Find the exact value of the expression whenever it is defined. (If an answer is undefined, enter UNDEFINED.)

Question

Find the exact value of the expression whenever it is defined. (If an answer is undefined, enter UNDEFINED.)

a) sin(arcsin 1/2 + arccos 0)

b) tan(arctan 5/3 + arccos 8/17)

Answer 1

a) .5sqr3, b) -5/3

a) sin(arcsin(1/2) + arccos0)

use trig identity sin(x+y) = sinxcosy +sinycosx

= sin(arcsin.5)cosarcos0 + sin(arccos0)cosarsin.5

= .6(0) + 1(sqr3/2)

= 0 + sqr3/2

= .5sqr3

check the answer

sin(30 + 90)

= sin120

= sqr3/2

=.5sqr3

b) tan(arctan5/3 + arccot8/17)

(this is slightly different from the posted problem which has arccos8/17, not arccot8/17)

use trig identity tan(x+y) = (tanx+ tany)/(1-tanxtany)

(5/3 + 17/8)/(1- (5/3)(17/8))

= (91/24)/(1-85/24)

= 91/-61

=-91/61

check the answer

use a calculator with an inverse trig function to find arctan5/3 and arcot8/17

arctan5/3=about 59 degrees

arccot8/17=arctan17/8= about 65 degrees

tan(59 +65 ) =tan124 =about -1.49 = about -91/61

for the posted problem with

tan(arctan5/3 + arccos8/17)

arccos8/17=about 62 degrees

tan(59+62) = tan121 = about -1.6643= about -5/3= -5(17)/3(17)= -85/51

the most simple exact answer is:

tan(arctan5/3 + arccos8/17)= -5/3 or - 1 2/3

the angle whose cosine = 8/17 has adjacent side=8, hypotenuse=17

and opposite side = sqr(17^2-8^2) = sqr(289-64)=sqr225 =15

tangent of the angle= opposite side/adjacent side = 15/8

tan(arctan5/3 + arccos8/17)

= tan(arctan5/3+arctan15/8)

= (5/3 + 15/8)/(1-(5/3)(15/18))

= (85/24)/(1-75/24)

= (85/24)/-(51/24)

= 85/-51

=-85/51

=-5/3

Answer 2

a. sin(Arcsin(1/2) + Arccos0) = sin(π/6 + π/2) = sin(2π/3) = √3 / 2

b. tan(Arctan(5/3) + Arccos(8/17)) =

[tan(Arctan5/3) + tan(Arccos(8/17)] / [1 - tan(Arctan(5/3))tan(Arccos(8/17)]

tan(Arctan(5/3)) = 5/3

If θ = Arccos(8/17), then cosθ = 8/17 and θ is acute. So, sinθ = √(1 - cos²θ) = 15/17

Therefore, tanθ = sinθ/cosθ = 15/8.

So, we have [5/3 + 15/8] / [1 - (5/3)(15/8)] = (85/24) / (-51/24) = -85/51