Trigonometry PPT Maths Class 11

 Page 1


Reciprocal Identities 
csc x = 
1
sin x
        sec x = 
1
cos x
           cot x = 
1
tan x
  
Quotient Identities 
tan x = 
sin x
cos x
        cot x = 
cos x
sin x
  
Pythagorean Identities 
sin
2
x + cos
2
x = 1             tan
2
x + 1 = sec
2
x   1 + cot
2
x = csc
2
x 
Even-Odd Identities  
sin(-x) = -sin x    cos(-x) = cos x   tan(-x) =-tan x 
csc(-x) =? -csc x    sec(-? x) = sec x   cot(-x) =-cot x  
Basic Trigonometric Identities
Page 2


Reciprocal Identities 
csc x = 
1
sin x
        sec x = 
1
cos x
           cot x = 
1
tan x
  
Quotient Identities 
tan x = 
sin x
cos x
        cot x = 
cos x
sin x
  
Pythagorean Identities 
sin
2
x + cos
2
x = 1             tan
2
x + 1 = sec
2
x   1 + cot
2
x = csc
2
x 
Even-Odd Identities  
sin(-x) = -sin x    cos(-x) = cos x   tan(-x) =-tan x 
csc(-x) =? -csc x    sec(-? x) = sec x   cot(-x) =-cot x  
Basic Trigonometric Identities
Verify the identity: sec x cot x = csc x.
Solution The left side of the equation contains the more complicated 
expression. Thus, we work with the left side. Let us express this side of the 
identity in terms of sines and cosines. Perhaps this strategy will enable us to 
transform the left side into csc x, the expression on the right. 
Apply a reciprocal identity: sec x = 1/cos x
and a quotient identity:  cot x = 
cos x
/
sin x
.
Divide both the numerator and the 
denominator by cos x, the common factor.
Example
secxcotx =
1
cosx
?
cosx
sinx
=
1
sinx
=cscx
Page 3


Reciprocal Identities 
csc x = 
1
sin x
        sec x = 
1
cos x
           cot x = 
1
tan x
  
Quotient Identities 
tan x = 
sin x
cos x
        cot x = 
cos x
sin x
  
Pythagorean Identities 
sin
2
x + cos
2
x = 1             tan
2
x + 1 = sec
2
x   1 + cot
2
x = csc
2
x 
Even-Odd Identities  
sin(-x) = -sin x    cos(-x) = cos x   tan(-x) =-tan x 
csc(-x) =? -csc x    sec(-? x) = sec x   cot(-x) =-cot x  
Basic Trigonometric Identities
Verify the identity: sec x cot x = csc x.
Solution The left side of the equation contains the more complicated 
expression. Thus, we work with the left side. Let us express this side of the 
identity in terms of sines and cosines. Perhaps this strategy will enable us to 
transform the left side into csc x, the expression on the right. 
Apply a reciprocal identity: sec x = 1/cos x
and a quotient identity:  cot x = 
cos x
/
sin x
.
Divide both the numerator and the 
denominator by cos x, the common factor.
Example
secxcotx =
1
cosx
?
cosx
sinx
=
1
sinx
=cscx
Solution We start with the more complicated side, the left side. Factor out 
the greatest common factor, cos x, from each of the two terms. 
cos x - cos x sin
2
x = cos x(1 - sin
2
x) Factor cos x from the two terms.
Use a variation of sin
2
x + cos
2
x = 1. 
Solving for cos
2
x, we obtain cos
2
x = 
1 – sin
2
x.
= cos x · cos
2
x
Multiply. = cos
3
x
We worked with the left and arrived at the right side. Thus, the identity is 
verified. 
Verify the identity: cosx - cosxsin
2
x = cos
3
x..
Example
Page 4


Reciprocal Identities 
csc x = 
1
sin x
        sec x = 
1
cos x
           cot x = 
1
tan x
  
