Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Trigonometric, Identities & Formulas


Trigonometric, Identities & Formulas
Relation between angle and sides of the triangle, values of trigonometric functions at standard angles, transformation and trigonometric identities

TRIGONOMETRIC IDENTITIES & FORMULAS FOR  9TH, 10TH STANDARD
https://dinesh51.blogspot.com

Trigonometry: 
It is the branch of mathematics dealing with the relations of the sides and angles of right triangles.
\[sin\theta =\frac{Perpendicular}{Hypotenuse}=\frac{P}{H}\]\[Cos\theta =\frac{Base}{Hypotenuse}=\frac{B}{H}\]\[Tan\theta =\frac{Perpendicular}{Base}=\frac{P}{B}\]\[Cot\theta =\frac{Base}{Perpendicular}=\frac{B}{P}\]\[Sec\theta =\frac{Hypotenuse}{Base}=\frac{H}{B}\]\[Cosec\theta =\frac{Hypotenuse}{Perpendicular}=\frac{H}{P}\]From these six formulas we also find that \[sin\theta =\frac{1}{cosec\theta }\Rightarrow cosec\theta =\frac{1}{sin\theta }\]\[cos\theta =\frac{1}{sec\theta }\Rightarrow sec\theta =\frac{1}{cos\theta }\]\[tan\theta =\frac{1}{cot\theta }\Rightarrow cot\theta =\frac{1}{tan\theta }\]\[Tan\theta =\frac{Sin\theta }{Cos\theta }\; \; and\; \; \; Cot\theta =\frac{Cos\theta }{Sin\theta }\] Shortcut method to learn these formulas
First of all learn this line: "Pandit Badri Parsad Har Har Bole Sona Chandi Tole" and then write these words in a tabular form as shown below

Sona

Chandi

Tole

Pandit

Badri

Parsad

Har

Har

Bole

Convert  the first letter of the each word as shown  below, and in the last row write the reciprocal of sinθ, cosθ, and tanθ

sin θ

cos θ

tan θ

P

B

P

H

H

B

cosec θ

sec θ

cot θ


VALUE OF THE TRIGONOMETRIC FUNCTIONS WITH STANDARD ANGLE

0o

30o

45o

60o

90o

Sin

0

1/2

\[\frac{1}{\sqrt{2}}\]

\[\frac{\sqrt{3}}{2}\]

1

Cos

1

\[\frac{\sqrt{3}}{2}\]

\[\frac{1}{\sqrt{2}}\]

1/2

0

Tan

0

\[\frac{1}{\sqrt{3}}\]

1

\[\sqrt{3}\]

\[\infty\]

Cot

\[\infty\]

\[\sqrt{3}\]

1

\[\frac{1}{\sqrt{3}}\]

0

sec

1

\[\frac{2}{\sqrt{3}}\]

\[\sqrt{2}\]

2

\[\infty\]

cosec

\[\infty\]

2

\[\sqrt{2}\]

\[\frac{2}{\sqrt{3}}\]

1


Method of finding these values

Sin0o

Sin30o

Sin45o

Sin60o

Sin90o

0

1

2

3

4

\[\frac{0}{4}\]

\[\frac{1}{4}\]

\[\frac{2}{4}\]

\[\frac{3}{4}\]

\[\frac{4}{4}\]

0

\[\frac{1}{4}\]

\[\frac{1}{2}\]

\[\frac{3}{4}\]

1

\[\sqrt{0}\]

\[\sqrt{\frac{1}{4}}\]

\[\sqrt{\frac{1}{2}}\]

\[\sqrt{\frac{3}{4}}\]

\[\sqrt{1}\]

0

\[\frac{1}{2}\]

\[\frac{1}{\sqrt{2}}\]

\[\frac{\sqrt{3}}{2}\]

1


Steps For finding the  values of sinθ 
1) Write counting from 0 to 4

2) Divide all the numbers by 4 and simplify these numbers

3) Taking square root of all these numbers

4) The values we get are the values on the sin function at different standard angles

For values of other trigonometric ratios
write all these values for sinθ in  the reverse order(from right to left)

For values of  tanθ use the formula tanθ = sinθ/cosθ

For values the values of cotθ use cotθ = 1/tanθ

For  the values of secθ use secθ = 1/cosθ


For the values of cosecθ use cosecθ = 1/sinθ

TRANSFORMATION OF TRIGONOMETRIC FUNCTIONS
 sin(90 - θ)   = cosθ,                        cos(90 - θ) =  sinθ,   
 tan(90 - θ)  = cotθ,                          cot(90 - θ) = tanθ,    
sec(90 - θ)   = cosecθ,                  cosec(90 - θ) = secθ

TRIGONOMETRIC IDENTITIES

In triangle ABC, by Pythagoras theorem

AB2 = AC2 +  BC2   ................  (1)   

