Trigonometric Functions - Unit Circle Approach Worksheet Page 2
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Definition
If θ is an angle in standard position and if the terminal side of θ intersects
the unit circle at (x
, y
), then
uc
uc
sin( θ ) = y
(sine of θ )
uc
cos( θ ) = x
(cosine of θ )
uc
The sine and cosine functions belong to a class of functions that we call the
trigonometric functions.
Find the exact values of the sine and cosine functions for the given
angles:
Ex. 1a
π
Ex. 1b
330˚
3 π
Ex. 1c
Ex. 1d
240˚
4
Solution:
a)
Since the terminal side of θ passes through (– 1, 0), then
sin( θ ) = 0 and cos( θ ) = – 1.
3
1
b)
Since the terminal side of θ passes through (
, –
), then
2
2
1
3
sin( θ ) = –
and cos( θ ) =
.
2
2
2
2
c)
Since the terminal side of θ passes through (–
,
),
2
2
2
2
then sin( θ ) =
and cos( θ ) = –
.
2
2
1
3
d)
Since the terminal side of θ passes through (–
, –
), then
2
2
3
1
sin( θ ) = –
and cos( θ ) = –
.
2
2
There are four other trigonometric functions we can define in reference to
our unit circle. We can first talk about the slope of the terminal side of θ in
standard position. Two points on that line are (0, 0) and (x
, y
), so the
uc
uc
y
y
−0
uc
uc
slope m =
=
. This value again depends on the value of θ , so we
x
x
−0
uc
uc
will define the tangent function (tan( θ )) to be equal to the slope of the
terminal side. The other three trigonometric functions, the cosecant
(csc( θ )), the secant (sec( θ )), and the cotangent (cot( θ )),are the reciprocals
of the first three trigonometric functions sine, cosine, and tangent.
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