If sin ⁡x=7/25, and 0<x<π/2, what is tan⁡(x−π/4)?

Question

If sin ⁡x=7/25, and 0<x<π/2, what is tan⁡(x−π/4) in fractional form?

Answer 1

If sin ⁡x=7/25, and 0<x<π/2, what is tan⁡(x−π/4)?

Sin^-1(7/25) = 16.26°

Tan(16.26 - 45°) = Tan(-28.74°) = -0.5484

Answer 2

sinx=7/25

therefore you have a right triangle with the adjacent side equal to 24 (25^2-7^2=625-49=576 and √576=24)

therefore cosx=24/25

∏/4=45°

use the tan(a-b) identity

tan(x-45°)=(sinxcos45-cosxsin45)/(cosxcos45+sinxsin45)

tan(x-45°)=[(7/25)(√2/2)-(24/25)(√2/2)]/[(24/25)(√2/2)+(7/25)(√2/2)

tan(x-45°)=[(7√2/50)-24√2/50)]/[(24√2/50)+7√2/50)]

tan(x-45°)=(-17√2/50)/(31√2/50)

tan(x-45°)=(-17√2/50)*(50/31√2)

tan(x-45°)=-17√2/31√2

tan(x-45°)=-17/31