#2) A) a = 1.35 (across from the 15° angle) ∠C = 50.07° (the angle at the top of the triangle) ∠B = 114.93°
B) ∠A = 83° b = 10.77 (across from angle B) a = 15.11 (across from angle A)
Explanation #1) A) Since b is across from the 25° angle and we have the hypotenuse, we have the information for the sine ratio (opposite/hypotenuse): sin 25 = b/25
Multiply both sides by 25: 25*sin 25 = (b/25)*25 25*sin 25 = b 10.57 = b
B) We will first use the cosine ratio. Side a is the side adjacent to the angle and we have the hypotenuse, and the cosine ratio is adjacent/hypotenuse: cos 25 = a/25
Multiply both sides by 25: 25*cos 25 = (a/25)*25 25*cos 25 = a 22.66 = a
Now we will use the Pythagorean theorem. We know from part a that side b = 10.57, and the figure has a hypotenuse of 25: a²+(10.57)² = 25² a² + 111.7249 = 625
Subtract 111.7249 from both sides: a²+111.7249-111.7249=625-111.7249 a² = 513.2751
Take the square root of both sides: √a² = √513.2751 a = 22.66
#2) A) Let A be the 15° angle, B be the angle to the right and C be the angle at the top of the triangle. This means side a is across from angle A, side B is across from angle B, and side c is across from angle C.
Using the law of cosines, a²=3²+4²-2(3)(4)cos(15) a²=9+16-24cos(15) a²=25-24cos(15) a²=1.8178
Take the square root of both sides: √a² = √1.8178 a = 1.3483≈1.35
Now we can use the Law of Sines to find angle C: sin 15/1.35 = sin C/4
Cross multiply: 4*sin 15 = 1.35* sin C
Divide both sides by 1.35: (4*sin 15)/1.35 = (1.35*sin C)/1.35 (4*sin 15)/1.35 = sin C
Take the inverse sine of both sides: sin⁻¹((4*sin 15)/1.35) = sin⁻¹(sin C) sin⁻¹((4*sin 15)/1.35) = C 50.07 = C
To find angle B, add angle A and angle C together and subtract from 180: B=180-(50.07+15) = 180-65.07 = 114.93
B) To find angle A, add angle B and angle C together and subtract from 180: 180-(52+45) = 180-97 = 83
Now use the Law of Sines to find side b (across from angle B): sin 52/12 = sin 45/b
Cross multiply: b*sin 52 = 12*sin 45
Divide both sides by sin 52: (b*sin 52)/(sin 52) = (12*sin 45)/(sin 52) b = 10.77
Find side a using the Law of Sines: sin 83/a = sin 52/12
Cross multiply: 12*sin 83 = a*sin 52
Divide both sides by sin 52: (12*sin 83)/(sin 52) = (a*sin 52)/(sin 52) 15.11 = a
