Domain and Range of Trigonometric Functions:

PYQ-2023-Function-Q5

Basic Trigonometric Function Formulae:

$\quad $ By using a right-angled triangle as a reference, the trigonometric functions and identities are derived:

• $ \sin \theta= \frac{\text{Opposite Side}}{Hypotenuse} $

• $ \cos \theta= \frac{\text{Adjacent Side}}{Hypotenuse} $

• $ \tan \theta= \frac{\text{Opposite Side}}{\text{Adjacent Side}}$

• $ \sec \theta= \frac{Hypotenuse}{\text{Adjacent Side}} $

• $ \csc \theta= \frac{Hypotenuse}{\text{Opposite Side}} $

• $ \cot \theta= \frac{\text{Adjacent Side}}{\text{Opposite Side}}$

Pythagorean Identities:

PYQ-2023-Trigonometric_Equations-Q2, PYQ-2023-Trigonometric_Ratios-Q2, PYQ-2023-Trigonometric_Ratios-Q4, PYQ-2023-Definite_Integration-Q16

• $ \sin^2 \theta + \cos^2 \theta = 1$

• $ \tan^2 \theta + 1 = \sec^2 \theta$

• $ 1 + \cot^2 \theta = \csc^2 \theta$

Even and Odd Formulae:

PYQ-2023-Trigonometric_Ratios-Q1

• $ \sin (-\theta) = -\sin \theta$

• $ \csc (-\theta) = -\csc \theta$

• $ \cos (-\theta) = \cos \theta$

• $ \sec (-\theta) = \sec \theta$

• $ \tan (-\theta) = -\tan \theta$

• $ \cot (-\theta) = -\cot \theta$

Periodic Formulae:

PYQ-2023-Trigonometric_Ratios-Q3

$\quad$ If $\mathrm{n}$ is an integer

• $ \sin(\theta + 2\pi n) = \sin \theta$

• $ \csc(\theta + 2\pi n) = \csc \theta$

• $ \cos(\theta + 2\pi n) = \cos \theta$

• $ \sec(\theta + 2\pi n) = \sec \theta$

• $ \tan(\theta + \pi n) = \tan \theta$

• $ \cot(\theta + \pi n) = \cot \theta$

Reciprocal Identities:

PYQ-2023-Indefinite_Integration-Q2, PYQ-2023-Trigonometric_Ratios-Q3

$\quad$ The Reciprocal Identities are given as:

• $ \csc \theta=\frac{1}{\sin \theta}$

• $ \sec \theta=\frac{1}{\cos \theta}$

• $ \cot \theta=\frac{1}{\tan \theta}$

• $ \sin \theta=\frac{1}{\csc \theta}$

• $ \cos \theta=\frac{1}{\sec \theta}$

• $ \tan \theta=\frac{1}{\cot \theta}$

Periodicity Identities (in Radians):

PYQ-2023-Trigonometric_Equations-Q2

$\quad$ These formulas are used to shift the angles by $\pi / 2, \pi, 2 \pi$, etc. They are also called co-function identities.

• $ \sin(\frac{\pi}{2} - A) = \cos A$ and $\cos(\frac{\pi}{2} - A) = \sin A$

• $ \sin(\frac{\pi}{2} + A) = \cos A$ and $\cos(\frac{\pi}{2} + A) = -\sin A$

• $ \sin(\frac{3\pi}{2} - A) = -\cos A$ and $\cos(\frac{3\pi}{2} - A) = -\sin A$

• $ \sin(\frac{3\pi}{2} + A) = -\cos A$ and $\cos(\frac{3\pi}{2} + A) = \sin A$

• $ \sin(\pi - A) = \sin A$ and $\cos(\pi - A) = -\cos A$

• $ \sin(\pi + A) = -\sin A$ and $\cos(\pi + A) = -\cos A$

• $ \sin(2\pi - A) = -\sin A$ and $\cos(2\pi - A) = \cos A$

• $ \sin(2\pi + A) = \sin A$ and $\cos(2\pi + A) = \cos A$

Sum & Difference Identities:

PYQ-2023-Definite_Integration-Q10

• $ \sin (x+y)=\sin (x) \cos (y)+\cos (x) \sin (y)$

• $ \cos (x+y)=\cos (x) \cos (y)-\sin (x) \sin (y)$

• $ \tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \cdot \tan y}$

• $ \sin (x-y)=\sin (x) \cos (y)-\cos (x) \sin (y)$

• $ \cos (x-y)=\cos (x) \cos (y)+\sin (x) \sin (y)$

• $ \tan (x-y)=\frac{\tan x-\tan y}{1+\tan x \cdot \tan y}$

Double Angle Identities:

PYQ-2023-Trigonometric_Equations-Q2, PYQ-2023-Indefinite_Integration-Q2, PYQ-2023-Trigonometric_Ratios-Q2, PYQ-2023-Definite_Integration-Q23

• $ \sin (2 x)=2 \sin x \cdot \cos x=\frac{2 \tan x}{1+\tan^2 x}$

• $ \cos 2 x=\cos^2 x-\sin^2 x=\frac{1-\tan^2 x}{1+\tan^2 x}$

• $ \cos (2 x)=2 \cos^2(x)-1=1-2 \sin^2(x)$

• $ \tan 2 x=\frac{2 \tan x}{1-\tan^2 x}$

• $ \sec 2 x=\frac{\sec^2 x}{2-\sec^2 x}$

• $ \csc 2 x=\frac{\sec x \cdot \csc x}{2}$

Triple Angle Identities:

