Download Common Values of Trig Functions

Standard Values for Trig Functions
...and ends here
(terminal side)
For any angle (which we will call θ) in standard position
(so its vertex is at the origin and it is measured from the
positive x-axis, as in this drawing),
θ
Angle starts here
(initial side)...
Vertex is here
we can find, for any point on the terminal side, values
of x, y and r as shown,
r
y
x
y
csc θ
=
r
x
cos θ =
sec θ
=
r
y
tan θ =
cot θ
=
x
sin θ
=
and we have definitions which state that:
r
y
r
x
x
y
Using geometry and algebra, we can find the values for some special angles very quickly. The angles that have
terminal sides on an axis are called “quadrantal angles” and can be found using any value for r. Consider a 90°
angle. It looks like this:
Since the point is not to the right or the left, the x coordinate must be zero.
If y can be any number, let’s pretend it is 5. Then the distance from
the origin to the point would be exactly 5, but r is that distance, so
r would be 5 as well. Thus=
x 0,=
y 5 and
=
r 5. This means that
y 5
r 5
sin 90°= = = 1
csc 90°= = = 1
r 5
y 5
x
=
r
y
tan 90°= =
x
cos 90°=
0
= 0
5
5
= Undefined
0
r
=
x
x
cot 90°= =
y
sec 90°=
(x, y)
r
90°
5
= Undefined
0
0
= 0
5
You could do the same calculations for 0°, 180°, 270° and 360° (notice that x and y are sometimes negative, but
r is a distance and therefore always positive).
Angles which are not on the axes can be drawn as triangles (as in the drawing on the previous page), with sides
of length x, y and r. The problem is that we need to know how these sides are related to one another, and we do
not usually know that. There are two “special triangles” for which we have this information. These are the
30­60­90 triangle and the 45­45­90 triangle. Depending on your geometry textbook, you should have seen two
of these triangles before:
30°
3
45°
45°
30°
2
2x
x 3
60°
1
2
x 2
45°
60°
1
x
x
45°
1
x
(If you learned it with the x’s, then, if x were 1, you would get the triangle with numbers alone.) If we place
these triangles on a coordinate system, we end up with values of x, y and r that we can use.
r=2
1= y
3=y
30°
45°
60°
x= 3
x =1
x =1
y 1
=
r 2
x
3
cos 30°= =
r
2
y
=
r
1
2
2
sin 30°=
sin 45°=
y
=
x
...and so on
x
1
2
=
=
r
2
2
y 1
tan 45° =
= = 1
x 1
...and so on
tan 30°=
r=2
1= y
r= 2
1
=
3
=
2
cos 45°=
3
3
y
3
=
r
2
x 1
cos 60°= =
r 2
sin 60°=
y
=
x
...and so on
tan 60°=
3
=
1
3
So, we could find all of the functions of these angles from the special triangles. But what about bigger angles?
We can find the functions of angles in other quadrants by using “reference angles.” A reference angle is the
smallest angle between the terminal side and the nearest x-axis. Since we measure angles from the positive xaxis, the actual angle and the reference angle are the same in the first quadrant. Elsewhere, they look like this:
the reference angle
θ (the angle)
θ (the angle)
the reference angle
θ (the angle)
the reference angle
Reference angles allow us to use the values we already found for 30°, 45° and 60° angles to find many others
without having to calculate any more. For the same size reference angle in any quadrant, there is an identical
triangle having the same x, y and r values (except for positives and negatives). For a 30° reference angle, the
triangles would look like this:
150°
30°
All four triangles are identical, so the lengths of the sides will all
be the same. Since r is a length, it will not change at all. Since x
and y are coordinates, they have positive and negative values. So,
30°
30°
30°
30°
triangles that go to the left will have negative x values, and
triangles that go down will have negative y values. Since nothing
else changes, we can find the values of the trig functions by adding
signs to the functions we already found for 30°.
210°
330°
sin 30°=
y 1
=
r 2
cos 30°=
x
=
r
3
2
tan 30°=
y
=
x
1
=
3
3
3
sin 150° =
y 1
=
r 2
cos 150° =
x − 3
3
=
=−
r
2
2
tan 150° =
3
y
1
=
=−
3
x − 3
sin 210° =
y −1
1
=
=−
r
2
2
cos 210° =
x − 3
3
=
=−
r
2
2
tan 210° =
−1
y
3
=
=
x − 3
3
sin 330° =
y −1
1
=
=−
r
2
2
cos 330° =
x
3
3
=
=
r
2
2
tan 330° =
y −1
3
=
=−
x
3
3
We could continue this process and find all six trigonometric functions for all angles with reference angles of
30°, 45° and 60°.