Step-by-step explanation:
13)
remember 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).
12/sin(90) = DC/sin(47)
12/1 = DC/sin(47)
DC = 12×sin(47) = 8.776244419... ≈ 8.8
14)
so, we have a right-angled triangle.
the ladder (20 ft) is the Hypotenuse (the baseline opposite of the 90° angle). the ground distance (17 ft) of the ladder to the wall and the height on the wall are the legs.
first let's get the third side (the height in the wall) by using Pythagoras :
c² = a² + b²
c being the Hypotenuse.
20² = 17² + height²
400 = 289 + height²
height² = 111
height = sqrt(111) = 10.53565375... ft
now, again the law of sine :
20/sin(90) = sqrt(111)/sin(angle)
20 = sqrt(111)/sin(angle)
sin(angle) = sqrt(111)/20 = 0.526782688...
angle (ladder with ground) = 31.78833062...° ≈ 32°
15)
cos > 0 and csc < 0.
as csc = 1/sin, this means sin < 0.
sin (and therefore csc) is negative only in the 3rd and 4th quadrant.
cos is positive in the 1st and the 4th quadrant.
so, the answer is the 4th quadrant (bottom right).
16)
tan(315) = sin(315)/cos(315)
315° = 360 - 45 = 270 + 45
so, this is half of the 4th quadrant, and that means
sin(315) = -cos(315) (sine of 45° = cosine of 45°).
tan(315) = -cos(315)/cos(315) = -1
