Question

PLEASE HELP. Please show all work. Im giving 80 points to whoever answers it first.

1) Solve the following triangles: A = 21 degrees a = 10 b = 6 A = 48 degrees a = 32 b = 6
2) Explain which cases can result in two possible triangles.
3) Find the area and perimeter of the two triangles formed in question 1.
4) if A = (2,4) and B = (7, 3) find the magnitude and direction of the vector AB.
5) Explain in words and numerically how to find the dot product of <2, 4, -6> and <-7, 3, -2>
6) Express the following complex number in trigonometric form: 3 - 3i and find the 4th roots.

Answer 1

1.1) B=12.42°, C=146.58°, c=15.37

1.2) B=8°, C=124°, c=35.7

2) In the case of SSA, when the given angle A is acute (<90°), the side "a" (opposite to the given angle "A") is less than the other given side "b" (a<b), and a > b sinA.

3.1) Perimeter = 31.37 units, Area = 16.5 square units

3.2) Perimeter = 73.70 units, Area = 75.6 square units

4.1) Magnitude of vector AB=5.1

4.2) Direction of vector AB=348.69°

5.1) The dot product of two vectors is equal to the sum of the product of their components

5.2) a . b = 10

6.1) Trigonometric form: z = 3√2 (cos 315°+ i sin 315°)

6.2) The 4th roots are:

w1 = 18^(1/8) [cos(78.75°) + i sin(78.75°)]

w2 = 18^(1/8) [cos(168.75°) + i sin(168.75°)]

w3 = 18^(1/8) [cos(258.75°) + i sin(258.75°)]

w4 = 18^(1/8) [cos(348.75°) + i sin(348.75°)]

Explanation:

3.1) Perimeter: P = a+b+c → P = 10+6+15.37 → P = 31.37

Area: A = a b sinC / 2

Replacing the known values:

A = (10)(6) sin 146.58325418° / 2 → A = 60(0.55072472) / 2 →

A = 16.52174152 → A = 16.52 square units

3.2) Perimeter: P = a+b+c → P = 32+6+35.70 → P=73.70

Area: A = a b sin C /2

Replacing the known values:

A = (32)(6) sin 123.99036327° / 2 → A = 192(0.82913161) / 2 →

A = 75.59663485 → A = 75.6 square units