Trigonometrical ratios and Identities

1. An angle is positive, if it is measured in anti clockwise direction and is negative if it is measured in clock wise direction

2. Angle in radian $=\dfrac{\text { length of the arc }}{\text { Radius of the circle }}$

$\text { i.e; } \theta=\dfrac{\ell}{\mathrm{r}}$

3.

Note: $\dfrac{\mathrm{D}}{90}=\dfrac{\mathrm{G}}{100}=\dfrac{2 \mathrm{C}}{\pi}$ OR (see the graph below)

4. Basic trigonometrical identities

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

$\begin{aligned} & \Rightarrow \quad|\sin \theta| \leq 1 \text { and }|\cos \theta| \leq 1 \\ & \Rightarrow \quad-1 \leq \sin \theta \leq 1 \text { and }-1 \leq \cos \theta \leq 1 \end{aligned}$

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

$\begin{array}{ll} \Rightarrow & |\sec \theta| \geq 1 \\ \Rightarrow & \sec \theta \leq-1 \quad \text { or } \sec \theta \geq 1 \end{array}$

$\tan \theta$ may take any real value

(iii) $\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1$

$\Rightarrow|\operatorname{cosec} \theta| \geq 1$

$\Rightarrow \operatorname{cosec} \theta \leq-1 \quad$ or $\operatorname{cosec} \theta \geq 1$

$\cot \theta$ may take any real value.

5. Sign of Trigonometrical Ratio : To find the sign of a trigonometrical ratio, remember the sentence

Where A stands for all ratios are positive in 1 st quadrant

$\mathrm{S}$ stands for $\sin \&$ its reciprocal are positive in $2^{\text {nd }}$ quadrant

$T$ stands for tan & its reciprocal are positive in $3^{\text {rd }}$ quadrant

C stands for $\cos \&$ its reciprocal are positive in $4^{\text {rd }}$ quadrant

6. Domain & Range

7. Trigonometric ratios in terms of each of the other

8. Sum and Difference formula.

(i) $\quad \sin (\mathrm{A} \pm \mathrm{B})=\sin \mathrm{A} \cos \mathrm{B} \pm \cos \mathrm{A} \sin \mathrm{B}$

(ii) $\quad \cos (\mathrm{A} \pm \mathrm{B})=\cos \mathrm{A} \cos \mathrm{B} \bar{\mp} \sin \mathrm{A} \sin \mathrm{B}$

(iii) $\tan (\mathrm{A} \pm \mathrm{B})=\dfrac{\tan \mathrm{A} \pm \tan \mathrm{B}}{1 \mp \tan \mathrm{A} \tan \mathrm{B}}$

(iv) $\tan \left(\dfrac{\pi}{4} \pm \mathrm{A}\right)=\dfrac{1 \pm \tan \mathrm{A}}{1 \mp \tan \mathrm{A}}$

(v) $\quad \cot (\mathrm{A} \pm \mathrm{B})=\dfrac{\cot \mathrm{A} \cot \mathrm{B} \mp 1}{\cot \mathrm{B} \pm \cot \mathrm{A}}$

(vi) $\sin (\mathrm{A}+\mathrm{B}) \sin (\mathrm{A}-\mathrm{B})=\sin ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B}=\cos ^{2} \mathrm{~B}-\cos ^{2} \mathrm{~A}$

(vii) $\quad \cos (\mathrm{A}+\mathrm{B}) \cos (\mathrm{A}-\mathrm{B})=\cos ^{2} \mathrm{~A}-\sin ^{2} \mathrm{~B}=\cos ^{2} \mathrm{~B}-\sin ^{2} \mathrm{~A}$

9. Multiple and half angles

(i) $\sin 2 \theta=\left\{\begin{array}{l}2 \sin \theta \cos \theta \\ \dfrac{2 \tan \theta}{1+\tan ^{2} \theta}\end{array}\right.$

