Given that sina 23 and cosb 16 with a and b both in the in

Given that sin(a) = 2/3 and cos(b) = -1/6, with a and b both in the interval [pi/2, pi), find the exact values of sin(a + b) and cos(a - b). sin(a + b) = cos(a - b) = Recall the sine and cosine formulas for the sum and difference of two angles. How is the trigonometric Pythagorean identity used to find cos(a) trigonometric values in the appropriate formula and simplify. eBook Sum and Difference Identities for Sine Sum and Difference Identities for Cosine Learn by Example Find the exact value algebraically. Sin(105 degree)

Solution

sin a = 2/3

cos b = -1/6

cos a = base / hypotenuse

base = sqrt (3^2 - 2^2) = sqrt 5

cos a = - sqrt 5 / 3

sin b = perpendicular / hypotenuse

perpendicular = sqrt ( 6^2 - 1^2 ) = sqrt 35

sin b = sqrt 35 / 6

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

plugging the values in the formula

sin (a+b) = (2/3)(-1/6) + ( - sqrt 5 /3 ) ( sqrt 35 / 6 ) = -2/ 18 - sqrt 175 / 18 = ( -2 - 5sqrt 7) /18

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

plugging the values

cos (a-b) = (- sqrt 5 / 3)(-1/6) + (2/3)(sqrt 35/6) = sqrt 5 /18 + 10 sqrt 7 /18 = ( sqrt 5 + 10sqrt 7) /18