Any trigonometric function whose argument is
,
,
and
(hence
) can be written simply in terms of
. For example, you may have noticed that the cosine graph is identical to the sine graph except for a phase shift of
. From this we may expect that
.
Function values of
Investigation : reduction formulae for function values of
Function Values of
In the figure P and P' lie on the circle with radius 2. OP makes an angle
with the
-axis. P thus has coordinates
. If P' is the reflection of P about the
-axis (or the line
), use symmetry to write down the coordinates of P'.
Write down values for
,
and
.
Using the coordinates for P' determine
,
and
.
From your results try and determine a relationship between the function values of
and
.
Function values of
In the figure P and P' lie on the circle with radius 2. OP makes an angle
with the
-axis. P thus has coordinates
. P' is the inversion of P through the origin (reflection about both the
- and
-axes) and lies at an angle of
with the
-axis. Write down the coordinates of P'.
Using the coordinates for P' determine
,
and
.
From your results try and determine a relationship between the function values of
and
.
Investigation : reduction formulae for function values of
Function values of
In the figure P and P' lie on the circle with radius 2. OP makes an angle
with the
-axis. P thus has coordinates
. P' is the reflection of P about the
-axis or the line
. Using symmetry, write down the coordinates of P'.
Using the coordinates for P' determine
,
and
.
From your results try and determine a relationship between the function values of
and
.
It is possible to have an angle which is larger than
. The angle completes one revolution to give
and then continues to give the required angle. We get the following results:
Note also, that if
is any integer, then
Write
as the function of an acute angle.
where we used the fact that
. Check, using your calculator, that these values are in fact equal:
Evaluate without using a calculator:
Reduction formulae
Write these equations as a function of
only:
Write the following trig functions as a function of an acute angle:
Determine the following without the use of a calculator:
Determine the following by reducing to an acute angle and using special angles. Do not use a calculator:
Function values of
When the argument of a trigonometric function is
we can add
without changing the result. Thus for sine and cosine
Function values of
Investigation : reduction formulae for function values of
Function values of
In the figure P and P' lie on the circle with radius 2. OP makes an angle
with the
-axis. P thus has coordinates
. P' is the reflection of P about the line
. Using symmetry, write down the coordinates of P'.
Using the coordinates for P' determine
,
and
.
From your results try and determine a relationship between the function values of
and
.
Function values of
In the figure P and P' lie on the circle with radius 2. OP makes an angle
with the
-axis. P thus has coordinates
. P' is the rotation of P through
. Using symmetry, write down the coordinates of P'. (Hint: consider P' as the reflection of P about the line
followed by a reflection about the
-axis)
Using the coordinates for P' determine
,
and
.
From your results try and determine a relationship between the function values of
and
.
Complementary angles are positive acute angles that add up to
. e.g.
and
are complementary angles.
Sine and cosine are known as
co-functions . Two functions are called co-functions if
whenever
(i.e.
and
are complementary angles). The other trig co-functions are secant and cosecant, and tangent and cotangent.
The function value of an angle is equal to the co-function of its complement (the co-co rule).
Thus for sine and cosine we have
Write each of the following in terms of
using
and
.
Function values of
and
.
These results may be proved as follows:
and likewise for
Summary
The following summary may be made
second quadrant
or
first quadrant
or
all trig functions are positive
third quadrant
fourth quadrant
These reduction formulae hold for any angle
. For convenience, we usually work with
as if it is acute, i.e.
.
When determining function values of
,
and
the functions never change.
When determining function values of
and
the functions changes to its co-function (co-co rule).
Function values of
Angles in the third and fourth quadrants may be written as
with
an acute angle. Similar rules to the above apply. We get