Using trigonometric ratios of the allied angles we can find the trigonometric ratios of any angle.

**Trigonometric ratios**

Let X' OX and Y' OY be two lines at right
angles to each other as in the figure. We call X' OX and Y' OY as X axis and Y
axis respectively. Clearly these axes divided the entire plane into four equal
parts called “Quadrants” .

The parts XOY, YOX' , X'OY' and Y'OX are known
as the first, second, third and the fourth quadrant respectively.

In the first quadrant both *x* and *y* are positive. So
all trigonometric ratios are positive. In the second quadrant (90° < θ <180°) *x* is
negative and *y* is positive. So
trigonometric ratios sin θ and cosec θ are positive. In the third quadrant (180° < θ <
270°) both *x* and *y *are negative. So trigonometric ratios
tan θ and cot θ are positive. In the fourth quadrant* *(270° < θ < 360° ) *x* is positive and *y* is negative. So trigonometric ratios cos θ and sec θ are positive.

Two angles are said to be allied angles when
their sum or difference is either zero or a multiple of 90° . The angles -*θ* , 90°
± *θ*, 180° ± *θ*,, 360° ± *θ* etc .,are angles allied to
the angle θ. Using trigonometric ratios of the allied angles we can find the
trigonometric ratios of any angle.

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11th Business Mathematics and Statistics(EMS) : Chapter 4 : Trigonometry : Trigonometric ratios | Definition, Formula, Solved Example Problems, Exercise | Mathematics

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