Also, $\sqrt{1 \pm \sin 2 \theta}=|\cos \theta \pm \sin \theta|$

(ii) $\cos 2 \theta=\left\{\begin{array}{l}\cos ^{2} \theta-\sin ^{2} \theta \\ 2 \cos ^{2} \theta-1 \\ 1-2 \sin ^{2} \theta \\ \dfrac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}\end{array}\right.$

Also $1+\cos 2 \theta=2 \cos ^{2} \theta$

$1-\cos 2 \theta=2 \sin ^{2} \theta$

(iii) $\tan 2 \theta=\dfrac{2 \tan \theta}{1-\tan ^{2} \theta}$

Also $\dfrac{1-\cos 2 \theta}{\sin 2 \theta}=\tan \theta$

$\dfrac{1-\cos 2 \theta}{1+\cos 2 \theta}=\tan ^{2} \theta$

(iv) $\sin 3 \theta=\left\{\begin{array}{l}3 \sin \theta-4 \sin ^{3} \theta \\ 4 \sin \left(\dfrac{\pi}{3}-\theta\right) \sin \theta \sin \left(\dfrac{\pi}{3}+\theta\right)\end{array}\right.$

or $\sin ^{3} \theta=\dfrac{3 \sin \theta-\sin 3 \theta}{4}$

(v) $\quad \cos 3 \theta=\left\{\begin{array}{l}4 \cos ^{3} \theta-3 \cos \theta \\ 4 \cos \left(\dfrac{\pi}{3}-\theta\right) \cos \theta \cos \left(\dfrac{\pi}{3}+\theta\right)\end{array}\right.$

or $\cos ^{3} \theta=\dfrac{\cos 3 \theta+3 \cos \theta}{4}$

(vi) $\tan 3 \theta=\left\{\begin{array}{l}\dfrac{3 \tan \theta-\tan ^{3} \theta}{1-3 \tan ^{2} \theta} \\ \tan \left(\dfrac{\pi}{3}-\theta\right) \tan \theta \tan \left(\dfrac{\pi}{3}+\theta\right)\end{array}\right.$

10. Transformation formulae

(i) $\quad \sin \mathrm{C}+\sin \mathrm{D}=2 \sin \left(\dfrac{\mathrm{C}+\mathrm{D}}{2}\right) \cos \left(\dfrac{\mathrm{C}-\mathrm{D}}{2}\right)$

$\sin \mathrm{C}-\sin \mathrm{D}=2 \cos \left(\dfrac{\mathrm{C}+\mathrm{D}}{2}\right) \sin \left(\dfrac{\mathrm{C}-\mathrm{D}}{2}\right)$

$\cos \mathrm{C}+\cos \mathrm{D}=2 \cos \left(\dfrac{\mathrm{C}+\mathrm{D}}{2}\right) \cos \left(\dfrac{\mathrm{C}-\mathrm{D}}{2}\right)$

$\cos \mathrm{C}-\cos \mathrm{D}=-2 \sin \left(\dfrac{\mathrm{C}+\mathrm{D}}{2}\right) \sin \left(\dfrac{\mathrm{C}-\mathrm{D}}{2}\right)$

(ii) $\quad 2 \sin \mathrm{A} \cos \mathrm{B}=\sin (\mathrm{A}+\mathrm{B})+\sin (\mathrm{A}-\mathrm{B})$

$2 \cos A \sin B=\sin (A+B)-\sin (A-B)$

$2 \cos \mathrm{A} \cos \mathrm{B}=\cos (\mathrm{A}+\mathrm{B})+\cos (\mathrm{A}-\mathrm{B})$

$2 \sin A \sin B=\cos (A-B)-\cos (A+B)$

(ii) $\quad \tan \mathrm{A} _{ \pm} \tan \mathrm{B}=\dfrac{\sin (\mathrm{A} \pm \mathrm{B})}{\cos \mathrm{A} \cos \mathrm{B}}$

