a Evaluate the expression under the given conditions sin ta

(a) Evaluate the expression under the given conditions.

sin( ); tan = 4/3, theta in quadrant 3, sin =- square root 10/10, in Quadrant IV ,

(b)Use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine.

cos^4x

sin ( theta - fi ) = sin theta cos fi - cos theta sin fi

tan theta = 4/3

sin theta = -4/5

cos theta = - 3/5

sin fi = - sqrt 10 / 10

cos fi = sqrt 90 / 10

plugging the values in the formula

sin ( theta - fi ) = sin theta cos fi - cos theta sin fi = (-4/5)(sqrt 90/10) - ( -3/5)(-sqrt 10/10)

= -4sqrt 90 / 50 - 3 sqrt 10 / 50

= -.9486

b) cos^4x can be written as

cos^2x * cos^2 x = (cos^2x)^2

cos^2x = ( 1+ cos 2x ) /2

(cos^2x)^2 = {( 1+ cos 2x ) /2 }^2 = 1/4 ( 1+ cos^2 2x + 2 cos 2x )

again plugging the value of cos^2 2x

1/4 ( 1+ (1+ cos 4x)/2 + 2 cos 2x )

cos ^4x = 1/8 ( 3 + cos 4x + 4 cos 2x )