Trigonometric Functions and Identities
2.0 Trigonometric functions
2.0 Trigonometric functions
The trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its side. In previous topic trigonometric ratios, we were restricted to acute angle only. In this topic, we will discuss those trigonometric ratios in terms of trigonometric functions in which there is no restriction on the angle.
There are $6$ main trigonometric functions:
- Sine (sin)
- Cosine (cos)
- Tangent (tan)
- Secant (sec)
- Cosecant (cosec)
- Cotangent (cot)
Consider a circle of radius $1$ unit with center at origin $(O)$ of the coordinate axes. Let $A(a,b)$ be any point on the circle with angle $XOA=x$ radian, i.e., the length of arc $XA=x$ as shown in the figure.
We define $cosx=a$ and $sinx=b$. Since $\Delta OMA$ is a right angle triangle, using pythagoras theorem, we can write $$\begin{equation} \begin{aligned} O{A^2} = O{M^2} + M{A^2}\quad {\text{where }}OA = {\text{radius = 1 unit}} \\ 1 = {a^2} + {b^2} \\ \Rightarrow {\cos ^2}x + {\sin ^2}x = 1 \\\end{aligned} \end{equation} $$
Since one complete revolution subtends an angle of $2\pi$ radian at the centre of the circle, $$\angle XOY = \frac{\pi }{2},\angle XOX' = \pi ,\angle XOY' = \frac{{3\pi }}{2}$$
The coordinates of the points $X$, $Y$, $X'$ and $Y'$ are, respectively, $(1,0)$, $(0,1)$, $(–1,0)$ and $(0,–1)$. Therefore, we can say that$$\begin{equation} \begin{aligned} \cos {0^ \circ } = 1,\quad \sin {0^ \circ } = 0 \\ \cos \frac{\pi }{2} = 0,\quad \sin \frac{\pi }{2} = 1 \\ \cos \pi = - 1,\quad \sin \pi = 0 \\ \cos \frac{{3\pi }}{2} = 0,\quad \sin \frac{{3\pi }}{2} = - 1 \\ \cos 2\pi = 1,\quad \sin 2\pi = 0 \\\end{aligned} \end{equation} $$
Now, if we take one complete revolution from the point $A$, we again come back to same point $A$. Thus, we also observe that if $x$ increases (or decreases) by any integral multiple of $2\pi$, the values of sine and cosine functions do not change. Thus,$$\begin{equation} \begin{aligned} \sin \left( {2n\pi + x} \right) = \sin x,\;n \in Z \\ \cos \left( {2n\pi + x} \right) = \cos x,\;n \in Z \\\end{aligned} \end{equation} $$Using the above results, we can say that $$\begin{equation} \begin{aligned} \sin x = 0 \Rightarrow x = 0, \pm \pi , \pm 2\pi , \pm 3\pi ,...,n\pi \quad {\text{where }}n{\text{ is an integer}} \\ \cos x{\text{ = 0}} \Rightarrow x = \pm \frac{\pi }{2}, \pm \frac{{3\pi }}{2}, \pm \frac{{5\pi }}{2},...,(2n + 1)\frac{\pi }{2}\quad {\text{where }}n{\text{ is an integer}} \\\end{aligned} \end{equation} $$
| $$0^\circ $$ | $$30^\circ {\text{ or }}\frac{\pi }{6}$$ | $$45^\circ {\text{ or }}\frac{\pi }{4}$$ | $$60^\circ {\text{ or }}\frac{\pi }{3}$$ | $$90^\circ {\text{ or }}\frac{\pi }{2}$$ | $$180^\circ {\text{ or }}\pi $$ | $$270^\circ {\text{ or }}\frac{3\pi }{2}$$ | $$360^\circ {\text{ or }}2\pi $$ | |
| sin | $0$ | $\frac{1}{2}$ | $\frac{1}{{\sqrt 2 }}$ | $\frac{{\sqrt 3 }}{2}$ | $1$ | $0$ | $-1$ | $0$ |
| cos | $1$ | $\frac{{\sqrt 3 }}{2}$ | $\frac{1}{{\sqrt 2 }}$ | $\frac{1}{2}$ | $0$ | $-1$ | $0$ | $1$ |
| tan | $0$ | $\frac{1}{{\sqrt 3 }}$ | $1$ | $\sqrt 3 $ | not defined | $0$ | not defined | $0$ |
| cosec | not defined | $2$ | $\sqrt 2 $ | $\frac{2}{{\sqrt 3 }}$ | $1$ | not defined | $-1$ | not defined |
| sec | $1$ | $\frac{2}{{\sqrt 3 }}$ | $\sqrt 2 $ | $2$ | not defined | $-1$ | not defined | $1$ |
| cot | not defined | ${\sqrt 3 }$ | $1$ | $\frac{1}{{\sqrt 3 }}$ | $0$ | not defined | $0$ | not defined |
