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For real numbers a and b, where a ≠ 0 and b ≠ 0, we can find numbers α, β, γ, δ and R such that $a \, ext{cos}(x) + b \, ext{sin}(x)$ can be written in the following 4 forms: $$egin{align*} R \, ext{sin}(x + \alpha) \\ R \, ext{sin}(x - \beta) \\ R \, ext{cos}(x + \gamma) \\ R \, ext{cos}(x - \delta) \end{align*}$$ where $R > 0$ and $0 < \alpha, \beta, \gamma, \delta < 2\pi$ - HSC - SSCE Mathematics Extension 1 - Question 10 - 2024 - Paper 1
Question 10
For real numbers a and b, where a ≠ 0 and b ≠ 0, we can find numbers α, β, γ, δ and R such that $a \, ext{cos}(x) + b \, ext{sin}(x)$ can be written in the followi... show full transcript
Worked Solution & Example Answer:For real numbers a and b, where a ≠ 0 and b ≠ 0, we can find numbers α, β, γ, δ and R such that $a \, ext{cos}(x) + b \, ext{sin}(x)$ can be written in the following 4 forms: $$egin{align*} R \, ext{sin}(x + \alpha) \\ R \, ext{sin}(x - \beta) \\ R \, ext{cos}(x + \gamma) \\ R \, ext{cos}(x - \delta) \end{align*}$$ where $R > 0$ and $0 < \alpha, \beta, \gamma, \delta < 2\pi$ - HSC - SSCE Mathematics Extension 1 - Question 10 - 2024 - Paper 1
Step 1
What is the value of α + β + γ + δ?
Answer
To solve for the sum of angles, we recognize that the forms represent different phase shifts of the trigonometric functions. Each phase shift corresponds to a rotation.
Given that each form relates to sine and cosine of the same angle with shifts of α, β, γ, and δ being defined within the intervals, we know:
- The total rotation (the full cycle) for each sine and cosine function concerning the unit circle is a complete 360 degrees, or .
- Since we can relate the forms back to one another through these phase shifts, the sum of all four angles must encapsulate one full cycle.
Thus, we conclude that:
Therefore, the correct choice is C. 2π.