C09 Trigonometric Equations
对应大纲的内容
| MM4.6 | Solution of simple trigonometric equations in a given interval (this may involve the use of the identities in 4.5).
For example: $\tan \theta = -\dfrac{1}{\sqrt{3}}$ for $-\pi < x < \pi$, $\sin^{2}\left(2x + \dfrac{\pi}{3}\right) = \dfrac{1}{2}$ for $-2\pi $ $<$ $x$ $<$ $2\pi$, $12 \cos^{2} x + 6 \sin x – 10 = 2$ for $0^\circ < x < 360^\circ$. |
学习目标
- Master domain transformations for composite angles by adjusting boundary conditions before mapping unit circle roots to solve equations like $\sin \left(2 x+\frac{\pi}{3}\right)$ $=$ $0.5$, ensuring all valid solutions are captured.
- Execute the zero-cancellation protocol by utilising non-standard factorisation (e.g., $\sin x(\cos x-1)=0$ ) instead of dividing by variables, ensuring no roots are lost during algebraic simplification.
- Synthesise quadratic trigonometric filters by applying Pythagorean identities to factorise disguised quadratics, rigorously filtering results against the absolute $[-1, 1]$ range limits of sine and cosine functions.
- Map universal periodic symmetries by exploiting the periodic nature of trigonometric graphs to identify all valid solutions within a continuous interval, moving fluidly between domains without relying solely on calculator principal values.
建议时间
| Task | No. of Questions | Suggested Time |
|---|---|---|
| Examples | 9 | 180 min |
| Quiz | 6 | 25 min |
| Practice A | 20 | 40 min |
| Practice B | 20 | 60 min |
| Total | 55 | approx. 5-6 hrs |
Practice:
Practice:
