For real numbers a and b, where a ≠ 0 and b ≠ 0, we can find numbers α, β, γ, δ and R such that a cos x + b sin x can be written in the following 4 forms: R sin(x + α) R sin(x − β) R cos(x + γ) R cos(x − δ) where R > 0 and 0 < α, β, γ, δ < 2π - HSC - SSCE Mathematics Extension 1 - Question 10 - 2024 - Paper 1

To find the value of α + β + γ + δ, we consider the periodic properties of the trigonometric functions involved in the transformations.

Each of the angles α, β, γ, and δ can be interpreted in the context of the equations given:

1. R sin(x + α) represents a phase shift in the sine function and essentially modifies the angle of the sine wave by α.

2. R sin(x − β) also alters the sine function, but in the opposite direction, thus combining with the previous term's effect.

3. R cos(x + γ) relates to the cosine function with a phase shift, which ultimately changes the sine representation.

4. R cos(x − δ) similarly adjusts the cosine wave.

The general observation is that:

• The sum of angles α, β, γ, and δ must cover the complete cycle of the trigonometric functions represented in the full range.
• Given the periodic nature of these functions and their respective rotations, it can be concluded that their net effect results in summing to a value of 2π.

Thus, we conclude that:

$ext{α + β + γ + δ} = 2 ext{π}$