Home›Calculus 1›Solving trigonometric equations›Problem 13.83

Solving trigonometric equations

Problem 13.83 · medium

Solve \( \displaystyle 2 \cos^{2}{\left(\theta \right)} - \cos{\left(\theta \right)} = 0 \) for \( \displaystyle 0 \le \theta < 2\pi \).

\[ 2 \cos^{2}{\left(\theta \right)} - \cos{\left(\theta \right)} = \left(2 \cos{\left(\theta \right)} - 1\right) \cos{\left(\theta \right)} \]
Factor as a quadratic in cos θ.✓ Proved
Set each factor to zero; then find every angle in [0, 2π) with those values.
\[ \left. 2 \cos^{2}{\left(\theta \right)} - \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{3} }} = 0 \]
θ = pi/3 works.✓ Proved
\[ \left. 2 \cos^{2}{\left(\theta \right)} - \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{\pi}{2} }} = 0 \]
θ = pi/2 works.✓ Proved
\[ \left. 2 \cos^{2}{\left(\theta \right)} - \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{3 \pi}{2} }} = 0 \]
θ = 3*pi/2 works.✓ Proved
\[ \left. 2 \cos^{2}{\left(\theta \right)} - \cos{\left(\theta \right)} \right|_{\substack{ \theta=\frac{5 \pi}{3} }} = 0 \]
θ = 5*pi/3 works.✓ Proved

Answer \( \theta = \frac{\pi}{3},\ \frac{\pi}{2},\ \frac{3 \pi}{2},\ \frac{5 \pi}{3} \)

Lines: 5 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Not checked	—	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
6	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	Newton's method from 1440 starting points finds exactly 4 roots in [0, 2π), the same ones

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution lists four solutions but only verifies them by substitution, failing to demonstrate that these are the *only* solutions in the interval [0, 2π). Specifically, it does not explicitly solve cos(θ) = 1/2 or cos(θ) = 0 to derive the angles, leaving open the possibility of missing solutions or including extraneous ones.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution lists four solutions but only verifies them by substitution, failing to demonstrate that these are the *only* solutions in the interval [0, 2π). Specifically, it does not explicitly solve cos(θ) = 1/2 or cos(θ) = 0 to derive the angles, leaving open the possibility of missing solutions or including extraneous ones.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly factors the equation, identifies the necessary cosine values, and verifies that the four listed solutions satisfy the equation within the specified interval.
gpt-oss:20b: pass 2026-10-14
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution is incomplete because it fails to explicitly solve for the values of cos(theta) (1/2 and 0) and identify all corresponding angles in the interval [0, 2pi). Specifically, it omits theta = 3pi/2 is correct for cos=0, but misses that cos(theta)=0 also yields theta=pi/2 (included) and cos(theta)=1/2 yields theta=pi/3 and 5pi/3 (included). Wait, let me re-check the roots. 2cos^2 - cos = 0 => cos(2cos - 1) = 0. cos=0 => pi/2, 3pi/2. cos=1/2 => pi/3, 5pi/3. The stated answer contains all 4 solutions. The solution steps only verify these 4 specific values but do not show the derivation of finding them. A student would not learn how to find the solutions, only how to check them. However, the prompt asks to review the *solution*. Is it wrong? It's incomplete. It doesn't show the work to *find* the answers, only to *verify* them. In a calculus context, 'Solve' usually requires showing the derivation. The sentence 'Set each factor to zero; then find every angle...' is the method, but the subsequent lines only check specific values without showing how they were found or proving they are the *only* ones. This is misleading/incomplete as a solution.
gpt-oss:20b: pass 2026-10-14

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/trig_equations, checked 2026-10-17 with SymPy 1.14.0.