Explanation:
remember, the sum of all angles in a triangle is always 180°.
1. you mean find the angle at A, yes ?
180 = angle A + angle B + angle C =
= angle A + 38 + 90
angle A = 180 - 38 - 90 = 90 - 38 = 52°
2. Pythagoras
AB² = 11² + 8.5² = 121 + 72.25 = 193.25
AB = 13.90143877... in
3. sine/cosine ratio is tangent.
sin(38) × AB = 8.5 in
cos(38) × AB = 11 in
sin(38)/cos(38) = tan(38) = 8.5/11 = 0.772727272...
sin(52) × AB = 11 in
cos(52) × AB = 8.5 in
sin(52)/cos(52) = tan(52) = 11/8.5 = 1.294117647...
4. they are equal. as sin(x) = cos(90-x).
5. they are equal. as cos(x) = sin(90-x).
6. 90°. again, the sum of all angles must be 180°. so, when one angle is all by itself already 90°, then there are only 90° left for the other 2 angles.
7. not sure what you mean by "kissing angles" but
a. sin(40°) = cos(90-40) = cos(50°) = 0.64278761...
b. cos(27°) = sin(90-27) = sin(63°) = 0.891006524...
c. sin(90°) = cos(90-90) = cos(0°) = 1
d. cos(65°) = sin(90 - 65) = sin(25°) = 0.422618262...