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All trigonometric roots on an interval — 0606

List every permitted periodic solution and include required endpoints.

Notes and key methods

Find the reference solutions in one period, add the appropriate period multiples, and keep only inputs in the stated interval. If the angle is nx, first solve in that angle and then divide every solution by n.

A closed interval may include both endpoints. Avoid counting the same solution twice when two families meet at an endpoint. Checking each proposed value in the original equation catches missed scaling factors.

Check the conditions

Three worked examples

Example 1 · Trigonometric equations over an interval

Solve \(\sin{\left(2 x \right)} = 0\) for \(0 \leq x\) and \(x \leq 5 \pi\).

  1. Give all solutions, using a clearly stated integer parameter.
  2. Find the smallest positive root divided by π.
  3. Find the largest root divided by π.
  4. Find the sum of all roots divided by π.
Show the reasoning

(a)

The sine is zero when \(\displaystyle nx=j\pi\); the stated x interval gives \(\displaystyle 0\le j\le 10\).

Answer: \(\displaystyle x= \frac{j\pi }{2}, \quad j= 0, 1, \ldots , 10\)

(b)

Use \(\displaystyle nx=\pi\) for the first positive zero.

\(\displaystyle \frac{1}{2}\)

Answer: \(\displaystyle \frac{1}{2}\).

(c)

The upper endpoint is also a root.

\(5\)

Answer: \(5\).

(d)

Sum \(\displaystyle \frac{j}{n}\) over the complete index range.

\(\displaystyle \frac{55}{2}\)

Answer: \(\displaystyle \frac{55}{2}\).

Example 2 · Trigonometric equations over an interval

Solve \(\sin{\left(3 x \right)} = 0\) for \(0 \leq x\) and \(x \leq 2 \pi\).

  1. Give all solutions, using a clearly stated integer parameter.
  2. Find the smallest positive root divided by π.
  3. Find the largest root divided by π.
  4. Find the sum of all roots divided by π.
Show the reasoning

(a)

The sine is zero when \(\displaystyle nx=j\pi\); the stated x interval gives \(\displaystyle 0\le j\le 6\).

Answer: \(\displaystyle x= \frac{j\pi }{3}, \quad j= 0, 1, \ldots , 6\)

(b)

Use \(\displaystyle nx=\pi\) for the first positive zero.

\(\displaystyle \frac{1}{3}\)

Answer: \(\displaystyle \frac{1}{3}\).

(c)

The upper endpoint is also a root.

\(2\)

Answer: \(2\).

(d)

Sum \(\displaystyle \frac{j}{n}\) over the complete index range.

\(7\)

Answer: \(7\).

Example 3 · Trigonometric equations over an interval

Solve \(\sin{\left(4 x \right)} = 0\) for \(0 \leq x\) and \(x \leq 3 \pi\).

  1. Give all solutions, using a clearly stated integer parameter.
  2. Find the smallest positive root divided by π.
  3. Find the largest root divided by π.
  4. Find the sum of all roots divided by π.
Show the reasoning

(a)

The sine is zero when \(\displaystyle nx=j\pi\); the stated x interval gives \(\displaystyle 0\le j\le 12\).

Answer: \(\displaystyle x= \frac{j\pi }{4}, \quad j= 0, 1, \ldots , 12\)

(b)

Use \(\displaystyle nx=\pi\) for the first positive zero.

\(\displaystyle \frac{1}{4}\)

Answer: \(\displaystyle \frac{1}{4}\).

(c)

The upper endpoint is also a root.

\(3\)

Answer: \(3\).

(d)

Sum \(\displaystyle \frac{j}{n}\) over the complete index range.

\(\displaystyle \frac{39}{2}\)

Answer: \(\displaystyle \frac{39}{2}\).

Common mistakes

Practise with answer checking

Original exam-style problems

Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.

Examination practice · 2020–2026

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