$\cot \mathrm{A} \pm \cot \mathrm{B}=\dfrac{\sin (\mathrm{B} \pm \mathrm{A})}{\sin \mathrm{A} \sin \mathrm{B}}$

(iv) $\quad \cot A-\tan A=2 \cot 2 \mathrm{~A}$

$\cot A+\tan A=2 \operatorname{cosec} 2 A=\dfrac{1}{\sin A \cos A}$

(v) $\quad 1 \pm \tan \mathrm{A} \tan \mathrm{B}=\dfrac{\cos (\mathrm{A} \mp \mathrm{B})}{\cos \mathrm{A} \cos \mathrm{B}}$

(vi) $\quad \cos \mathrm{A} \pm \sin \mathrm{A}=\sqrt{2} \sin \left(\dfrac{\pi}{4} \pm \mathrm{A}\right)=\sqrt{2} \cos \left(\dfrac{\pi}{4} \mp \mathrm{A}\right)$

11. Three angles

$\sin (A+B+C)=\cos A \cos B \cos C(\tan A+\tan B+\tan C-\tan A \tan B \tan C)$

$\cos (\mathrm{A}+\mathrm{B}+\mathrm{C})=\cos \mathrm{A} \cos \mathrm{B} \cos \mathrm{C}(1-\tan \mathrm{A} \tan \mathrm{B}-\tan B \tan \mathrm{C}-\tan C \tan \mathrm{A})$

$\tan (A+B+C)=\dfrac{\tan A+\tan B+\tan C-\tan A \tan B \tan C}{1-\tan A \tan B-\tan B \tan C-\tan C \tan A}$

12. Trigonometrical series

(i) $\quad \sin \mathrm{A}+\sin (\mathrm{A}+\mathrm{d})+\sin (\mathrm{A}+2 \mathrm{~d})+$ $+\sin (\mathrm{A}+(\mathrm{n}-1) \mathrm{d})=\dfrac{\sin \left(\mathrm{A}+\dfrac{(\mathrm{n}-1) \mathrm{d}}{2}\right) \sin \left(\dfrac{\mathrm{nd}}{2}\right)}{\sin \left(\dfrac{\mathrm{d}}{2}\right)}$

(ii) $\cos \mathrm{A}+\cos (\mathrm{A}+\mathrm{d})+\cos (\mathrm{A}+2 \mathrm{~d})+\ldots \ldots \ldots+\cos (\mathrm{A}+(\mathrm{n}-1) \mathrm{d})=\dfrac{\cos \left(\mathrm{A}+\dfrac{(\mathrm{n}-1) \mathrm{d}}{2}\right) \sin \left(\dfrac{\mathrm{nd}}{2}\right)}{\sin \left(\dfrac{\mathrm{d}}{2}\right)}$

(iii) $\cos \mathrm{A} \cdot \cos 2 \mathrm{~A} \cdot \cos 2^2 \mathrm{~A}$ ……………… $\cos 2^{\mathrm{n}-1} \mathrm{~A}=\dfrac{\sin \left(2^{\mathrm{n}} \mathrm{A}\right)}{2^{\mathrm{n}} \sin \mathrm{A}}$

(iv) $\quad(2 \cos \theta-1)(2 \cos 2 \theta-1)\left(2 \cos 2^{2} \theta-1\right) \ldots \ldots \ldots . .\left(2 \cos 2^{\mathrm{n}-1} \theta-1\right)=\dfrac{2 \cos 2^{\mathrm{n}} \theta+1}{2 \cos \theta+1} ; \mathrm{n} \in \mathrm{N}$

(v) $\tan \theta+2 \tan 2 \theta+2^{2} \tan 2^{2} \theta+2^{3} \tan 2^{3} \theta+\ldots \ldots \ldots . .+2^{\mathrm{n}} \tan 2^{\mathrm{n}} \theta+2^{\mathrm{n}+1} \cot 2^{\mathrm{n}+1} \theta=\cot \theta ; \forall \mathrm{n} \in \mathrm{N}$

(vi) $\sqrt{2+\sqrt{2+\sqrt{2+\ldots \ldots .+\sqrt{2+2 \cos 2^{\mathrm{n}} \theta}}}}=2 \cos \theta, \mathrm{n} \in \mathrm{N}$ where there are $\mathrm{n}$ square root signs on left hand side.

