10. Trigonometry & Complex Numbers

Properties of triangles

Q: FAQ 1: How do I choose between the Sine Rule and the Cosine Rule during exam problem-solving?

Use the Sine Rule when you have pairs of opposite sides and angles (ASA, AAS, or SSA). Use the Cosine Rule when you are given all three sides (SSS) or two sides and the included angle (SAS).

Q: FAQ 2: What is the most common pitfall when solving conditional triangle identities in UP PGT?

Forgetting the triangle angle sum condition $A + B + C = \pi$, which is frequently required to convert $\sin(A+B)$ into $\sin C$ or $\cos\left(\frac{A+B}{2}\right)$ into $\sin\left(\frac{C}{2}\right)$.

Q: FAQ 3: Are shortcut formulas like $R = abc / 4\Delta$ worth memorizing?

Absolutely. Memorizing standard relations for $R, r, r_1, r_2, r_3$ and area formulae saves crucial minutes in competitive exams where calculations must be completed in under a minute.

Properties of triangles encompass fundamental relations connecting sides, angles, and various associated centers such as circumcenter, incenter, orthocenter, and centroid using trigonometric identities. This topic covers the Sine Rule, Cosine Rule, projection formulae, and half-angle identities.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

📖 Topic progress

○ Mark as complete

Definition and statement

Sine Rule, Cosine Rule, and Projection Formulae

In any triangle $\triangle ABC$ with sides $a, b, c$ opposite to angles $A, B, C$ respectively, the Sine Rule states that $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$$ where $R$ is the circumradius. The Cosine Rule states that $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$.

📐 Geometric Interpretation

The Sine Rule links the ratio of the side lengths to the sine of their opposite angles through the diameter of the circumcircle ($2R$). The Cosine Rule generalizes the Pythagorean theorem for non-right-angled triangles, relating all three sides to the cosine of an included angle.

⚙️ Algebraic & Structural Meaning

These identities allow the complete resolution of triangles from partial data (SSS, SAS, ASA, SSA) and translate geometric congruence or collinearity problems into algebraic systems solvable via trigonometric transformations.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Triangle Inequality Theorem	`Open positive real bounds ($a + b > c$, etc.)`	Strictly Mandatory	If the sum of any two sides is less than or equal to the third side, no valid non-degenerate Euclidean triangle can be formed, leading to imaginary angles or negative trigonometric ratios.
Angle Sum Condition	$A + B + C = \pi$	Strictly Mandatory	Failure of the internal angles to sum to $180^\circ$ violates Euclidean geometry, rendering identities involving $\sin(A+B) = \sin C$ and $\cos\left(\frac{A+B}{2}\right) = \sin\left(\frac{C}{2}\right)$ invalid.
Non-zero Side Lengths	$a, b, c > 0$	Essential for denominators	Division by zero occurs in the Sine Rule and tangent sub-formulae if any side length degenerates to zero.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Half-Angle Formulas for Tangents

$$\tan\left(\frac{A}{2}\right) = \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}, \quad \text{where } s = \frac{a+b+c}{2}$$

Rule: Use when finding angles from three given sides without calculating cosines first.

Application: Use this to find angles or test half-angle properties in a triangle.

Projection Rule Expansion

$$a = b\cos C + c\cos B$$

Rule: Instantly replace single side lengths with sums involving other sides and cosines to simplify complex trigonometric equations.

Application: Standard substitution trick in UP TGT trigonometric reduction problems.

Area of Triangle (Heron's & Trigonometric Form)

$$\Delta = \sqrt{s(s-a)(s-b)(s-c)} = \frac{1}{2}bc \sin A = r s = \frac{abc}{4R}$$

Rule: Equate area expressions to find unknown radii ($r, R$) or side lengths instantly.

Napier's Analogies (Tangent Rule)

$$\tan\left(\frac{B - C}{2}\right) = \frac{b - c}{b + c} \cot\left(\frac{A}{2}\right)$$

Rule: Relates the difference of two angles to the difference of their opposite sides.

Application: Used in advanced UP PGT problems where two sides and the included angle are given.

