(a)
A = 2x
B = 10°
C = 70°

tanC = tan(70°) = 2.74747741945
tanB = tan(10°) = 0.1763269807

cot(2x-10°) = cot(A-B)

= (cotB•cotA + 1)/(cotB - cotA)

= (cosB•cosA + sinB•sinA)
/(cosB•sinA - sinB•cosA)

= (1 + tanB•tanA)
/(tanA - tanB)

= tan(C)

1 + tanB•tanA = tan(C)•(tanA - tanB)

[tan(C)-tan(B)] • tanA
= 1+tan(B)•tan(C)


tanA
= [1+tan(B)•tan(C)] / [tan(C)-tan(B)]
= 1.48445439791 ÷ 2.57115043875
= 0.57735026917

A₁ = 2x = 30°；A₂ = 2x = 210°

x₁ = 15°；x₂ = 195°
x₃ = 105° ；x₄ = 285°

(b)

A = ³⁄₂ x
B = 80°

2•cosec(³⁄₂ x - 80°) = -4
⇨
cosec(A - B) = -2
⇨
sin(A - B) = -½
⇨
A-B = asin( -½ ) = 180° + 30° = 210°

OR

A-B = asin( -½ ) = 360° - 30° = 330°


x₁ = (210° + 80°) × ⅔ ≈ 193.33°
x₂ = (330° + 80°) × ⅔ ≈ 273.33°

(c)
sec(m) = 1/cos(m)
cosec(m) = 1/sin(m)

sin(40°) = cos(2x+55°)
≈ 0.64278760968

sin(40°) = cos(90°-40°) = cos(2x+55°)
2x+55° = 90°-40° = 50°
OR
2x+55° = 360°-50° = 310°

x₁ = (310° - 55°) ÷ 2 = 127.5°
x₂ = 360° + (50° - 55°) ÷ 2 = 357.5°

(d)

-sin(120°)
= sin(180° + 120°) = sin(300°)
= sin(360° - 120°) = sin(240°)

½ x + 70° = 300° → x₁ = 460°
→ 100° ✗

½ x + 70° = 240° → x₂ = 340° ✓