Quotient Identities 
tan x = 
sin x
cos x
        cot x = 
cos x
sin x
  
Pythagorean Identities 
sin
2
x + cos
2
x = 1             tan
2
x + 1 = sec
2
x   1 + cot
2
x = csc
2
x 
Even-Odd Identities  
sin(-x) = -sin x    cos(-x) = cos x   tan(-x) =-tan x 
csc(-x) =? -csc x    sec(-? x) = sec x   cot(-x) =-cot x  
Basic Trigonometric Identities
Verify the identity: sec x cot x = csc x.
Solution The left side of the equation contains the more complicated 
expression. Thus, we work with the left side. Let us express this side of the 
identity in terms of sines and cosines. Perhaps this strategy will enable us to 
transform the left side into csc x, the expression on the right. 
Apply a reciprocal identity: sec x = 1/cos x
and a quotient identity:  cot x = 
cos x
/
sin x
.
Divide both the numerator and the 
denominator by cos x, the common factor.
Example
secxcotx =
1
cosx
?
cosx
sinx
=
1
sinx
=cscx
Solution We start with the more complicated side, the left side. Factor out 
the greatest common factor, cos x, from each of the two terms. 
cos x - cos x sin
2
x = cos x(1 - sin
2
x) Factor cos x from the two terms.
Use a variation of sin
2
x + cos
2
x = 1. 
Solving for cos
2
x, we obtain cos
2
x = 
1 – sin
2
x.
= cos x · cos
2
x
Multiply. = cos
3
x
We worked with the left and arrived at the right side. Thus, the identity is 
verified. 
Verify the identity: cosx - cosxsin
2
x = cos
3
x..
Example
Guidelines for Verifying 
Trigonometric Identities
1. Work with each side of the equation independently of the 
other side. Start with the more complicated side and 
transform it in a step-by-step fashion until it looks 
exactly like the other side.
2. Analyze the identity and look for opportunities to apply 
the fundamental identities. Rewriting the more 
complicated side of the equation in terms of sines and 
cosines is often helpful.
3. If sums or differences of fractions appear on one side, 
use the least common denominator and combine the 
fractions.
4. Don't be afraid to stop and start over again if you are not 
getting anywhere. Creative puzzle solvers know that 
strategies leading to dead ends often provide good 
problem-solving ideas.
Page 5


Reciprocal Identities 
csc x = 
1
sin x
        sec x = 
1
cos x
           cot x = 
1
tan x
  
Quotient Identities 
tan x = 
sin x
cos x
        cot x = 
cos x
sin x
  
Pythagorean Identities 
sin
2
x + cos
2
x = 1             tan
2
x + 1 = sec
2
x   1 + cot
2
x = csc
2
x 
Even-Odd Identities  
sin(-x) = -sin x    cos(-x) = cos x   tan(-x) =-tan x 
csc(-x) =? -csc x    sec(-? x) = sec x   cot(-x) =-cot x  
Basic Trigonometric Identities
Verify the identity: sec x cot x = csc x.
Solution The left side of the equation contains the more complicated 
expression. Thus, we work with the left side. Let us express this side of the 
identity in terms of sines and cosines. Perhaps this strategy will enable us to 
transform the left side into csc x, the expression on the right. 
Apply a reciprocal identity: sec x = 1/cos x
and a quotient identity:  cot x = 
cos x
/
sin x
.
Divide both the numerator and the 
denominator by cos x, the common factor.
Example
secxcotx =
1
cosx
?
cosx
sinx
=
1
sinx
=cscx
Solution We start with the more complicated side, the left side. Factor out 
the greatest common factor, cos x, from each of the two terms. 
cos x - cos x sin
2
x = cos x(1 - sin
2
x) Factor cos x from the two terms.
Use a variation of sin
2
x + cos
2
x = 1. 
Solving for cos
2
x, we obtain cos
2
x = 
1 – sin
2
x.
= cos x · cos
2
x
Multiply. = cos
3
x
We worked with the left and arrived at the right side. Thus, the identity is 
verified. 
Verify the identity: cosx - cosxsin
2
x = cos
3
x..
Example
Guidelines for Verifying 
Trigonometric Identities
1. Work with each side of the equation independently of the 
other side. Start with the more complicated side and 
transform it in a step-by-step fashion until it looks 
exactly like the other side.
2. Analyze the identity and look for opportunities to apply 
the fundamental identities. Rewriting the more 
complicated side of the equation in terms of sines and 
cosines is often helpful.
3. If sums or differences of fractions appear on one side, 
use the least common denominator and combine the 
fractions.
4. Don't be afraid to stop and start over again if you are not 
getting anywhere. Creative puzzle solvers know that 
strategies leading to dead ends often provide good 
problem-solving ideas.
Example
• Verify the identity: 
csc(x) / cot (x) = sec (x)
x x
x
x
x
x
x
x
x
x
x
cos
1
cos
sin
sin
1
cos
1
sin
cos
sin
1
sec
cot
csc
= ?
=
=
Solution: