Explanation:
first of all, there is the rule for all types of triangles : the sum of all angles in a triangle is always 180°.
and for right-angled triangles there is also Pythagoras :
c² = a² + b²
with c being the Hypotenuse (the side opposite of the 90° angle), and a, b being the legs.
so, with these rules we can get all missing elements :
180 = 90 + angle Q + angle R
180 = 90 + 70 + angle R
20° = angle R
30² = 24² + x²
900 = 576 + x²
324 = x²
x = sqrt(324) = 18
the 6 trigonometric ratios (functions) of any angle are
sine
cosine
tangent
cotangent
secant
cosecant
there is something very wrong with the numbers.
a triangle with the sides 30, 24, 18 cannot have the angles 90°, 70°, 20°.
the angles are with these sides
P = 90°
Q = 36.87°
R = 53.13°
or a triangle with the angles 90°, 70°, 20° and the Hypotenuse = 30 has the legs
PQ = 10.261
PR = 28.191
the given sides do not fit to the given angles.
I will give you now the trigonometric ratios of Q based on the calculated side lengths 30, 24, 18 and ignore the specified angle sizes :
sin(Q) = opposite/ Hypotenuse = 18/30 = 3/5 = 0.6
Q = 36.87°
cos(Q) = adjacent/ Hypotenuse = 24/30 = 4/5 = 0.8
Q = 36.87°
tan(Q) = opposite/ adjacent = sin(Q)/cos(Q) = 18/24 =
= 3/4 = 0.75 Q = 36.87°
cot(Q) = adjacent/ opposite = 24/18 = 4/3 = 1.33333...
Q = 36.87°
sec(Q) = Hypotenuse/ adjacent = 30/24 = 5/4 = 1.25
Q = 36.87°
csc(Q) = Hypotenuse/ opposite = 30/18 = 5/3 = 1.6666...
Q = 36.87°
if the sides of the right-angled triangle are truly 30, 24, 18, then Q = 36.87° (confirmed by all 6 trigonometric ratios), and above are all 6 trigonometric ratios for that angle.