CONDITIONAL TRIGONOMETRIC IDENTITIES PROBLEMS

Trigonometric identities are true for all admissible values of the angle involved. There are some trigonometric identities which satisfy the given additional conditions. Such identities are called conditional trigonometric identities.

Abbreviations used : 

L.H.S -----> Left hand side

R.H.S -----> Right hand side

Problem 1 :

If A + B + C = 180°, prove that 

sin2A + sin2B + sin2C  =  4sinAsinBsin C

Solution :

L.H.S :

=  sin2A + sin2B + sin2C

Use the formula of (sin C + sin D).

=  2sin[(2A + 2B)/2]cos[(2A - 2B)/2] + sin2C

=  2sin(A + B)cos(A - B) + sin2C -----(1)

A + B + C  =  180°

A + B  =  180 - C

sin(A + B)  =  sin(180 - C)  =  sinC

(1)-----> =  2sinCcos(A - B) + 2sinCcosC

=  2sinC[cos(A - B) + cosC]

=  2sinC{cos(A - B) + cos[180 - (A + B)]}

=  2sinC{cos(A - B) + cos(A + B)} -----(2)

cosC - cosD  =  -2sin(C + D)/2sin(C - D)/2

Here C  =  A - B and D =  A  + B

C  + D  =  A - B + A + B  =  2A

C  - D  =  A - B - A - B  =  -2B

(2)----->  =  2sinC[-2sin(2A/2)sin(-2B/2)]

=  2sinC[2sinAsinB]

=  4sinAsinBsinC

=  R.H.S

Problem 2 :

If A + B + C = 180°, prove that 

cosA + cosB - cosC  =  -1 + 4cos(A/2)cos(B/2)sin(C/2)

Solution :

L.H.S :

=  cosA + cosB - cosC

=  2cos(A + B)/2cos(A - B)/2 - cosC

=  2cos(180 - C)/2cos(A - B)/2 - cosC

=  2cos[90 - (C/2)]cos(A - B)/2 - cosC

=  2sin(C/2)cos(A - B)/2 - [1 - 2sin2(C/2)]

=  2sin(C/2)cos(A - B)/2 - 1 + 2sin2(C/2)

=  -1 + 2sin(C/2)cos(A - B)/2 + 2sin2(C/2)

=  -1 + 2sin(C/2){cos(A - B)/2 + sin[180 - (A + B)]/2}

=  -1 + 2sin(C/2){cos(A - B)/2 + sin[90 - (A + B)/2]}

=  -1 + 2sin(C/2)[cos(A - B)/2 + cos(A + B)/2)] -----(1)

Use the formula of (cosC + cosD).

Here C  =  (A - B) / 2 and D  =  (A + B)/2

C + D  =  2A/2  =  A

C - D  =  -2B/2  =  -B

(1)----->  =  -1 + 2sin(C/2)[2cos(A/2)cos(-B/2)]

=  -1 + 4sin(C/2)[cos(A/2)cos(B/2)]

=  -1 + 4cos(A/2)cos(B/2)sin(C/2)

=  R.H.S

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. Digital SAT Math Problems and Solutions (Part - 155)

    May 03, 25 08:08 AM

    digitalsatmath189.png
    Digital SAT Math Problems and Solutions (Part - 155)

    Read More

  2. AP Calculus AB Problems with Solutions (Part - 20)

    May 03, 25 07:47 AM

    apcalculusab18.png
    AP Calculus AB Problems with Solutions (Part - 20)

    Read More

  3. Digital SAT Math Problems and Solutions (Part - 154)

    Apr 30, 25 11:12 PM

    digitalsatmath187.png
    Digital SAT Math Problems and Solutions (Part - 154)

    Read More