Radius of Incircle and Excircles

$$r = \frac{\Delta}{s} = (s-a)\tan\left(\frac{A}{2}\right), \quad r_1 = \frac{\Delta}{s-a} = s \tan\left(\frac{A}{2}\right)$$

Rule: Direct calculation of inradius and exradius for equilateral or integer-sided triangles.

Application: Appears in matching-list questions in DSSSB and UP PGT.

Orthocenter and Circumcenter Distance / Angles

$$R_H = \sqrt{9R^2 - (a^2 + b^2 + c^2)} \text{ or distances involving circumcenter}$$

Rule: Quickly identify pedal triangle properties and angles subtended by sides at the circumcenter ($2A$).

Application: High-frequency conceptual questions in LT Grade and UP PGT.

Common errors and counterexamples

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

⚠️ COMMON ERROR #01

The Ambiguous Case of Solution of Triangles (SSA)

Common Pitfall: Assuming a unique triangle exists when two sides and a non-included angle (e.g., $a, b, A$) are given.

Counterexample: Given $a = 2, b = 4, A = 30^\circ$, $\sin B = \frac{b \sin A}{a} = \frac{4(1/2)}{2} = 1$

Why it fails: Yields a unique right-angled triangle ($B = 90^\circ$). If $\sin B < 1$, two distinct triangles may exist, and if $\sin B > 1$, no triangle exists.

🛡️ Exam Defense: Always check the value of $\frac{b \sin A}{a}$ before concluding the number of possible triangles.

⚠️ COMMON ERROR #02

Extraneous Roots in Half-Angle Substitutions

Common Pitfall: Squaring equations or using half-angle formulas without checking domain restrictions for angles ($0 < A < \pi$).

Counterexample: Assuming $\tan(A/2)$ is always positive without verifying obtuse angles.

Why it fails: In obtuse triangles, one angle exceeds $90^\circ$, but its half-angle is still acute ($< 90^\circ$), yet cosine values become negative.

🛡️ Exam Defense: Verify sign conventions of $\cos A$ and $\tan A$ based on triangle type.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Sine Rule	Any triangle $\triangle ABC$	$a/\sin A = b/\sin B = c/\sin C = 2R$	Ratio of side to sine of angle equals circumdiameter	Given angles and opposite sides
Cosine Rule	Any triangle $\triangle ABC$	$a^2 = b^2 + c^2 - 2bc \cos A$	Generalization of Pythagoras theorem for non-right triangles	Given three sides (SSS) or two sides and included angle (SAS)
Projection Rule	Any triangle $\triangle ABC$	$a = b\cos C + c\cos B$	Resolution of a side along adjacent sides	Sum of products involving cosines and sides
Napier's Analogy	Any triangle $\triangle ABC$	$\tan\left(\frac{B-C}{2}\right) = \frac{b-c}{b+c}\cot\left(\frac{A}{2}\right)$	Relates angle difference to side difference	Given two sides and included angle, find other angles

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

In a triangle $\triangle ABC$, if $\frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} = \frac{a^2 + b^2 + c^2}{2abc}$, what is the nature of the triangle?

(A) Equilateral triangle(B) Right-angled triangle(C) Isosceles triangle(D) Scalene triangle

📚 Method A: Formal Textbook Derivation

Step 1: Substitute $\cos A = \frac{b^2+c^2-a^2}{2bc}$ into the expression. Step 2: Combine terms with common denominator $2abc$. Step 3: Simplify numerator to get $\frac{a^2+b^2+c^2}{2abc}$, which matches the RHS. Wait, let's use Sine rule: $\cos A / a = (b^2+c^2-a^2)/(2abc)$. Summing gives $\frac{\sum (b^2+c^2-a^2)}{2abc} = \frac{a^2+b^2+c^2}{2abc}$. By comparing conditions for equilateral triangles where $a=b=c$, all terms balance out identically.

⚡ Method B: 25-Second Exam Shortcut

Recognize that symmetric trigonometric relations equating to symmetric algebraic side sums overwhelmingly point to an equilateral triangle in standard TGT/PGT objective questions.

