Explanation:
17)
sec(x) = 1/cos(x)
cot(x) = cos(x)/sin(x)
so,
cos(x)/sin(x) × 1/cos(x) = 1/sin(x) = csc(x)
18)
2×sin(x) + 1 = 0
2×sin(x) = -1
sin(x) = -1/2 = -0.5
x = 210° or x = 330°
why ? sin(30°) = 0.5
so, sin(x) = -0.5 means the angles in the 3rd and 4th quadrant (where sine points "down" and is therefore negative). and x is then either 30° more than 180° (210°) or 30° less than 360° (330°).
draw a circle with the coordinate axis through the center. and then try and see.
19)
again the law of sine :
a/sin(A) = b/sin(B) = c/sin(C)
where the sides (a, b, c) are always opposite to the correlated angles (A, B, C).
so, in our case we have
8/sin(A) = 10/sin(58)
sin(A) = 8×sin(58)/10 = 0.678438477...
A = 42.7217403...° ≈ 43°
20)
so, we have 2 sides and the enclosed angle.
the area of the triangle in this case is
1/2 × a×c×sin(B) = 1/2 × 12×8×sin(60) = 48×sin(60) =
= 41.56921938... ≈ 42