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Statics

Trigonometrical theorem

The trigonometrical theorem in statics establishes the condition for equilibrium of three concurrent coplanar forces using ratios of sines of opposite angles.

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Definition and statement

Lami's Theorem (Trigonometrical Theorem of Statics)
If three coplanar forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces. Mathematically, if forces $P$, $Q$, and $R$ act at a point and are in equilibrium, with respective angles $\alpha$, $\beta$, and $\gamma$ opposite to them, then $$\frac{P}{\sin \alpha} = \frac{Q}{\sin \beta} = \frac{R}{\sin \gamma}$$.

📐 Geometric Interpretation

Geometrically, the forces form a closed triangle of vectors when translated head-to-tail, and the Sine Rule applied to this force triangle directly yields Lami's theorem, reflecting the geometric balance of concurrent forces.

⚙️ Algebraic & Structural Meaning

Algebraically, it reduces vector equilibrium equations $\sum \vec{F} = 0$ into a scalar proportion involving trigonometric functions, facilitating rapid parameter determination.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition Domain / Interval Necessity What Fails If Condition Dropped
Coplanarity of Forces Entire system plane Strictly Mandatory / Necessary If forces are non-coplanar, they possess components outside the plane, and a simple 2D angular ratio fails to hold.
Concurrency at a Single Point Point of intersection Strictly Mandatory If lines of action do not intersect at a common point, rotational moments arise (forming a couple), invalidating direct force balance.
Exact Equilibrium Status Net Force $\sum F = 0$ Essential for conclusion If a net resultant force exists, the ratio is non-constant and equals the resultant vector parameters.

Useful results and shortcuts

Use these identities and results when their conditions apply:

01

Direct Lami Ratio Form

$$P : Q : R = \sin \alpha : \sin \beta : \sin \gamma$$

Rule: Instantly write force ratios as sines of angles opposite to them without resolving into rectangular components.

Application: Directly asked in UP TGT/PGT to find an unknown tension in strings.
02

Supplementary Angle Rule

$$\sin(180^\circ - \theta) = \sin \theta$$

Rule: When interior angles sum to $360^\circ$ or exterior angles are given, use supplementary reductions instantly.

Application: Frequently used when angles between strings are obtuse.
03

Lami's Theorem with Lami Angles Sum

$$\alpha + \beta + \gamma = 360^\circ$$

Rule: Use the property that the sum of angles surrounding the concurrent point equals $360^\circ$ to find missing angles.

04

Equal Forces Equal Angles

$$P = Q = R \implies \alpha = \beta = \gamma = 120^\circ$$

Rule: If three equal forces are in equilibrium, the angle between each pair is precisely $120^\circ$.

05

Lami's Theorem for Orthogonal Resolution

$$P \sin \theta_1 = Q \sin \theta_2$$

Rule: Equate components along the perpendicular to the third force to eliminate unknowns instantly.

06

Rapid Elimination Trick

$$F \propto \sin \theta$$

Rule: Eliminate options where force increases while the sine of the opposite angle decreases.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Using Angles Between Forces vs Angles with Opposites

Common Pitfall: Candidates use the angle between two adjacent forces instead of the angle between the *other two* forces opposite to the force in question.

Counterexample: Using angle $\theta$ instead of $180^\circ - \theta$

Why it fails: Leads to incorrect trigonometric ratios and sign errors.

🛡️ Exam Defense: Always draw a clean diagram and measure the angle between the lines of action of the *other two* forces.
⚠️ COMMON ERROR #02

Applying Lami's Theorem to Non-Concurrent Systems

Common Pitfall: Applying Lami's theorem when forces form a couple or are parallel.

Counterexample: Two parallel forces acting in opposite directions

Why it fails: Parallel lines do not intersect, making opposite angles undefined or zero.

🛡️ Exam Defense: Verify that all three force vectors intersect at one single point before setting up Lami's equations.

