Trigonometrical Functions or Trigonometrical Ratios

Definition

Trigonometrical functions or trigonometrical ratios are the ratio of sides of a right-angled triangle taken two at a time. This can further be understood by the explanation given below.

Let ABC be a right-angled triangle in which $ \displaystyle \angle C$ is a right angle. Let $ \displaystyle \angle A$ be an acute angle and let $ \displaystyle AC=b,BC=p\text{ and }AB=h\text{ (w}\text{.r}\text{.t }\angle A\text{)}$.

Then,

$ \displaystyle {\sin A=\dfrac{p}{h}=\dfrac{{BC}}{{AB}};}$ $ \displaystyle {\cos A=\dfrac{b}{h}=\dfrac{{AC}}{{AB}}}$

$ \displaystyle {\tan A=\dfrac{p}{b}=\dfrac{{BC}}{{AC}};}$ $  \displaystyle {\cot A=\dfrac{b}{p}=\dfrac{{AC}}{{BC}}}$

$ \displaystyle {\sec A=\dfrac{h}{b}=\dfrac{{AB}}{{AC}};}$ $ \displaystyle {\text{cosec}A=\dfrac{h}{p}=\dfrac{{AB}}{{BC}}}$

The abbreviations stand for sine, cosine, tangent, cotangent, secant, and cosecant of A respectively. These functions of angle A are called trigonometrical functions or trigonometrical ratios.

Cartesian co-ordinate system

The definition given here does not differ from the one given earlier. But it is more scientific, since it undoubtedly gives us information about the sign of the trigonometrical ratios, whether positive or negative, according to the different measurement of the angles. Here they are being defined and analyzed by means of co-ordinates.

Let $ \displaystyle {X}’OX$ and $ \displaystyle YO{Y}’$ be the two co-ordinate axes. Let us suppose that $ \displaystyle {X}’OX$ is the initial line and OP is a revolving line which makes an $latex \displaystyle \angle POX$. It is obvious that if OP is rotated around O, then we shall get a circle whose center will be O.

Let the co-ordinate of P be (x, y). Though P is in the first quadrant in the figure but actually there is no such restriction; P may be in other quadrants as well. As stated earlier, the quadrant in which P lies will determine the signs of x and y, positive or negative.

Therefore, if PM is perpendicular to OX from P, then OM = x and PM = y.

Let $ \displaystyle OP=r$ and $ \displaystyle \angle POX=\theta $.

Now from right angled $ \displaystyle \Delta POM$,

$ \displaystyle {\sin \theta =\dfrac{{PM}}{{OP}}=\dfrac{y}{r};}$ $ \displaystyle {\cos ec\text{ }\theta =\dfrac{r}{y}\text{ when }y\ne 0}$

$ \displaystyle {\cos \theta =\dfrac{{OM}}{{OP}}=\dfrac{x}{r};}$ $ \displaystyle {\sec \theta =\dfrac{r}{x}\text{ when }x\ne 0}$

$ \displaystyle {\tan \theta =\dfrac{{PM}}{{OM}}=\dfrac{y}{x}\text{ when }x\ne 0}$

$ \displaystyle {\text{and }\cot \theta =\dfrac{{OM}}{{PM}}=\dfrac{x}{y}}$ $ \displaystyle {\text{when }y\ne 0}$

We leave $ \displaystyle \tan \theta $ undefined at $ \displaystyle x=0$. Similarly we leave $ \displaystyle \cot \theta $ undefined at $ \displaystyle y=0$. All the trigonometrical ratios defined above are called trigonometrical functions.

