Determine the exact value of the expression cos(3pi/16)cos(pi/16) – sin(3pi/16)sin(pi/16)?

4 Answers

• Nah, that's just the right side of the cosine of a sum:

  cos (a+b) = (cos a)(cos b) - (sin a)(sin b)

  ....with a=3pi/16 and b=pi/16. the result (the left side) is:

  cos (3pi/16 + pi/16) = cos (pi/4) = 1/√2

• We should recognize that this is the the cosine addition identity, cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

  cos(3pi/16)cos(pi/16) - sin(3pi/16)sin(pi/16)

  =cos(3pi/16 + pi/16)

  =cos(4pi/16)

  =cos(pi/4)

  =1/√2

• cos(3pi/16)cos(pi/16) - sin(3pi/16)sin(pi/16)

  =cos A cosB - sinA sinB [Say , A=3pi/16, B=pi/16 ]

  =cos(A+B)

  =Cos{(3pi/16)+(pi/16)}

  =cos(4pi/16)

  =cos (pi/4)

  =cos 45 deg

  =1 / sqrt2

• cos(a + b) = cosa cosb - sina sinb

  cos(3π/16 + π/16) = cos(π/4) = √(2)/2