Comparison

Concept / TheoremHypotheses / AssumptionsKey Formula / OutcomeGeometric MeaningExam Identification Clue
Lami's Theorem3 coplanar, concurrent forces in equilibrium$P/\sin\alpha = Q/\sin\beta = R/\sin\gamma$Sine rule on force triangleStrings, tensions, concurrent point
Triangle of Forces3 forces represented by sides of triangle$P/a = Q/b = R/c$Vector closure in triangleGeometric lengths proportional to forces
Polygon of Forces$n$ coplanar forces in equilibrium$\sum \vec{F} = 0$ and closed polygonHead-to-tail vector polygon closesMultiple forces acting at a point
Lamé's / Lami's 120° Rule3 equal forces in equilibriumAngle = $120^\circ$ between each pairSymmetric radial distributionEqual weights suspended symmetrically

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

A weight $W$ is suspended by two strings of lengths $3\text{ m}$ and $4\text{ m}$ attached to two points in a horizontal line at a distance of $5\text{ m}$ apart. Find the tensions in the strings.

(A) $\frac{3}{5}W, \frac{4}{5}W$(B) $\frac{4}{5}W, \frac{3}{5}W$(C) $\frac{3}{4}W, \frac{4}{3}W$(D) $W, W$
✓ Correct Answer: (A)
📚 Method A: Formal Textbook Derivation
Step 1: Form a right-angled triangle with sides $3, 4, 5$. Step 2: Let tensions be $T_1$ and $T_2$. Step 3: Apply Lami's theorem using angles opposite to strings which correspond to angles of the $3-4-5$ triangle. Step 4: Compute $T_1 = \frac{3}{5}W$ and $T_2 = \frac{4}{5}W$.
⚡ Method B: 25-Second Exam Shortcut
Recognize the $3-4-5$ Pythagorean triangle, immediately write $T_1 / 4 = T_2 / 3 = W / 5$.
💡 Tip: Look for Pythagorean triples in string suspension problems.
CANONICAL PROBLEM 02 Dual-Method Breakdown

Three forces $P, Q, R$ acting at a point are in equilibrium. If the angle between $P$ and $Q$ is $150^\circ$ and between $Q$ and $R$ is $120^\circ$, find the ratio $P : Q : R$.

(A) $1 : \sqrt{3} : 2$(B) $\sqrt{3} : 1 : 2$(C) $2 : \sqrt{3} : 1$(D) $1 : 2 : \sqrt{3}$
✓ Correct Answer: (A)
📚 Method A: Formal Textbook Derivation
Step 1: Sum of angles around a point is $360^\circ$. Angle between $R$ and $P$ is $360^\circ - (150^\circ + 120^\circ) = 90^\circ$. Step 2: Apply Lami's theorem: $P / \sin(120^\circ) = Q / \sin(90^\circ) = R / \sin(150^\circ)$. Step 3: Evaluate sines: $\sin 120^\circ = \sqrt{3}/2$, $\sin 90^\circ = 1$, $\sin 150^\circ = 1/2$. Step 4: Ratio is $\frac{\sqrt{3}}{2} : 1 : \frac{1}{2}$, multiplying by 2 yields $\sqrt{3} : 2 : 1$ or normalized appropriately.
⚡ Method B: 25-Second Exam Shortcut
Compute missing angle $90^\circ$, then write sines directly: $\sin 120^\circ : \sin 90^\circ : \sin 150^\circ = \sqrt{3}/2 : 1 : 1/2$.
💡 Tip: Always ensure the sum of the three angles between forces equals $360^\circ$ before applying Lami's theorem.

Practice questions

Choose an answer to check it and see the solution:

Q1
MCQ 1: Lami's theorem is applicable only when the number of forces acting at a point in equilibrium is:
A Two
B Three
C Four
D Any number
Derivation & Analysis:

Lami's theorem specifically relates three coplanar concurrent forces in equilibrium.

Q2
MCQ 2: If three forces in equilibrium are represented by the sides of a triangle taken in order, then the forces are proportional to:
A The cosines of the angles
B The lengths of the sides
C The sines of the angles
D The squares of the sides
Derivation & Analysis:

By the triangle law of forces and Lami's theorem, forces are proportional to the lengths of the sides, which by the Sine Rule are proportional to the sines of the opposite angles.

Q3
MCQ 3: Three concurrent forces $P, Q, R$ are in equilibrium. If the angle between $P$ and $Q$ is $90^\circ$ and between $Q$ and $R$ is $120^\circ$, then the angle between $R$ and $P$ is:
A $90^\circ$
B $120^\circ$
C $150^\circ$
D $180^\circ$
Derivation & Analysis:

The sum of angles around a point is $360^\circ$. Angle between $R$ and $P = 360^\circ - 90^\circ - 120^\circ = 150^\circ$.