PYQ-2023-Definite_Integration-Q24

• $ \sin 3 x=3 \sin x-4 \sin^3 x$

• $ \cos 3 x=4 \cos^3 x-3 \cos x$

• $ \tan 3 x=\frac{3 \tan x-\tan^3 x}{1-3 \tan^2 x}$

Half Angle Identities:

• $ \sin \frac{x}{2}= \pm \sqrt{\frac{1-\cos x}{2}}$

• $ \cos \frac{x}{2}= \pm \sqrt{\frac{1+\cos x}{2}}$

• $ \tan \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos (x)}{1+\cos (x)}}$

Product Identities:

PYQ-2023-Trigonometric_Equations-Q1

• $ \sin x \cdot \cos y=\frac{\sin (x+y)+\sin (x-y)}{2}$

• $ \cos x \cdot \cos y=\frac{\cos (x+y)+\cos (x-y)}{2}$

• $ \sin x \cdot \sin y=\frac{\cos (x-y)-\cos (x+y)}{2}$

Sum to Product Identities:

• $ \sin x+\sin y=2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}$

• $ \sin x-\sin y=2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}$

• $ \cos x+\cos y=2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}$

• $ \cos x-\cos y=-2 \sin \frac{x+y}{2} \sin \frac{x-y}{2}$

Important Trigonometric Ratios:

PYQ-2023-Trigonometric_Ratios-Q3

• $ \sin \mathrm{n} \pi=0 \quad ; \quad \cos \mathrm{n} \pi=(-1) \quad ; \tan \mathrm{n} \pi=0, \quad$ where $\mathrm{n} \in \mathrm{I}$

• $ \sin 15^{\circ}$ or $\sin \frac{\pi}{12}=\frac{\sqrt{3}-1}{2 \sqrt{2}}=\cos 75^{\circ}$ or $\cos \frac{5 \pi}{12}$

• $ \cos 15^{\circ} \text { or } \cos \frac{\pi}{12}=\frac{\sqrt{3}+1}{2 \sqrt{2}}=\sin 75^{\circ}$ or $\sin \frac{5 \pi}{12}$

• $ \tan 15^{\circ}=\frac{\sqrt{3}-1}{\sqrt{3}+1}=2 \sqrt{3}=\cot 75^{\circ} $

• $ \tan 75^{\circ}=\frac{\sqrt{3}+1}{\sqrt{3}-1}=2+\sqrt{3}=\cot 15^{\circ}$

• $ \sin \frac{\pi}{10} $ or $\sin 18^{\circ}=\frac{\sqrt{5}-1}{4} \cos 36^{\circ} \hspace{1mm} $ or $ \hspace{1mm} \cos \frac{\pi}{5}=\frac{\sqrt{5}+1}{4}$

Range of Trigonometric Expression:

PYQ-2023-Trigonometric_Ratios-Q1, PYQ-2023-Trigonometric_Ratios-Q2

$$-\sqrt{a^{2}+b^{2}} \leq a \sin \theta+b \cos \theta \leq \sqrt{a^{2}+b^{2}}$$

Sine Series:

$$ \sin \alpha+\sin (\alpha+\beta)+\sin (\alpha+2 \beta)+\ldots \ldots+\sin (\alpha+\overline{n-1} \beta)=\frac{\sin \frac{n \beta}{2}}{\sin \frac{\beta}{2}}\sin \left(\alpha+\frac{n-1}{2} \beta\right)$$

Cosine Series:

$$\cos \alpha+\cos (\alpha+\beta)+\cos (\alpha+2 \beta)+\ldots \ldots+\cos (\alpha+\overline{n-1} \beta)=\frac{\sin \frac{n \beta}{2}}{\sin \frac{\beta}{2}}=\cos \left(\alpha+\frac{n-1}{2} \beta\right)$$

Trigonometric Equations:

PYQ-2023-Trigonometric_Ratios-Q1

• Principal Solutions

  Solutions which lie in the interval $[0,2 \pi)$ are called Principal solutions.

• General Solution PYQ-2023-Trigonometric_Equations-Q1 PYQ-2023-Trigonometric_Equations-Q2

• $ \sin \theta=\sin \alpha \Rightarrow \theta=\mathrm{n} \pi+(-1)^{\mathrm{n}} \alpha $ where $ \alpha \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right], \mathrm{n} \in \mathrm{I} .$

• $\cos \theta=\cos \alpha \Rightarrow \theta=2 \mathrm{n} \pi \pm \alpha \ $ where $ \alpha \in[0, \pi], \mathrm{n} \in \mathrm{I}.$

• $ \tan \theta = \tan \alpha \Rightarrow \theta = n \pi + \alpha $ where $ \alpha \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $, $ n \in \mathbb{Z} $.

• $\sin ^{2} \theta=\sin ^{2} \alpha, \cos ^{2} \theta=\cos ^{2} \alpha, \tan ^{2} \theta=\tan ^{2} \alpha \Rightarrow \theta=\mathrm{n} \pi \pm \alpha$.