13. Greatest and least value of $\operatorname{asin} \theta \pm b \cos \theta$ is $\sqrt{a^{2}+b^{2}}$ and $-\sqrt{a^{2}+b^{2}}$ respectively

i.e. $-\sqrt{a^{2}+b^{2}} \leq \operatorname{asin} \theta \pm b \cos \theta \leq \sqrt{a^{2}+b^{2}}$

Also $\sin ^{2} \theta+\operatorname{cosec}^{2} \theta \geq 2$

$\cos ^{2} \theta+\sec ^{2} \theta \geq 2$

$\tan ^{2} \theta+\cot ^{2} \theta \geq 2$

14. Method of componendo and dividendo.

If $\dfrac{p}{q}=\dfrac{a}{b}$ then by componendo and dividendo we can write $\dfrac{p-q}{p+q}=\dfrac{a-b}{a+b}$ or $\dfrac{p+q}{p-q}=\dfrac{a+b}{a-b}$

Periodicity

All the six trigonometric functions are periodic $\sin \theta, \cos \theta, \operatorname{cosec} \theta, \sec \theta$ are periodic with period $2 \pi$ where $\tan \theta$ and $\cot \theta$ are period $\mathrm{c}$ with period $\pi$

Solved Examples

1. If $f(\theta)=\sin ^{4} \theta+\cos ^{4} \theta+1$, then the range of $f(\theta)$ is

(a) $\left[\dfrac{3}{2}, 2\right]$

(b) $\left[1, \dfrac{3}{2}\right]$

(c) $[1,2]$

(d) None of these

2. If $4 \mathrm{n} \alpha=\pi$, then the value of $\cot \alpha \cdot \cot 2 \alpha \cdot \cot 3 \alpha$ . $\cot (2 \mathrm{n}-1) \alpha$ is

(a) 1

(b) -1

(c) $\infty$

(d) None of these

3. If $\tan \dfrac{\alpha}{2}$ and $\tan \dfrac{\beta}{2}$ are roots of the equation $8 x^{2}-26 x+15=0$ then $\cos (\alpha+\beta)=$

(a) $-\dfrac{627}{725}$

(b) $\dfrac{627}{725}$

(c) 1

(d) None of these

4. If $\cos (x-y), \cos x, \cos (x+y)$ are in H.P, then $\left|\cos x \cdot \sec \dfrac{y}{2}\right|$ equals

(a) 1

(b) 2

(c) $\sqrt{2}$

(d) None of these

5. If $\tan \dfrac{\pi}{9}, x, \tan \dfrac{15 \pi}{18}$ are in A.P and $\tan \dfrac{\pi}{9}, y, \tan \dfrac{7 \pi}{18}$ are in A.P, then

(a) $2 x=y$

(b) $x=y$

(c) $x=2 y$

(d) None of these

6. The value of $\sin \dfrac{\pi}{n}+\sin \dfrac{3 \pi}{n}+\sin \dfrac{5 \pi}{n}+$. $\mathrm{n}$ terms is

(a) 1

(b) 0

(c) $\dfrac{\pi}{2}$

(d) None of these

7. $\sum\limits _{\mathrm{r}=1}^{\mathrm{n}-1} \cos ^{2} \dfrac{\mathrm{r} \pi}{\mathrm{n}}=$

