Trigonometric Functions

Relation between different sides and angles of a right angled triangle are called trigonometric ratios.

Trigonometric ratios can be represented as

$\sin \theta =\frac{Perpendicular}{Hypotenuse}=\frac{BC}{AC}$

$\cos \theta =\frac{Base}{Hypotenuse}=\frac{AB}{AC}$

$\tan \theta =\frac{Prerpendicular}{Base}=\frac{BC}{AB},\cos ec\theta =\frac{1}{\sin \theta }$

$\sec \theta =\frac{1}{\cos \theta },\cot \theta =\frac{\cos \theta }{\sin \theta }=\frac{1}{\tan \theta }$

Trigonometric (or Circular) Functions

Let $X^{\prime }OX$and $YO{Y}^{\prime }$ be the coordinate axes. Taking $O$ as the centre and a unit radius draw a circle, cutting the coordinate axes at $A,B,A^{\prime }$and $B^{\prime }$ as shown in the figure.

$\left[\therefore \angle AOP=\frac{arcAP}{radiusOP}=\frac{\theta }{1}=\theta {}^{\text{o}},USING\theta =\frac{l}{r}\right]$

Now, the six circular functions may be defined as under

(i) $\cos \theta =x$ (ii) $\sin \theta =y$ (iii) $\sec \theta =\frac{1}{x},x\ne 0$

(iv) $\cos ec\theta =\frac{1}{y},y\ne 0$ (v) $\tan \theta =\frac{y}{x},x\ne 0$ (iv) $\cot \theta =\frac{x}{y},y\ne 0$

Domain and Range Table

Range of Modulus Functions

$\left|\sin \theta \right|\le 1,\left|\cos \theta |\le 1,\right|\sec \theta |\ge 1,|\cos ec\theta |\ge 1$ for all values of $\theta$, for which the functions are defined.

Trigonometric Identities

An equation involving trigonometric functions which is true for all those angles for which the functions are defined is called trigonometrical identity . Some identities are

1. $\sin \theta =\frac{1}{\cos ec\theta }or\cos ec\theta =\frac{1}{\sin \theta }$
2. $\cos \theta =\frac{1}{\sec \theta }or\sec \theta =\frac{1}{\cos \theta }$
3. $\cot \theta =\frac{1}{\tan \theta }=\frac{\cos \theta }{\sin \theta }or\tan \theta =\frac{1}{\cot }=\frac{\sin \theta }{\cos \theta }$
4. ${\cos }^2\theta +{\sin }^2\theta =1or1-{\cos }^2\theta ={\sin }^2\theta or1-{\sin }^2\theta ={\cos }^{2}0$
5. $1+{\tan }^2\theta ={\sec }^2\theta or{\tan }^2\theta ={\sec }^2\theta -1or{\sec }^2\theta -{\tan }^2\theta =1$
6. $1+{\cot }^2\theta =\cos e{c}^2\theta or{\cot }^2\theta =\cos e{c}^2\theta -1or\cos e{c}^2\theta -{\cot }^2\theta =1$

Transformation of One Trigonometric Ratio to Another Trigonometric Ratios

Sign of Trigonometric Ratios

Trigonometric Ratios of Some standard Angles

Trigonometric Ratios of Some Special Angles

Trigonometric Ratios of Allied Angles

Two angles are said to be allied when t heir sum or difference is either zero or a multiple of $90{}^{\text{o}}$. The angles $-\theta ,90{}^{\text{o}}\pm \theta ,180{}^{\text{o}}\pm \theta ,270{}^{\text{o}}\pm \theta ,360{}^{\text{o}}-\theta$etc. , are angles allied to the angle $\theta$,if $\theta$is measured in degrees.

Trigonometric Periodic Functions

A function $f\left(x\right)$ is said to be periodic, if there exists a real number $T>0$ such that $f\left(x+T\right)=f\left(x\right)$ for all $x.T$ is called the periodic.

Key Points

1. $\sin \theta ,\cos \theta ,\cos ec\theta$and $\sec \theta$ has a period of $2\pi$.
2. $\tan \theta ,\cot \theta$has a period of $\pi .$
3. Period of $sink\theta$ is $\frac{2\pi }{k}$.
4. Period of $\tan k\theta$ is $\frac{\pi }{k}\cdot$
5. Period of ${\sin }^n\theta ,{\cos }^n\theta ,{\sec }^n\theta$ and $\cos e{c}^n\theta$ is $2\pi ,$if $n$ is odd and, $\pi$if $n$ is even.
6. Period of ${\tan }^n\theta ,{\cot }^n\theta$is $\pi$, if $n$ is even or odd.
7. Period of $\left|\sin \theta \right|,\left|\cos \theta \right|,\left|\tan \theta \right|,\left|\cot \theta \right|,\left|\sec \theta \right|,$and $\left|\cos ec\theta \right|$ is $\pi$.
8. Period of $\left|\sin \theta \right|+\left|\cos \theta \right|,\left|\tan \theta \right|+\left|\cot \theta \right|$and $\left|\sec \theta \right|+\left|\cos ec\theta \right|$is $\frac{\pi }{2}.$

Maximum and Minimum Values of Trigonometric Expressions

1. Maximum value of $\alpha \cos \theta \pm b\sin \theta =\sqrt{a^2+b^2}$

Minimum value of $\alpha \cos \theta \pm b\sin \theta =-\sqrt{a^2+b^2}$

2. Maximum value of $\alpha \cos \theta \pm b\sin \theta +c=c+\sqrt{a^2+b^2}$

Minimum value of $\alpha \cos \theta \pm b\sin \theta +c=c-\sqrt{a^2+b^2}$