In a triangle ABC, ∠A = 75° and ∠B = 45°. What is 2a - b equal to?

Triangle Angle Sum Property:

• In a triangle, the sum of all interior angles is always equal to 180º.
• Given angles can be used to find the unknown angle using the formula: \( \angle A + \angle B + \angle C = 180^\circ \)
• Once all angles are known, trigonometric ratios can be used to relate the sides of the triangle.
• Law of Sines:
• The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
• Here, a, b, and c are the sides opposite to angles A, B, and C respectively.

Calculation:

Given,

Angle A = 75º

Angle B = 45º

Using the triangle angle sum property, find angle C:

⇒ ∠C = 180º - (∠A + ∠B)

⇒ ∠C = 180º - (75º + 45º)

⇒ ∠C = 180º - 120º

⇒ ∠C = 60º

Now, using the Law of Sines:

\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

⇒ \(a = \frac{sin 75°}{sin 60°}.c = \frac{\sqrt3 +1}{\sqrt6}.c\)

⇒ \(b = \frac{sin 45°}{sin 60°}.c = \frac{2c}{\sqrt6}\)

Now

⇒  2a - b = \(\frac{2\sqrt3 c}{\sqrt6} = \sqrt2 c\)

∴ Option (b) is correct.