(a) $\dfrac{\mathrm{n}}{2}$

(b) $\dfrac{\mathrm{n}}{2}-\dfrac{1}{2}$

(c) $\dfrac{\mathrm{n}}{2}-1$

(d) None of these

Exercise

1. Let $\theta \in\left(0, \dfrac{\pi}{4}\right)$ and $\mathrm{t} _{1}=(\tan \theta)^{\tan \theta}, \mathrm{t} _{2}=(\tan \theta)^{\cot \theta}, \mathrm{t} _{3}=(\cot \theta)^{\tan \theta} \theta$ and $\mathrm{t} _{4}=(\cot \theta)^{\cot \theta}$, then

(a) $\mathrm{t} _{1}>\mathrm{t} _{2}>\mathrm{t} _{3}>\mathrm{t} _{4}$

(b) $\mathrm{t} _{4}>\mathrm{t} _{3}>\mathrm{t} _{1}>\mathrm{t} _{2}$

(c) $\mathrm{t} _{3}>\mathrm{t} _{1}>\mathrm{t} _{2}>\mathrm{t} _{4}$

(d) $t _{2}>t _{3}>t _{1}>t _{4}$

2.* For a positive integer n, let $f _{\mathrm{n}}(\theta)=\left(\tan \dfrac{\theta}{2}\right)$

$(1+\sec \theta)(1+\sec 2 \theta)\left(1+\sec ^{2} \theta\right)$ $\left(1+\sec 2^{n} \theta\right)$, then

(a) $f _{2}\left(\dfrac{\pi}{16}\right)=1$

(b) $f _{3}\left(\dfrac{\pi}{32}\right)=1$

(c) $f _{4}\left(\dfrac{\pi}{64}\right)=1$

(d) $f _{5}\left(\dfrac{\pi}{128}\right)=1$

3. Two rays are drawn through a point $\mathrm{A}$ at an angle $30^{\circ}$. A point $\mathrm{B}$ is taken on one of them at a distance ’ $\mathrm{a}$ ’ from the point $\mathrm{A}$. A perpendicular is drawn from the point $\mathrm{B}$ to the other ray, and another perpendicular is drawn from its foot to $\mathrm{AB}$ to meet $\mathrm{AB}$ at another point from where the similar process is repeated indefinitely. The length of the resulting infinite polygonal line is

(a) $\mathrm{a}(2+\sqrt{3})$

(b) $\mathrm{a}(2-\sqrt{3})$

(c) a

(d) None of these

4. If $\cos ^{6} \alpha+\sin ^{6} \alpha+\operatorname{ksin}^{2} 2 \alpha=1(0<\alpha<\pi / 2)$, then $k$ is

(a) $3 / 4$

(b) $1 / 4$

(c) $1 / 3$

(d) $1 / 8$

5. In an acute angled triangle $A B C$, the least value of $\sec A+\sec B+\sec C$ is

(a) 6

(b) 8

(c) 3

(d) none of these

6. The sum to infinite tems of the series $\cos \dfrac{\pi}{3}+\dfrac{1}{2} \cos \dfrac{2 \pi}{3}+\dfrac{1}{3} \cos \dfrac{3 \pi}{3}+\ldots \ldots \ldots \ldots .$. is

(a) 0

(b) 2

(c) 1

(d) 6

7. If $\alpha=\dfrac{2 \pi}{7}$, then the value of $\tan \alpha \tan 2 \alpha+\tan 2 \alpha \tan 4 \alpha+\tan 4 \alpha \tan \alpha$ is

(a) 0

(b) -7

(c) $\dfrac{13}{4}$

(d) 2

8. If $\sin (y+z-x), \sin (z+x-y) \sin (x+y-z)$ be is A.P., then $\tan x, \tan y, \tan z$ are in

(a) A.P.

(b) G.P

(c) H.P.

