Step-by-step explanation:
11.
x is sin(60) in a circle with radius of 8.
x = sin(60)×8 = 6.92820323... ≈ 6.9
12.
x is sin(30) in a circle with radius "diagonal length".
6 is cos(30) in the same circle.
6 = cos(30)×r
r = 6/cos(30) = 6.92820323...
x = sin(30)×r = 3.464101615... ≈ 3.5
13.
similar to 12.
but we refer to the angle at the bottom right vertex.
remember, the sum of all angles in a triangle is always 180°.
so, the angle over there at the right is
180 - 90 - 68 = 22°
x = sin(22)×r
25 = cos(22)×r
r = 25/cos(22) = 26.96336857...
x = sin(22)×r = 10.10065565... ≈ 10.1
14.
9 is the sine of the complementary angle of 49 in a circle with radius x.
the complementary angle is the angle that adds up with the original angle to 90° and is therefore
90 - 49 = 41°
9 = sin(41)×x
x = 9/sin(41) = 13.71827778... ≈ 13.7
18.
we use the extended Pythagoras :
c² = a² + b² - 2ab×cos(C)
with c being the side opposite of the angle C.
so we have
LK² = 16² + 24² - 2×16×24×cos(29) =
= 256 + 576 - 768×cos(29) =
= 832 - 671.7079351... = 160.2920649...
LK = 12.66065026... ≈ 12.7
19.
law of sine
a/sin(A) = b/sin(B) = c/sin(C)
with the sides being always opposite to the corresponding angles.
to find the angle at H we need to find the angle at G first, because we don't know GF yet.
so,
28/sin(76) = 18/sin(G)
sin(G) = 18×sin(76)/28 = 0.623761538...
G = 38.59134528...°
remember the 180° rule in triangles.
H = 180 - 76 - G = 65.40865472...° ≈ 65.4°
20.
GF is now calculated using the law of sine again.
28/sin(76) = GF/sin(H)
GF = 28×sin(H)/sin(76) = 26.23980579... ≈ 26.2
