section 64 4 Sum and Difference Fomulas arccos 4 4754 find t

section 64 4 Sum and Difference Fomulas +arccos )4 47-54, find the value of the esprssion without7 s1. the solutions of the equation in the nterval 9.2 55-58, evalaate the trigomwertrie function lah et using SSalra adaliate the tr_trie function -t-.la(..., withoet using a calculator.onnrtrie 79. sanding waves The equation of a\"wwr is. bained by ading the diplacemems of waves weling s opponite directioes (see figure Assume that each of the vaes has amplitude A, pend T, and wineleagh cos - models for these waves are Eseries 59-62, use right triangles to evaluate the y-Acm2!)-dn®Am2r( e sin-i- show thai 00s-18 17 netric and In Evercises 63-70, verify the identity. etric a th 1-tan csy)+cose y-2cos x cos y

Solution

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47 sin ( sin^-1(1) +cos^-1(1) )

As per inverse trig: sin^-1x + cos^-1x = pi/2

So,sin ( sin^-1(1) +cos^-1(1) ) = sin( pi/2) =1

48. sin ( sin^-(1) - cos^-1(-1))

Now Let cos^-1(-1) = y ---> cosy = -1 ---> y = pi

sin^-1(1) = pi/2

So, sin ( sin^-(1) - cos^-1(-1)) = sin( pi/2 - pi) =sin(-pi/2) = -1

50. cos[ cos^-1(-1) -cos^-1(1) ]

cos^-1(-1) = pi ; cos^-1(1) = 0

So,cos[ cos^-1(-1) -cos^-1(1) ] = cos(pi) = -1

52 cos [ cos^-1(-1/2) + sin^-1(1) ]

sin^-1(1) = pi/2

cos^-1(-1/2) = pi -pi/3 = 2pi/3

cos[ 2pi/3 +pi/2] = -sqrt(3)/2

54. tan[ cos^-1(sqrt(2)/2 -sin^-1(0)]

sin^-1(0) =0 ; cos^-1(sqrt(2)/2 = pi/4

So,tan[ cos^-1(sqrt(2)/2 -sin^-1(0)] = tanpi/4 =1