Trigonometrical Ratios are Constant for the Same Angle (Uniqueness Theorem)

Let S be any point on the bounding line OP. Let $ \displaystyle \angle XOP=\theta $ (here we take $ \displaystyle \theta $ to be an acute angle). If the co-ordinates of S be $ \displaystyle \left( {{{x}_{1}},{{y}_{1}}} \right)$, then according to the definition, from the $ \displaystyle \Delta SON$,

$ \displaystyle \begin{array}{l}\sin \theta =\dfrac{{{{y}_{1}}}}{{{{r}_{1}}}};\text{ }\cos \theta =\dfrac{{{{x}_{1}}}}{{{{r}_{1}}}}\\\tan \theta =\dfrac{{{{y}_{1}}}}{{{{x}_{1}}}}\text{ etc}\text{.}\end{array}$

Since, the triangles OSN and OPM are similar,

therefore $ \displaystyle \dfrac{{SN}}{{OS}}=\dfrac{{PM}}{{OP}}\text{ i}\text{.e}\text{. }\sin \theta =\dfrac{y}{r}=\dfrac{{{{y}_{1}}}}{{{{r}_{1}}}}$ and similarly $ \displaystyle \text{cos}\theta =\dfrac{x}{r}=\dfrac{{{{x}_{1}}}}{{{{r}_{1}}}}\text{ etc}\text{.}$

But S is any arbitrary point on OP. Similarly it can be shown that wherever we take any point OP, the trigonometrical ratios with respect to that point will be equal to $ \displaystyle \dfrac{y}{r},\dfrac{x}{r},\text{ etc}$. Respectively i.e. $ \displaystyle \sin \theta =\dfrac{y}{r},\text{ }\cos \theta =\dfrac{x}{r},\text{ etc}$.

Therefore, it follows from the above results that the trigonometrical ratios remain constant for the same angle i.e. so long as the angle does not change. Thus for a given angle, trigonometrical ratios are unique.

Signs of Trigonometrical Functions

Let O be the center of a circle whose radius is r. Let P be any point on the circumference of the circle such that OP makes an angle $ \displaystyle \theta $ with the initial line OX.. Let co-ordinates of P be $ \displaystyle \left( {x,y} \right)$ i.e. $ \displaystyle OM=x\text{ and }PM=y$.

Then, from the right angled $ \displaystyle \Delta OMP$,

$ \displaystyle \begin{array}{l}O{{M}^{2}}+P{{M}^{2}}=O{{P}^{2}}\\\text{i}\text{.e}\text{. }{{x}^{2}}+{{y}^{2}}={{r}^{2}}\end{array}$

Again, from the definition,

$ \displaystyle \sin \theta =\dfrac{y}{r},\text{ }\cos \theta =\dfrac{x}{r},\text{ }\tan \theta =\dfrac{y}{x}\text{ etc}$

Since, r is always taken to be positive, therefore the signs of the trigonometrical ratios will depend upon the co-ordinates $ \displaystyle \left( {x,y} \right)$ of the point P, i.e. on the quadrant in which $ \displaystyle \theta $ lies.

For example, if $ \displaystyle \theta $ is in the second quadrant, then $ \displaystyle x\text{ is }-ve\left( {x<0} \right)$ and $  \displaystyle y\text{ is +}ve\left( {y>0} \right)$.

$ \displaystyle \therefore \sin \theta =+,\text{ }\cos \theta =-,\text{ }\tan \theta =-,\text{ etc}\text{.}$

The above result can be summarised as follows:

In the first quadrant

$ \displaystyle x>0,y>0,r>0$

$ \displaystyle \therefore $ All the trigonometrical ratios are positive.

In the second quadrant

$ \displaystyle x<0,y>0,r>0$

$ \displaystyle \therefore $ Only sine and its reciprocal cosec are positive.

In the third quadrant

$ \displaystyle x<0,y<0,r>0$

$ \displaystyle \therefore $ Only tan and its reciprocal cot are positive.

In the fourth quadrant

$ \displaystyle x>0,y<0,r>0$

$ \displaystyle \therefore $ Only cos and sec are positive

We can express the above results in a tabular form or  symbolic figure below which will tell s as to which of the trigonometrical ratios (and hence their reciprocal ratios also)are positive in which quadrants: