Factor the expression completely 2 csc2 w 9 csc w 10 2 cs

Factor the expression completely 2 csc^2 w + 9 csc w + 10 = (2 csc w +) (csc w +) Use Appropriate identity to find Exact value of the expression sin (-13 pi/12) squareroot 2 + squareroot 6/4 -squareroot 2 - squareroot 6/4 squareroot 2 - squareroot 6/4 - squareroot 2 + squareroot 6/4 Use a confunction identity to fill in blank cos (65 degree) = sin () degree Use the appropriate identity to find the exact value of the expression cos 13 pi/12

Solution

1) 2 csc^2 w + 9csc w + 10

find two factors of 2*10 that is 20 such that they add up to become 9

two factors of 20 are 5 and 4

splitting the mid term in terms of 5 and 4

2 csc^2 w + 4csc w + 5 csc w + 10

2 csc w ( csc w + 2) + 5 ( csc w + 2)

factors are

( 2 csc w + 5) ( csc w + 2)

2) sin ( -13pi/ 12)

-13pi / 12 = 2pi/12 - 15pi/12 = pi/6 - 5pi/4

sin ( pi/6 - 5pi/4 ) = sin pi/6 cos 5pi/4 - cos pi/6 sin 5pi/4

                         = 1/2 * (-1/sqrt 2) - sqrt 3 / 2 ( -1/sqrt 2)

                         = - 1/ 2sqrt 2 + sqrt3 / 2 sqrt 2

                         = ( 2sqrt 6 - 2 sqrt 2 ) / 8

= (sqrt 6 - sqrt 2 )/ 4 or (- sqrt 2 + sqrt 6 ) / 4 ( option d)

c) cos 65 degrees = cos ( 90 - 25 )

cos ( 90 - theta ) = sin theta

hence , cos ( 90 - 25 ) = sin 25

d) cos ( 13pi/12)

cos (13pi/12 ) = cos ( 4pi/12 + 9pi/12 ) = cos ( pi/3 + 3pi/4 )

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

cos ( pi/3 + 3pi/4 )= cos pi/3 cos 3pi/4 - sin pi/3 sin 3pi/4

plugging the values we get

cos ( 13pi/12) = - .9659