Short Tricks on Trigonometric Identities

Pythagorean Identities

  • sin2 θ + cos2 θ = 1
  •  sec2 θ – tan2 θ = 1
  •  cosec2 θ – cot2 θ = 1

Negative of a Function

  • sin (–x) = –sin x
  • cos (–x) = cos x
  • tan (–x) = –tan x
  • cosec (–x) = –cosec x
  • sec (–x) = sec x
  • cot (–x) = –cot x

TRICK-1

If A + B = 90o, Then

  • Sin A = Cos B
  • Sin2A + Sin2B = Cos2A + Cos2B = 1
  • Tan A = Cot B
  • Sec A = Cosec B

For example:    

If tan (x+y) tan (x-y) = 1, then find tan (2x/3)?

Solution:            

Tan A = Cot B, Tan A*Tan B = 1

So, A +B = 90o

(x+y)+(x-y) = 90o, 2x = 90o , x = 45o

Tan (2x/3) = tan 30o = 1/√3

TRICK-2

If A – B = 90o, Then

  • Sin A = Cos B
  • Cos A = – Sin B
  • Tan A = – Cot B

If A ± B = 180o, then

  • Sin A = Sin B
  • Cos A = – Cos B

If A + B = 180o                   

Then, tan A = – tan B

If A – B = 180o                    

Then, tan A = tan B

For example:    

Find the Value of tan 80o + tan 100o ?

Solution:

Since 80 + 100 = 180

Therefore, tan 80o + tan 100o = 1

TRICK-3

If A + B + C = 180o, then

Tan A + Tan B +Tan C = Tan A * Tan B *Tan C

sin θ * sin 2θ * sin 4θ = ¼ sin 3θ

cos θ * cos 2θ * cos 4θ = ¼ cos 3θ

For Example:

What is the value of cos 20o cos 40o cos 60o cos 80o?

Solution:

We know cos θ * cos 2θ * cos 4θ = ¼ cos 3θ

Now, (cos 20o cos 40o cos 80o ) cos 60o

¼ (Cos 3*20) * cos 60o

¼ Cos2 60o = ¼ * (½)2 = 1/16

TRICK-4

If   a sin θ + b cos θ = m     &    a cos θ – b sin θ = n; then a2 + b2 = m2 + n2

For Example:

If 4 sin θ + 3 cos θ = 2 , then find the value of  4 cos θ – 3 sin θ:

Solution:

Let 2 cos θ – 3 sin θ = x

By using formulae a2 + b2 = m2 + n2

42 + 32 = 22 + x2

⇒16 + 9 = 4 + x2

⇒X = √21

TRICK-5

If  sin θ +  cos θ = p     &     cosec θ –  sec θ = q;  then P – (1/p) = 2/q

For Example:

If sin θ + cos θ = 2 , then find the value of  cosec θ – sec θ:

Solution:

By using formulae:

P – (1/p) = 2/q

2-(1/2) = 3/2 = 2/q

Q = 4/3 or csc θ – sec θ = 4/3

TRICK-6

If a cot θ + b csc θ = m     &    a csc θ + b cot θ = n then b2 – a2  = m2 – n2

If cot θ + cos θ = x     &    cot θ – cos θ = y then x2 – y2 = 4 √xy

If tan θ + sin θ = x     &    tan θ – sin θ = y then x2 – y2 = 4 √xy

If

y = a2 sin2x + b2 csc2x + c

y = a2 cos2x + b2 sec2x + c

y = a2 tan2x + b2 cot2x + c

then,

ymin = 2ab + c

ymax = not defined

For Example:                    

If y = 9 sin2 x + 16 csc2 x +4 then ymin is:

Solution:            

For, y min = 2* √9 * √16 + 4

= 2*3*4 + 20 = 24 + 4 = 28

TRICK-7

If            

y = a sin x + b cos x + c

y = a tan x + b cot x + c

y = a sec x + b csc x + c

then,     ymin = + [√(a2+b2)] + c

ymax = – [√(a2+b2)] + c

For Example:                    

If y = 1/(12sin x + 5 cos x +20) then ymax is:

Solution:            

For, y max = 1/x min

= 1/- (√122 +52) +20 = 1/(-13+20) = 1/7

Sin2 θ, maxima value = 1, minima value = 0

Cos2 θ, maxima value = 1, minima value = 0

Important questions of Trigonometric identities

(1)Value of image005 is

(a)image002

(b)image004

(c)image003

(d)None of these

Ans.(a)

image005 is equal to

image006

(2)If image007is acute and image008 then image009 is equal to

(a)

(b)3

(c)  2

(d)  4

Ans.  (c)

If sum of the inversely proportional value is 2

i.e if . image013 thenimage014

image015

 

so image009=2

or we can put image007= 45°

(3)The simplified value ofimage019 is

(a)-1

(b)0

(c)sec2x

(d)1

Ans. (d)

The simplified value of

image019is obtained by  putting x=y=45°

image022

(4) Find the value of image024

(a)  1

(b)  -1

(c)  2

(d)  -2

Ans. (c)

put image017

image026

(5) If image033 then image028 is equal

(a)7/4

(b) 7/2

(c)5/2

(d)5/4

Ans. (d)

image012 as we know thatimage034

j

on solving we get secimage007= 5/4

Note:if x+y=a

and x-y=b

then x=(a+b)/2 and y=(x-y)/2

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