## Trigonometric Functions

• ### Trigonometric Functions

The trigonometric functions are functions of angles. In the blogpost ‘Angle and Its Measurement‘, we studied what an angle is and how it is measured. An important concept to be recalled is that of coterminal angles. These are the angles, whose initial and terminal rays are identical and their magnitudes differ by an integral multiple of $360^o$ or $2 \pi ^c$.

There are 6 basic trigonometric functions:

1) Sine (abbr. sin)

2) Cosine (abbr. cos)

3) Tangent (abbr. tan)

4) Cosecant (abbr. cosec)

5) Secant (abbr. sec)

6) Cotangent (abbr. cot)

• ### Definitions

Consider a circle (radius $r$) and a point $P$ on it. Fix the $X$ and $Y$ axes such that the origin is the center of the circle. Let $\theta$ be the angle made by $OP$ with positive $X$ axis.

Recall: Length of perpendicular from a point on $X$ ($Y$) axis gives the magnitude of $y$ ($x$) coordinate of the point.

Let coordinates of $P$ be $x$ and $y$. Then,

${sin (\theta) = \frac {y}{r}, \ cos (\theta) = \frac {x}{r}}$

${tan (\theta) = \frac {y}{x}, x \ne 0, \ cot (\theta) = \frac {x}{y}, y \ne 0}$

${cosec (\theta) = \frac {r}{y}, y \ne 0 , \ sec (\theta) = \frac{r}{x}, x \ne 0}$

As the location of point $P$ changes (along the circle), $\theta$ changes and so the ratios. Coterminal angles have same trigonometric ratios.

It may also be noted that the ratios are independent of radius of circle.

• ### Alternative Definitions

${tan (\theta) = \frac {sin (\theta)}{cos (\theta)}, \ cot (\theta) = \frac {cos (\theta)}{sin (\theta)}}$

${cosec (\theta) = \frac {1}{sin (\theta)}, \ sec (\theta) = \frac {1}{cos (\theta)}}$

• ### Important Identities

For any angle $\theta$,

${sin^2 (\theta)+ cos^2 (\theta)=1}$

${1 + tan^2 (\theta)= sec^2 (\theta)}$

${1 + cot^2 (\theta) = cosec^2 (\theta)}$

These can be derived from the Pythagoras’ theorem.

• ### Domains and Ranges of Trigonometric Functions

1) $Sine$ Function: Domain $\mathbb R$ and Range $[-1,1]$

2) $Cosine$ Function: Domain $\mathbb R$ and Range $[-1,1]$

3) $Tangent$ Function: Domain $\mathbb R - \{ \theta | \ \theta = (2n-1)\frac {\pi}{2}, n \in \mathbb Z \}$ and Range $\mathbb R$

4) $Cotangent$ Function: Domain $\mathbb R - \{ \theta | \ \theta = n \pi, n \in \mathbb Z \}$ and Range $\mathbb R$

5) $Cosecant$ Function: Domain $\mathbb R - \{ \theta | \ \theta = {n \pi}, n \in \mathbb Z \}$ and Range $\mathbb R - (-1,1)$

6) $Secant$ Function: Domain $\mathbb R - \{ \theta | \theta = (2n-1)\frac {\pi}{2}, n \in \mathbb Z \}$ and Range $\mathbb R - (-1,1)$

• ### Standard Angles

The angles $0, \frac {\pi}{6}, \frac {\pi}{4}, \frac {\pi}{3}, \frac {\pi}{2}, \pi, \frac {3 \pi}{2}, 2 \pi$ are termed as the standard angles. Given below are their $sine$ and $cosine$ values. The other ratios can be found using these (if they are defined).

• ### Signs of Trigonometric Ratios in the Quadrants

Note: $r$ is always positive, $x$ is positive in $1st$ and $4th$ quadrants, $y$ is positive in $1st$ and $2nd$ quadrants.

• ### Periodicity

A characteristic of trigonometric functions is their periodicity. The values get repeated after a certain fixed interval. This can also be seen from the graphs.

A function $f(x)$ is said to be periodic, if $\forall x$ in its domain, $f(x+T)= f(x)$, where $T$ is the period.

The least positive value of $T$ is known as the fundamental period.

Sine, Cosine, Cosecant and Secant are periodic with fundamental period $2 \pi$.

Tangent and Cotangent are periodic with fundamental period $\pi$.

• ### Odd and Even Trigonometric Functions

A function $f(x)$ is said to be odd, when $\forall x$ in its domain,

${f(-x) = - f(x)}$

Sine, Tangent, Cosecant, Cotangent are odd functions.

A function $f(x)$ is said to be even, when $\forall x$ in its domain,

${f(-x) = f(x)}$

Cosine and Secant are even functions.