the first quadrant. In particular, it is true that
| trig ( θ ) | =
trig ( θ' ),
where trig denotes one of the six trigonometric functions. For example, suppose
we want to find sin (7π/6) . First, convert 7π/6 radians to degrees since we have
more intuition about degree measure. Since 7π/6 radians is the same as 210° , we
are trying to find sin (210°), where θ = 210° is located in quadrant III. Thus, θ' = 210° −180° = 30° , where θ' is the reference angle (a special angle). Since sin (30°) =sin
(π/6) = 1/
we know that sin (210°) = sin(7π/6)
is either equal to 1/2 or −1/2 . To determine which
one, we ask ourselves, is the sine function positive or negative in quadrant III?
To answer this question, we recall the definition of sin θ,
sin θ =
Since the y-coordinate of a point in quadrant III is negative, and r is defined to
be positive, sin θ (where θ lies in quadrant III) must be negative. Therefore, we
Finding the trigonometric functional values of angles that are not related to the special angles usually requires the use of a calculator. For example, the value of sin (π/ 17) can be approximated by using your calculator (be sure you are in radian mode).
In the next section we will explore the graphs of trigonometric
Graphing Trigonometric Functions