Dividing on both side by AB2

\[\frac{AB^{2}}{AB^{2}}=\frac{AC^{2}}{AB^{2}}+\frac{BC^{2}}{AB^{2}}\]\[1=\left ( \frac{AC}{AB} \right )^{2}+\left ( \frac{BC}{AB} \right )^{2}\]\[1=\left ( \frac{P}{H} \right )^{2}+\left ( \frac{B}{H} \right )^{2}\]\[1=sin^{2}\theta +cos^{2}\theta\]Dividing equation(1) by AC2  \[\frac{AB^{2}}{AC^{2}}=\frac{AC^{2}}{AC^{2}}+\frac{BC^{2}}{BC^{2}}\] \[\left ( \frac{AB}{AC} \right )^{2}=1+\left ( \frac{BC}{AC} \right )^{2}\]\[\left ( \frac{H}{P} \right )^{2}=1+\left ( \frac{B}{P} \right )^{2}\]\[cosec^{2}\theta =1+cot^{2}\theta \Rightarrow cosec^{2}\theta-cot^{2}\theta=1\] Dividing on both side by BC2\[\frac{AB^{2}}{BC^{2}}=\frac{AC^{2}}{BC^{2}}+\frac{BC^{2}}{BC^{2}}\]\[\left(\frac{AB}{BC} \right )^{2}=\left ( \frac{AC}{BC} \right )^{2}+1\]\[\left(\frac{H}{B} \right )^{2}=\left ( \frac{P}{B} \right )^{2}+1\]\[sec^{2}\theta =tan^{2}\theta +1\Rightarrow sec^{2}\theta -tan^{2}\theta =1\]
So we have the following results
\[Sin^{2}\theta +Cos^{2}\theta = 1,\; \; \; Sin^{2}\theta=1-Cos^{2}\theta,\; \; \; Cos^{2}\theta =1-Sin^{2}\theta\]
\[Sec^{2}\theta -Tan^{2}\theta = 1,\; \; \; Sec^{2}\theta=1+Tan^{2}\theta,\; \; \; Tan^{2}\theta =Sec^{2}\theta-1\]
\[Cosec^{2}\theta -Cot^{2}\theta = 1,\; \; Cosec^{2}\theta=1+Cot^{2}\theta,\; \; \; Cot^{2}\theta = Cosec^{2}\theta-1\]
Geometrical Representation of Trigonometric Ratios with the 30o, 60o .

Let ΔABC is an equilateral triangle. Therefore its each angle is = 60o . Draw AD⊥ BC so that AD bisect the base as well as the vertex angle A.
Let each side of ΔABC is a. That is  AB = BC = CA = a and BD = a/2 and ㄥA= 30o 
In   ΔABD by Pythagoras theorem\[AD^{2}=\sqrt{AB^{2}-BD^{2}}\]\[AD^{2}=\sqrt{a^{2}-\left (\frac{a}{2} \right )^{2}}\]\[AD^{2}=\sqrt{\frac{4a^{2}-a^{2}}{4}}\]\[AD=\frac{\sqrt{3}a}{2}\]
\[Sin 60^{o}=\frac{P}{H}=\frac{\frac{\sqrt{3}}{2}a}{a}=\frac{\sqrt{3}}{2}\]\[Cos 60^{o}=\frac{B}{H}=\frac{a/2}{a}=\frac{1}{2}\]\[tan 60^{o}=\frac{P}{B}=\frac{\frac{\sqrt{3}}{2}a}{\frac{1}{2}a}=\sqrt{3}\]\[cot60^{o}=\frac{B}{P}=\frac{\frac{1}{2}a}{\frac{\sqrt{3}}{2}a}=\frac{1}{\sqrt{3}}\]\[Sec 60^{o}=\frac{H}{B}=\frac{a}{a/2}=2\]\[Cosec60^{o}=\frac{H}{P}=\frac{a}{\frac{\sqrt{3}}{2}a}=\frac{2}{\sqrt{3}}\]
Similarly we can find the geometrical representation of all trigonometric functions at 30o
Geometrical Representation of Trigonometric Ratios with the 45o.
In order to find the geometrical representation of trigonometric functions at 45o , we should make an isosceles right angled triangle . And then find the values with the same method as shown above.  
Here let each equal side of the triangle = a
Then by pythagoras theorem \[AB=\sqrt{BC^{2}+AC^{2}}\]\[AB =\sqrt{a^{2}+a^{2}} =\sqrt{2a^{2}}=\sqrt{2}a\]\[Sin45^{o} =\frac{P}{H}=\frac{a}{\sqrt{2}a} =\frac{1}{\sqrt{2}}\]\[Cos 45^{o} =\frac{B}{H}=\frac{a}{\sqrt{2}a}=\frac{1}{\sqrt{2}}\]\[Tan45^{o} =\frac{P}{B}=\frac{a}{a}=1\]\[Cot 45^{o} =\frac{B}{P}=\frac{a}{a}=1\]\[Cosec45^{o} =\frac{H}{P}=\frac{\sqrt{2}a}{a}=\sqrt{2}\]\[Sec45^{o} =\frac{H}{B}=\frac{\sqrt{2}a}{a}=\sqrt{2}\]
Angle of elevation:


Angle of Depression:

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TRIGONOMETRY-CBSE Mathematics


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