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Section 2: Trigonometry on the Unit Circle
THE UNIT CIRCLE
To help us understand the geometric meaning of the trigonometric
functions, it is helpful
to consider what sin and cos represent on the unit circle. A
unit circle is a circlewith radius 1 that is centered at the
origin. (i.e. this circle would have the equation
122 =+ yx ). The 4 quadrants are as labeled below. Angles are
measured counter-
clockwise starting from the positive x-axis.
CAST RULE
The CAST rule is used to help you remember the quadrants in
which )sin( )cos( and
)tan( are positive.
Quadrant 1 is represented by A therefore all three are positive
in that quadrant.
Quadrant 2 is represented by S therefore )sin( is positive in
that quadrant.
Quadrant 3 is represented by T therefore )tan( is positive in
that quadrant.
Quadrant 4 is represented by C therefore )cos( is positive in
that quadrant.
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SOHCAHTOA AND SPECIAL ANGLES
In trigonometry there are special angles at which you should
know the value of thevarious trigonometric functions. The two
special triangles below can be used to help you
find these values, but first we need to remember how sine and
cosine are defined based
on the sides of a right angle triangle. To help us remember we
use:
SOH CAH TOA
hypotenuse
opposite=sin
hypotenuse
adjacent=cos
adjacent
opposite=tan
Now, lets consider two special triangles:
Using the above triangles and SOHCAHTOA, we end up with the
following chart:
0
6
4
3
2
)sin( 0
2
1
2
1
2
3
1
)cos( 1
2
3
2
1
2
1
0
)tan( 0
3
1
1 3 Undefined
These values can be used to find the values for )csc( , )sec( ,
)cot( and can also be
used with the unit circle below (or CAST rule) to find values in
other quadrants.
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Example: Evaluate
6
5cos
Solution: First of all,6
5 is in the second quadrant so cosine has a negative value.
Next, use the unit circle to determine that6
5is related to the angle
6
.
Finally using the special triangle2
3
6cos =
Therefore,2
3
6
5cos =
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Example: Evaluate
4
5csc
Solution: First, we know4
5is in the third quadrant so sine has a negative value;
consequently, cosecant will also have a negative value.
Next, use the unit circle to determine that4
5is related to the angle
4
.
Finally using the special triangle
2
1
4
sin =
so
2
1
4
5sin =
Therefore, 24
5csc =
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For a detailed explanation of some more difficult examples,
check out the mini-clips!