(d) None of these

9.* $\dfrac{3+\cot 76^{\circ} \cot 16^{\circ}}{\cot 76^{\circ}+\cot 16^{\circ}}=$

(a) $\tan 16^{\circ}$

(b) $\cot 76^{\circ}$

(c) $\tan 46^{\circ}$

(d) $\cot 44^{\circ}$

10. If $x \cos \alpha+y \sin \alpha=2 a, x \cos \beta+y \sin \beta=2 a$ and $2 \sin \dfrac{\alpha}{2} \sin \dfrac{\beta}{2}=1$, then

(a) $\cos \alpha+\cos \beta=\dfrac{2 a x}{x^{2}+y^{2}}$

(b) $\cos \alpha \cos \beta=\dfrac{2 \mathrm{a}^{2}-\mathrm{y}^{2}}{\mathrm{x}^{2}+\mathrm{y}^{2}}$

(c) $y^{2}=4 a(a-x)$

(d) $\cos \alpha+\cos \beta=2 \cos \alpha \cos \beta$

11. Match the following:-

12. Let $\mathrm{A} _{0} \mathrm{~A} _{1} \mathrm{~A} _{2} \mathrm{~A} _{3} \mathrm{~A} _{4} \mathrm{~A} _{5}$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $\mathrm{A} _{0} \mathrm{~A} _{1}, \mathrm{~A} _{0} \mathrm{~A} _{2}$ and $\mathrm{A} _{0} \mathrm{~A} _{4}$ is

(a) $\dfrac{3}{4}$

(b) $3 \sqrt{3}$

(c) 3

(d) $\dfrac{3 \sqrt{3}}{2}$

13. If A, B, C, D are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity $\mathrm{k}$, then the value of

$4 \sin \dfrac{\mathrm{A}}{2}+3 \sin \dfrac{\mathrm{B}}{2}+2 \sin \dfrac{\mathrm{C}}{2}+\sin \dfrac{\mathrm{D}}{2}$ is

(a) $2 \sqrt{1-\mathrm{k}}$

(b) $2 \sqrt{1+\mathrm{k}}$

(c) $2 \sqrt{\mathrm{k}}$

(d) None of these

14. Read the paragraph and answer the following questions .

If $\alpha, \beta, \gamma, \delta$ are the solutions of the equation $\tan \left(\theta+\dfrac{\pi}{4}\right)=3 \tan 3 \theta$, no two of which have equal tangents, then

(i) The value of $\tan \alpha+\tan \beta+\tan \gamma+\tan \delta$ is

(a) $1 / 3$

(b) $8 / 3$

(c) $-8 / 3$

(d) 0

(ii) The value of $\tan \alpha \tan \beta \tan \gamma \tan \delta$ is

(a) $-1 / 3$

(b) -2

(c) 0

(d) None of these

(iii) The value of $\dfrac{1}{\tan \alpha}+\dfrac{1}{\tan \beta}+\dfrac{1}{\tan \gamma}+\dfrac{1}{\tan \delta}$ is

(a) -8

(b) 8

(c) $2 / 3$

(d) $1 / 3$

15.* Which of the following is / are correct?

(a) $(\tan x)^{\log _{c}(\sin x)}>(\cot x)^{\log _{c}(\sin x)} \hspace {2 cm}, \forall x \in(0, \pi / 4)$

(b) $4^{\log _{c}(\operatorname{cosec} x)}<5^{\log _{c}(\operatorname{cosec} x)}$ $\hspace {3 cm}$ $,\forall \mathrm{x} \in(0, \pi / 2)$

(c) $\left(\dfrac{1}{2}\right)^{\log _{c}(\cos x)}<\left(\dfrac{1}{3}\right)^{\log _{c}(\cos x)}$ $\hspace {2.4 cm}$,$\forall \mathrm{x} \in(0, \pi / 2)$

(d) $2^{\log _{e}(\tan x)}<2^{\log _{e}(\sin x)}$ $\hspace {3.5 cm}$,$\forall \mathrm{x} \in(0, \pi / 2)$