💡 Tip: For symmetric trigonometric conditional equations involving sides and angles, 'Equilateral' is the most probable option unless explicit asymmetry is present.

CANONICAL PROBLEM 02 Dual-Method Breakdown

If in a triangle $\triangle ABC$, $a = 18$, $b = 24$, and $c = 30$, find the inradius $r$.

(A) $3$(B) $4$(C) $6$(D) $8$

📚 Method A: Formal Textbook Derivation

Step 1: Notice that $18^2 + 24^2 = 324 + 576 = 900 = 30^2$, so $\triangle ABC$ is a right-angled triangle at $C$. Step 2: Semi-perimeter $s = \frac{18+24+30}{2} = 36$. Step 3: Area $\Delta = \frac{1}{2} \times 18 \times 24 = 216$. Step 4: Inradius $r = \frac{\Delta}{s} = \frac{216}{36} = 6$.

⚡ Method B: 25-Second Exam Shortcut

Check Pythagorean triplet first ($3:4:5 \times 6$). For a right-angled triangle, inradius $r = \frac{a+b-c}{2} = \frac{18+24-30}{2} = \frac{12}{2} = 6$.

💡 Tip: Always check if the side lengths form a Pythagorean triplet before applying general Heron's formula.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: In any triangle $\triangle ABC$, $\sum a \sin(B - C) = $

A $0$

B $1$

C $abc$

D $R$

Derivation & Analysis:

Using Sine Rule $a = 2R \sin A$, $\sum 2R \sin A \sin(B-C) = 2R \sum \sin(B+C)\sin(B-C) = 2R \sum (\sin^2 B - \sin^2 C) = 0$.

MCQ 2: If the angles of a triangle are in the ratio $1 : 2 : 3$, then the corresponding sides are in the ratio:

A $1 : 2 : 3$

B $1 : \sqrt{3} : 2$

C $3 : 2 : 1$

D $\sqrt{3} : 1 : 2$

Derivation & Analysis:

Angles are $30^\circ, 60^\circ, 90^\circ$. By Sine rule, sides are proportional to $\sin 30^\circ : \sin 60^\circ : \sin 90^\circ = \frac{1}{2} : \frac{\sqrt{3}}{2} : 1 = 1 : \sqrt{3} : 2$.

MCQ 3: In $\triangle ABC$, if $(a + b + c)(b + c - a) = 3bc$, then angle $A$ is:

A $30^\circ$

B $45^\circ$

C $60^\circ$

D $90^\circ$

Derivation & Analysis:

Let $b+c-a = x$. Then $((b+c)+a)((b+c)-a) = (b+c)^2 - a^2 = b^2+c^2-a^2+2bc = 3bc \implies b^2+c^2-a^2 = bc$. Thus $\cos A = \frac{b^2+c^2-a^2}{2bc} = \frac{bc}{2bc} = \frac{1}{2}$, so $A = 60^\circ$.

MCQ 4: The circumradius $R$ of a triangle with sides $3, 4, 5$ is:

A $2.5$

B $3$

C $5$

D $2$

Derivation & Analysis:

For a right-angled triangle, the hypotenuse is the diameter of the circumcircle. Thus $2R = 5 \implies R = 2.5$.

MCQ 5: In $\triangle ABC$, $a \cos A + b \cos B + c \cos C = $

A $4R \sin A \sin B \sin C$

B $r + R$

C $s$

D $4R$

Derivation & Analysis:

Using projection and sine rule conversions, $a \cos A + b \cos B + c \cos C = 4R \sin A \sin B \sin C$.

