Important Short Tricks on Trigonometric Identities

Important Short Tricks on Trigonometric Identities

Pythagorean Identities

• sin2 θ + cos2 θ = 1
• tan2 θ + 1 = sec2 θ
• cot2 θ + 1 = csc2 θ

Negative of a Function

• sin (–x) = –sin x
• cos (–x) = cos x
• tan (–x) = –tan x
• csc (–x) = –csc x
• sec (–x) = sec x
• cot (–x) = –cot x                                                                                     If A + B = 90o, Then

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

If A – B = 90o, (A › B) 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

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

If   a sin θ + b cos θ = m &  a cos θ – b sin θ = n

then a2 + b2 = m2 + n2    

                                                          

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

Cos2 θ, maxima value = 1, minima value = 0                                                                                                             BY - ANSAR       www.fidgetacademy.blogspot.in