Edexcel IAL - Pure Maths 2- 6.2 Solving Trigonometric Equations- Study notes - New syllabus
Edexcel IAL – Pure Maths 2- 6.2 Solving Trigonometric Equations -Study notes- New syllabus
Edexcel IAL – Pure Maths 2- 6.2 Solving Trigonometric Equations -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 6.2 Solving Trigonometric Equations
Solution of Simple Trigonometric Equations in a Given Interval
To solve trigonometric equations in a given interval, we:
- Simplify the equation if required
- Find the general solutions using standard trigonometric values
- Restrict the solutions to the given interval
All solutions must lie within the stated domain.
General Strategy
| Step | Action |
| 1 | Isolate the trigonometric function |
| 2 | Find reference angles using standard values |
| 3 | Use CAST diagram (for sine and cosine) |
| 4 | Apply the given interval |
Example
Solve:
\( \sin\!\left(x + \dfrac{\pi}{2}\right) = \dfrac{3}{4}, \quad 0 < x < 2\pi \)
▶️ Answer / Explanation
Let \( \theta = x + \dfrac{\pi}{2} \).
\( \sin\theta = \dfrac{3}{4} \)
Reference angle:
\( \theta = \sin^{-1}\!\left(\dfrac{3}{4}\right) \)
Solutions for \( \theta \) in \( (0, 2\pi) \):
\( \theta = \sin^{-1}\!\left(\dfrac{3}{4}\right),\ \pi – \sin^{-1}\!\left(\dfrac{3}{4}\right) \)
Subtract \( \dfrac{\pi}{2} \):
Solutions:
\( x = \sin^{-1}\!\left(\dfrac{3}{4}\right) – \dfrac{\pi}{2},\quad \pi – \sin^{-1}\!\left(\dfrac{3}{4}\right) – \dfrac{\pi}{2} \)
Example
Solve:
\( \cos(x + 30^\circ) = \dfrac{1}{2}, \quad -180^\circ < x < 180^\circ \)
▶️ Answer / Explanation
\( \cos\theta = \dfrac{1}{2} \Rightarrow \theta = 60^\circ,\ 300^\circ \)
Let \( \theta = x + 30^\circ \).
\( x + 30^\circ = 60^\circ \Rightarrow x = 30^\circ \)
\( x + 30^\circ = 300^\circ \Rightarrow x = 270^\circ \)
Only \( x = 30^\circ \) lies in the interval.
Solution: \( x = 30^\circ \)
Example
Solve:
\( \tan 2x = 1, \quad 90^\circ < x < 270^\circ \)
▶️ Answer / Explanation
\( \tan\theta = 1 \Rightarrow \theta = 45^\circ + k180^\circ \)
So:
\( 2x = 45^\circ,\ 225^\circ \)
\( x = 22.5^\circ,\ 112.5^\circ \)
Only \( x = 112.5^\circ \) lies in the interval.
Solution: \( x = 112.5^\circ \)
Example
Solve:
\( 6\cos^2 x + \sin x – 5 = 0, \quad 0^\circ \le x < 360^\circ \)
▶️ Answer / Explanation
Use \( \cos^2 x = 1 – \sin^2 x \):
\( 6(1 – \sin^2 x) + \sin x – 5 = 0 \)
\( -6\sin^2 x + \sin x + 1 = 0 \)
\( 6\sin^2 x – \sin x – 1 = 0 \)
Factorise:
\( (3\sin x + 1)(2\sin x – 1) = 0 \)
\( \sin x = -\dfrac{1}{3},\ \dfrac{1}{2} \)
Solutions:
\( x = 30^\circ,\ 150^\circ,\ 200^\circ,\ 340^\circ \) (approx)
Example
Solve:
\( \sin^2\!\left(x + \dfrac{\pi}{6}\right) = \dfrac{1}{2}, \quad -\pi \le x < \pi \)
▶️ Answer / Explanation
\( \sin^2\theta = \dfrac{1}{2} \Rightarrow \sin\theta = \pm\dfrac{1}{\sqrt{2}} \)
\( \theta = \dfrac{\pi}{4},\ \dfrac{3\pi}{4},\ \dfrac{5\pi}{4},\ \dfrac{7\pi}{4} \)
Subtract \( \dfrac{\pi}{6} \):
Solutions:
\( x = \dfrac{\pi}{12},\ \dfrac{7\pi}{12},\ -\dfrac{11\pi}{12},\ -\dfrac{5\pi}{12} \)