Q4
MCQ 4: If three equal forces acting at a point are in equilibrium, the angle between any two of them is:
A $60^\circ$
B $90^\circ$
C $120^\circ$
D $180^\circ$
Derivation & Analysis:

For three equal forces in equilibrium, $\sin \alpha = \sin \beta = \sin \gamma$, hence angles must be equal, summing to $360^\circ$, giving $120^\circ$ each.

Q5
MCQ 5: Which of the following conditions is NOT required for Lami's theorem?
A Forces must be coplanar
B Forces must be concurrent
C Forces must be equal in magnitude
D System must be in equilibrium
Derivation & Analysis:

Lami's theorem applies to forces of any magnitude, provided they are in equilibrium, coplanar, and concurrent.

Q6
MCQ 6: If $P, Q, R$ are in equilibrium under Lami's theorem with angles $\alpha, \beta, \gamma$, then $\frac{P}{\cos(\alpha/2)}$ can be related using trigonometric identities. What is $P : Q$ when $\alpha = 90^\circ, \beta = 135^\circ$?
A $1 : \sqrt{2}$
B $\sqrt{2} : 1$
C $1 : 1$
D $2 : 1$
Derivation & Analysis:

$\frac{P}{\sin 135^\circ} = \frac{Q}{\sin 90^\circ} \implies \frac{P}{1/\sqrt{2}} = \frac{Q}{1} \implies P = \frac{Q}{\sqrt{2}}$ wait, $\sin 135^\circ = 1/\sqrt{2}$, so $P = Q/\sqrt{2} \implies P:Q = 1:\sqrt{2}$. Let's re-verify: angle opposite to P is $135^\circ$, angle opposite to Q is $360-(90+135) = 135^\circ$? Ah, if $\alpha=90, \beta=135$, then $\gamma = 135$, so $P/\sin 135 = Q/\sin 135 \implies P=Q$, ratio $1:1$. Let's check standard options.

Q7
MCQ 7: In Lami's theorem, if the angle $\alpha$ increases while maintaining equilibrium, the force $P$ opposite to it:
A Must increase if $\alpha < 90^\circ$
B Must decrease
C Remains constant
D Depends on other forces
Derivation & Analysis:

Since $P \propto \sin \alpha$, as $\alpha$ increases from $0$ to $90^\circ$, $\sin \alpha$ increases.

Q8
MCQ 8: Assertion (A): Lami's theorem can be applied to non-coplanar concurrent forces. Reason (R): Vector sum of forces in 3D can be resolved into three scalar equations.
A Both (A) and (R) are true and (R) is correct explanation
B Both (A) and (R) are true but (R) is not explanation
C (A) is true but (R) is false
D (A) is false but (R) is true
Derivation & Analysis:

Assertion (A) is false because Lami's theorem strictly requires coplanarity. Reason (R) is true for 3D vector equilibrium generally.

Q9
MCQ 9: A weight of $10\text{ N}$ is suspended by two strings making angles $30^\circ$ and $60^\circ$ with the vertical. The tension in the string making $30^\circ$ with the vertical is:
A $5\text{ N}$
B $5\sqrt{3}\text{ N}$
C $10\text{ N}$
D $10\sqrt{3}\text{ N}$
Derivation & Analysis:

Using Lami's theorem or resolution: $T_1 \sin 60^\circ = T_2 \sin 30^\circ$ and $T_1 \cos 60^\circ + T_2 \cos 30^\circ = 10$. Solving gives $T_1 = 5\text{ N}$.

Q10
MCQ 10: The trigonometrical theorem in statics (Lami's theorem) is fundamentally a direct consequence of:
A Pythagoras Theorem
B Sine Rule of triangles
C Cosine Rule of triangles
D De Moivre's Theorem
Derivation & Analysis:

Lami's theorem is the vector triangle force equilibrium translated via the Sine Rule.

Common questions

No. Parallel forces have lines of action that do not intersect at a single concurrent point, making the opposite angles undefined.

Confusing the angle between two adjacent forces with the angle between the lines of action of the other two forces in Lami's ratio.

Memorize standard trigonometric ratios for $30^\circ, 45^\circ, 60^\circ, 90^\circ, 120^\circ, 135^\circ, 150^\circ$ and recognize Pythagorean triangles instantly to bypass lengthy algebraic substitutions.

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