
Trigonometric Identities
A trigonometric identity is an equation involving trigonometric
functions that holds true for all angles. These identities are commonly
called Pythagorean identities, because they come from the Pythagorean
theorem.
In general, the Math IIC will test your knowledge of the
trigonometric identities by giving you a complex expression that
you have to simplify. These questions have more to do with memorizing
the identities and being good with algebraic substitution than they
do with the theoretical concepts of trigonometry.
For example:

To solve a problem like this, use the trigonometric identities
to simplify the trigonometric expression into sines and cosines.
After you have simplified the expression using the identities, it
is quite likely that the expressions will simplify further through
the canceling of terms. The simplification of the expression in
the example questions proceeds as follows.
Simplify the mess given to you by the problem, and you
get sin^{2}.
Here’s another example:

Other Trigonometric Identities
About once a test, you may encounter a particularly complicated
trigonometric expression. Such expressions can usually
be solved using the trigonometric identities we’ve already covered,
but the process will be laborious and difficult. However, if you
know the special trigonometric identities below, these questions
will be much easier and less timeconsuming.
We list the special trigonometric identities in the order
of usefulness, starting with the ones most likely to be used on
the test.
Sum and Difference Formulas
There are three identities for the sum of two angles:
There are also three identities for the difference of
two angles:
Try to use these identities in a sample problem:

Using the identities sin^{2}x +
cos^{2}x = 1 and 1 + tan^{2}x =
sec^{2}x, the expression
simplifies in the following way:
Here’s another example to practice with:

If sin a =
^{3}/_{5} and
the angle is in the second quadrant, then cos a =
–^{4}/_{5} .
Also, if cos b = –^{3}
/_{5} and b is
in the second quadrant, then sin b =
^{4}/_{5} .
See the figure below:
Now you can use the subtraction identity to see that = (–^{4}
/_{5} )(–
^{3}/_{5} )
+ (^{3}/_{5} )(
^{4}/_{5} )
= ^{12}/_{25 } +
^{12}/_{25} =
^{24}/_{25} .
If you are familiar with the formula for cos(a – b)
and remember the Pythagorean triple 345, you can save time on
a question like this. Otherwise, you have to calculate the arcsine
of ^{3}/_{5} and
the arccosine of –^{3}/
_{5} in order to find and , respectively. Then you have to
subtract from and take the cosine of the resulting
angle.
Double Angle Formulas
There are three double angle formulas.
Half Angle Formulas
Of all trigonometric identities, the six halfangle formulas
appear most infrequently on the Math IIC.
