Introduction
This cosec cot equation is a classic O Level A Maths trigonometry question because it tests identities, algebraic manipulation, and solving for angles in a given range. Many students panic when they see \(\cosec^2 \theta\) and \(\cot \theta\) together, but once you convert the expression into one trig function, this cosec cot equation becomes much more manageable.
The Question / Scenario Explanation
Source: TRIGO SOLVING O Level 2025 Paper 1
Question (as shown): Solve the equation \(2\cosec^2 \theta – \cot \theta – 3 = 0\) for \(0^\circ \leq \theta \leq 360^\circ\).
Step-by-Step Solution / Explanation
Step 1: Use the trig identity
For this cosec cot equation, the key identity is:
\(\cosec^2 \theta = \cot^2 \theta + 1\)
Substitute this into the equation:
\(2(\cot^2 \theta + 1) – \cot \theta – 3 = 0\)
Step 2: Simplify into a quadratic in \(\cot \theta\)
Expand and collect like terms:
\(2\cot^2 \theta + 2 – \cot \theta – 3 = 0\)
\(2\cot^2 \theta – \cot \theta – 1 = 0\)
Now the cosec cot equation has been turned into a quadratic expression in \(\cot \theta\).
Step 3: Factorise the quadratic
\(2\cot^2 \theta – \cot \theta – 1 = 0\)
\((2\cot \theta + 1)(\cot \theta – 1) = 0\)
So:
\(2\cot \theta + 1 = 0\) or \(\cot \theta – 1 = 0\)
This gives:
\(\cot \theta = -\frac{1}{2}\) or \(\cot \theta = 1\)
Step 4: Solve \(\cot \theta = 1\)
If \(\cot \theta = 1\), then:
\(\tan \theta = 1\)
The reference angle is:
\(\theta = 45^\circ\)
Since tangent is positive in Quadrants I and III:
\(\theta = 45^\circ, 225^\circ\)
Step 5: Solve \(\cot \theta = -\frac{1}{2}\)
If \(\cot \theta = -\frac{1}{2}\), then:
\(\tan \theta = -2\)
First find the reference angle:
\(\alpha = \tan^{-1}(2) \approx 63.4^\circ\)
Tangent is negative in Quadrants II and IV, so:
\(\theta = 180^\circ – 63.4^\circ = 116.6^\circ\)
\(\theta = 360^\circ – 63.4^\circ = 296.6^\circ\)
Step 6: Write the full solution set
The four solutions for this cosec cot equation are:
\(\theta = 45^\circ,\ 116.6^\circ,\ 225^\circ,\ 296.6^\circ\)
✅ Final Answer: \(\theta = 45^\circ,\ 116.6^\circ,\ 225^\circ,\ 296.6^\circ\)
Key Concepts Students Must Know
- For a cosec cot equation, a very useful identity is \(\cosec^2 \theta = \cot^2 \theta + 1\).
- When two trig functions appear together, try converting everything into one function first.
- After simplifying, many trig equations become quadratics that can be factorised.
- Once you get values of \(\tan \theta\) or \(\cot \theta\), use quadrant rules carefully to find all angles in the required range.
Exam Tips / Common Mistakes
Exam Tips
- Memorise key identities like \(\cosec^2 \theta = \cot^2 \theta + 1\).
- In a cosec cot equation, rewrite everything in terms of one trig function before solving.
- Factorise carefully and solve each branch separately.
- Always check the interval given, here \(0^\circ \leq \theta \leq 360^\circ\).
- Use CAST or quadrant signs properly when finding multiple angle solutions.
Common Mistakes
- Using the wrong identity, such as mixing up \(\cosec^2 \theta\) with \(1 + \tan^2 \theta\).
- Factorising \(2\cot^2 \theta – \cot \theta – 1\) wrongly.
- Finding only the reference angle and forgetting the second solution in the correct quadrant.
- Writing \(\tan \theta = -2\) and giving only one angle instead of two.
- Stopping at \(\cot \theta = 1\) or \(\cot \theta = -\frac{1}{2}\) without converting to actual angle values.
Parent Insight
This cosec cot equation is a strong O Level A Maths example because it combines identity use, algebra, and angle reasoning in one question. Students who struggle with trigonometry often do not fail because the maths is impossible, but because they do not know how to start. With regular guided practice, children learn to recognise the identity first, simplify confidently, and then solve the angle part step by step.
Conclusion
To solve this cosec cot equation, we first used the identity \(\cosec^2 \theta = \cot^2 \theta + 1\), then simplified to \(2\cot^2 \theta – \cot \theta – 1 = 0\). After factorising, we solved \(\cot \theta = 1\) and \(\cot \theta = -\frac{1}{2}\), giving the final answers \(\theta = 45^\circ,\ 116.6^\circ,\ 225^\circ,\ 296.6^\circ\).
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