Reduction Formulae
In trigonometry angles are located in a polar plot with Cartesian axes superimposed.. It is often
convenient to express angles as offsets to either the horizontal (180 or 360 or 0 degree) line or the
90 and 270 degree line. The Cartesian axes divide the plot into 4 'quadrants'. The CAST acronym
may be used to find the sign of a given ratio and argument.
As the unit vector rotates on the polar plot, its projection on the 0-180 and 90-270 degree lines
form, together with the (always positive) unit vector, a right angled triangle from which the
trigonometric ratios may be determined.
The following graphics show the angle 30 degrees as an offset from first the horizontal (0-180
degrees) axis (Fig. 1) and then from the vertical (90-270 degrees) axis (Fig. 2). Studying the
images reveals several rules for quickly 'reducing' ratios with off-set arguments to ratios with simple
arguments.
Fig 1. Offset from horizontal (0 / 180) axis.
Fig. 2 Offset from vertical (90 / 270) axis
e.g.
Sin(90 – x) = Cos(x)
Rules for Determining Reduction.
We notice that, in general, our problem may be stated as
Trig-Function (Off-set Angle Argument) = <value> = Trig-Function(Simple Argument)
e.g. 1
Sin(270-x) = -Cos(x)
e.g. 2
Sin (180+x) = -Sin(x)
1. The value on the Left-hand Side (LHS) must equal the value on the Right-hand Side (RHS)
2. We take the offset angle 'x' in the LHS argument and make it our reference angle (always
positive)
3. The argument of the RHS function is the reference angle 'x'
4. The SIGN (polarity) of the RHS function is determined by the quadrant in which the LHS
argument lies and the resulting LHS function polarity from CAST.
5. If the LHS argument is an offset from the horizontal axis the trig function on the RHS is the
same as that on the LHS
6. If the LHS argument is an offset from the vertical axis, the trig function on the RHS is the
co-ratio of the trig function on the LHS
n.b.
sine → cosine
cosine → sine
tangent → cotangent
cotangent → tangent
7. You can check the result by substituting a small angle for the reference angle and evaluating
both sides
e.g.
Sin(270-x) = - cos(x)
Sin(270-30) = Sin(240) = -0.866, -cos(30) = -0.866
Examples:
Sin(180 – x) = Sin(x):
The reference angle is x (Rule 2)
180-x lies in the second quadrant where Sine is positive. The RHS function is also positive
(Rule 4)
180-x is an offset from horizontal axis so the trig function does not change (Rule 5)
We check the answer (Rule 7)
Sin(150) = Sin(30) = 0.5
Cos(270 + x) = Sin(x)
The reference angle is x (Rule 2)
270+x lies in the forth quadrant where Cosine is positive. The RHS function is also
positive (Rule 4)
270+x is an offset from vertical axis so trig function changes to the co-ratio, 'Sine' (Rule 6)
We check the answer (Rule 7)
Cos(270+30) = Cos(300) = Sin(30) = 0.5
Tan(90 + q) = -Cot(q)
The reference angle is q (Rule-2)
90+q lies in the second quadrant where Tangent is negative. The RHS function is also
negative (Rule-4)
90+q is an offset from vertical axis so trig function changes to the co-ratio, 'Cotangent'
(Rule-6)
We check the answer (Rule-7)
Tan(90+30) = Sin(90+30) =
-------------Cos(90+30)
Cos(30) = -Cot(30) = -1.73
-----------Sin(30)
or apply the Rules directly
Tan(90+30) = -cot(30) = -1.73
Finally note the following:
Sin(-x) = Sin(0-x) (forth quadrant) = -sin(x)
Cos(-x) = Cos(0-x)(forth quadrant) = cos(x)
Tan(-x) = Tan(0-x)(forth quadrant) = -tan (x)