Question

1. A) Is there a number x ∈ R, so cos x = 0.8 and sin x = 0.2? b) Same question for cos x = 0.8 and sin x = 0.6 2. Show that for any x ∈ R: a) sin^{3}x + cos^{3} = (sin x + cos x)(1- sin x * cos x) b) sin^{4} x + cos^{4} x = 1 - 2sin^{2} x * cos^{2}x

Answer 1

1Ai)
`x = 36.9^(o)`

1Aii)
`x = 11.5^(o)`

Explanation:

Question 1a

`cosx=0.8\\x= cos^(-1)(0.8)\\ x= 36.9^(o)`

`sinx=0.2\\x=sin^(-1)(0.2)\\ x=11.5^(o)`

Question 2a

`sin^(3)x+cos^(3)x = sin^(2)x(sinx)+cos^(2)x(cosx)\\ \\= (1-cos^(2)x)sinx+(1-sin^(2)x)cosx\\ \\= sinx-sinxcos^(2)x+cosx-sin^(2)xcosx\\ \\=sinx-sin^(2)xcosx+cosx-sinxcos^(2)x\\`

Factorize

`sinx(1-sinxcosx)+cosx(1-sinxcosx)\\\\=(sinx+cosx)(1-sinxcosx)\\\\=(sinx+cosx)(1-sinx*cosx)`

Question 2b

`sin^(4)x+ cos^(4)x=(sin^(2)x)^(2) + (cos^2}x)^(2) \\But , a^(2) +b^(2)=(a+b)^(2)-2ab\\ \\a = sin^(2)x\\ b = cos^(2)x\\\\Therefore, (sin^(2)x)^(2) + (cos^2}x)^(2) = (sin^(2)x+cos^(2)x)^(2)-2sin^(2)xcos^(2)x \\\\But, sin^(2)x+cos^(2)x = 1\\\\Therefore, (sin^(2)x+cos^(2)x)^(2)-2sin^(2)xcos^(2)x = 1 - 2sin^(2)xcos^(2)x\\\\= 1 - 2sin^(2)x*cos^(2)x`