Solve the following equation using the cofunction identity for θ where 0≤θ<π.

Question

Give your answer in radians as a fraction in terms of π.

sin(2θ+π/4)=cos(2θ−11π/4)

θ=?

Answer 1

First, cos(pi/4) = sin(pi/4)= cos(11*pi/4) = sin(11*pi/4) = sqrt(2)/2

because 11*pi/4 = pi/4 by multiplies of 2*pi.

sin (2t + pi/4) = cos( 2t - 11*pi/4) is given

By angle addition and subtraction formulae:

sin(2t) cos(pi/4) - sin(pi/4)cos(2t) = cos(2t)cos(11*pi/4) + sin(2t)sin(11*pi/4)

so then:

sin(2t) * sqrt(2)/2 - sqrt(2)/2 * cos(2t) = cos(2t)*sqrt(2)/2 + sin(2t)*sqrt(2)/2

factors out sqrt(2)/2 and then it cancels, so

sqrt(2)/2 [ sin(2t) - cos(2t) ] = sqrt(2)/2 [ cos(2t) + sin(2t)]

and then

sin(2t) - cos(2t) = cos(2t) + sin(2t)

sin(2t) cancels by subtraction:

-cos(2t) = cos(2t)

0 = 2 * cos(2t)

0 = cos(2t)

0 = cos(X), X = 2t

X= pi/2 since 0<=X<pi

2t = pi/2

t = pi/4