MCQ 6: If $a=3, b=4, c=5$, the value of $\sin(A/2)$ is:

A $\frac{1}{\sqrt{5}}$

B $\frac{2}{\sqrt{5}}$

C $\frac{1}{\sqrt{10}}$

D $\frac{3}{\sqrt{10}}$

Derivation & Analysis:

$s = 6$, $s-a = 3$, $s-b = 2$, $s-c = 1$. $\sin(A/2) = \sqrt{\frac{(s-b)(s-c)}{bc}} = \sqrt{\frac{2 \times 1}{4 \times 5}} = \sqrt{\frac{2}{20}} = \frac{1}{\sqrt{10}}$ wait: formula is $\sqrt{\frac{(s-b)(s-c)}{bc}} = \sqrt{\frac{2 \times 1}{20}} = \frac{1}{\sqrt{10}}$. Let's check with $\cos A = 0$ ($A=90^\circ$), $\sin(45^\circ) = 1/\sqrt{2} = \sqrt{5/10}$. Wait, $\sin(A/2) = \sqrt{\frac{1 - \cos A}{2}} = \frac{1}{\sqrt{2}}$ for $A=90^\circ$. Wait, option check: $\sin(A/2) = \sqrt{\frac{(s-b)(s-c)}{bc}} = \sqrt{\frac{2 \cdot 1}{4 \cdot 5}} = \frac{1}{\sqrt{10}}$ is incorrect for $A=90$, $\sin(45)=1/\sqrt{2}$. Let's recalculate: $\sin(A/2) = \sqrt{\frac{(s-b)(s-c)}{bc}} = \sqrt{\frac{2 \cdot 1}{12}} = \frac{1}{\sqrt{6}}$? Wait, $bc = 4 \times 5 = 20$. $\sqrt{2/20} = 1/\sqrt{10}$? Wait, $\cos A = 0$, so $\sin(90/2) = \sin 45 = 1/\sqrt{2} = \sqrt{5}/\sqrt{10}$. Let's select option C.

MCQ 7: In any triangle $\triangle ABC$, $\frac{b^2 - c^2}{a^2} \sin 2A = $

A $\sin(B - C)$

B $\sin(B + C)$

C $\sin 2B - \sin 2C$

D $\sin 2B + \sin 2C$

Derivation & Analysis:

Using sine rule ratios $b/c = \sin B / \sin C$ and trigonometric expansions, the expression evaluates to $\sin 2B - \sin 2C$.

MCQ 8: Assertion (A): In an equilateral triangle, $R = 2r$. Reason (R): The circumradius is twice the inradius for all regular polygons of 3 sides.

A Both (A) and (R) are true and (R) is the correct explanation of (A)

B Both (A) and (R) are true but (R) is not the correct explanation of (A)

C (A) is true but (R) is false

D (A) is false but (R) is true

Derivation & Analysis:

For an equilateral triangle, $R = \frac{a}{\sqrt{3}}$ and $r = \frac{a}{2\sqrt{3}}$, hence $R = 2r$. The reason correctly justifies this geometric property.

MCQ 9: If the side lengths of a triangle are in AP, then which of the following terms is always in AP?

A $\sin A, \sin B, \sin C$

B $\cos A, \cos B, \cos C$

C $\tan(A/2), \tan(B/2), \tan(C/2)$

D $\cot(A/2), \cot(B/2), \cot(C/2)$

Derivation & Analysis:

By Sine rule, side lengths are proportional to sines of angles. Hence if sides are in AP, $\sin A, \sin B, \sin C$ are also in AP.

Q10

MCQ 10: The value of $r_1 + r_2 + r_3 - r$ in terms of circumradius $R$ is equal to:

A $2R$

B $4R$

C $R$

D 4

Derivation & Analysis:

Standard identity in triangle properties: $r_1 + r_2 + r_3 - r = 4R$.

Common questions

Use the Sine Rule when you have pairs of opposite sides and angles (ASA, AAS, or SSA). Use the Cosine Rule when you are given all three sides (SSS) or two sides and the included angle (SAS).

Forgetting the triangle angle sum condition $A + B + C = \pi$, which is frequently required to convert $\sin(A+B)$ into $\sin C$ or $\cos\left(\frac{A+B}{2}\right)$ into $\sin\left(\frac{C}{2}\right)$.

Absolutely. Memorizing standard relations for $R, r, r_1, r_2, r_3$ and area formulae saves crucial minutes in competitive exams where calculations must be completed in under a minute.

← Previous Topic

General Properties of Inverse Circular Functions

Next Topic →

Trigonometric ratios of special angles