Angles in 2 triangles on each other

Question

I need to find angle b  If you can't see the image is of two right angled triangles , the lower's slant is the higher's base. Angle A and B are the angles between the hypotenuse and base of both. The right angle is opposite the hypotenuse for both and the side opposite the hypotoneuse for both is 3 units long. Also the base of the lower triangle is 8 units long. I got 70.65 to 2 dp but I would like a second opinion, please show all decimal places.

 

Answer 1

I think I get the picture as you've described. If I'm not mistaken the hypotenuse of the lower triangle (which forms the base of the upper one) has length sqrt(3^2+8^2)=sqrt(73) by Pythagoras. So the base of the upper triangle is sqrt(73), and the tangent of the angle between the hypotenuse and base is 3/sqrt(73)=0.3511. The angle in the upper triangle is tan^-1(0.3511)=19.35 deg. The corresponding angle in the lower triangle is tan^-1(3/8)=20.56 deg. The combined angle is 39.90. The other angle in the upper triangle would be 70.65 deg, which ties in with your calculation, and the other angle in the lower triangle is 69.44 deg. So I think we're singing from the same hymn sheet!