Chapter 12 Application of Trigonometry
Solution of Triangles: Complete Guide
Master triangle geometry, laws of sines/cosines/tangents, circles, and area calculations
📐 Introduction to Solution of Triangles
A triangle has six elements: three sides (a, b, c) and three angles (A, B, C). The process of finding unknown elements from known elements is called solution of a triangle.
Triangle ABC with sides a, b, c opposite to angles A, B, C respectively
In any triangle ABC: A + B + C = 180° (π radians)
This is the fundamental relationship that connects the three angles of any triangle.
Angle of Elevation & Depression
Angle of Elevation: When looking upward from horizontal.
Angle of Depression: When looking downward from horizontal.
Elevation = Looking UP (like elevating something)
Depression = Looking DOWN (like feeling depressed)
🔍 When Can We Solve a Triangle?
A triangle can be solved (constructed) when:
- Case 1: Two sides and the included angle are given (SAS)
- Case 2: One side and two angles are given (ASA or AAS)
- Case 3: Three sides are given (SSS)
- Case 4: Two sides and a non-included angle are given (SSA) – ambiguous case
Right Triangle: If one angle is 90°, use Pythagorean theorem and basic trig ratios (SOH-CAH-TOA).
Oblique Triangle: If no angle is 90°, use law of sines, cosines, or tangents.
📏 Law of Cosines
For any triangle ABC with sides a, b, c opposite to angles A, B, C respectively:
b² = a² + c² – 2ac cos B
c² = a² + b² – 2ab cos C
Alternative Form: cos A = (b² + c² – a²) / 2bc
1. SAS case: When two sides and included angle are known
2. SSS case: When all three sides are known (to find angles)
3. Special Note: Pythagorean theorem is a special case of law of cosines when angle = 90°
📐 Law of Sines
Where R is the circumradius (radius of circumcircle).
1. Use to find acute angles (< 90°) – smallest angle first
2. For angles > 90°, this law gives the acute reference angle
3. Smallest angle corresponds to smallest side
4. Largest angle corresponds to largest side
📊 Law of Tangents & Napier’s Analogies
Law of Tangents:
Napier’s Analogies (Tangent Rules):
tan[(A – C)/2] = (a – c)/(a + c) cot(B/2)
tan[(B – C)/2] = (b – c)/(b + c) cot(A/2)
Law of tangents is useful when two sides and the included angle are known, or when two angles and a side are known.
🔢 Half-Angle Formulas
cos(A/2) = √[s(s – a) / bc]
tan(A/2) = √[(s – b)(s – c) / s(s – a)]
Where s = (a + b + c)/2 (semi-perimeter)
Similar formulas exist for angles B/2 and C/2 by cyclic permutation.
🔺 Classification of Triangles
| Type | Definition | Properties |
|---|---|---|
| Scalene | All sides different | All angles different |
| Isosceles | Two sides equal | Two angles equal |
| Equilateral | All sides equal | All angles = 60° |
| Acute | All angles < 90° | Sum of squares of two sides > square of third side |
| Right | One angle = 90° | a² + b² = c² (Pythagoras) |
| Oblique | No angle = 90° | Use laws of sines/cosines |
Scalene = All different (like “scale” has different notes)
Isosceles = Two equal (think “I” has two equal legs)
Equilateral = All equal (equal lateral sides)
🎯 Important Triangle Results
For Isosceles Triangle:
- If a = b, then A = B
- If sin A = sin B, then triangle is isosceles or right angled
- If cos A = cos B, then triangle is isosceles
For Right Triangle:
- If a² + b² = c², then C = 90°
- Midpoint of hypotenuse is equidistant from all vertices
- Orthocenter is at the right angle vertex
- Circumcenter is at midpoint of hypotenuse
For Equilateral Triangle:
- If a = b = c, then A = B = C = 60°
- If sin A = sin B = sin C, then triangle is equilateral
- If cos A = cos B = cos C, then triangle is equilateral
- Circumradius R = a/√3, Inradius r = a/(2√3)
⭕ Circles Connected with Triangles
Circumcircle
Passes through all three vertices
Center: Circumcenter (intersection of perpendicular bisectors)
Radius R: a/(2 sin A) = b/(2 sin B) = c/(2 sin C)
Incircle
Inside triangle, touches all three sides
Center: Incenter (intersection of angle bisectors)
Radius r: Δ/s, where Δ = area, s = semi-perimeter
Excircle
Touches one side externally and other two extended
Center: Excenter (intersection of one internal and two external angle bisectors)
Radius rₐ: Δ/(s – a)
1. For any triangle: r = 4R sin(A/2) sin(B/2) sin(C/2)
2. rₐ = 4R sin(A/2) cos(B/2) cos(C/2)
3. 1/r = 1/rₐ + 1/r_b + 1/r_c
4. For equilateral triangle: R = 2r
📍 Special Points in Triangles
| Point | Definition | Location |
|---|---|---|
| Circumcenter | Center of circumcircle | Intersection of perpendicular bisectors |
| Incenter | Center of incircle | Intersection of angle bisectors |
| Excenter | Center of excircle | Intersection of one internal & two external angle bisectors |
| Orthocenter | Intersection of altitudes | In right triangle: at right angle vertex |
| Centroid | Intersection of medians | Divides each median in ratio 2:1 |
📐 Area of a Triangle (Δ)
1. When two sides and included angle are given:
2. When one side and two angles are given:
3. When three sides are given (Heron’s Formula):
Δ = √[s(s – a)(s – b)(s – c)]
4. Using circumradius R:
5. Using inradius r:
🎯 Area Calculation Examples
Example 1: a=5, b=6, C=30° ⇒ Δ = ½ × 5 × 6 × sin 30° = 7.5
Example 2: a=7, b=8, c=9 ⇒ s=12, Δ = √[12×5×4×3] = √720 ≈ 26.83
Example 3: a=10, B=60°, C=45° ⇒ A=75°, Δ = (100 × sin 60° × sin 45°) / (2 sin 75°) ≈ 30.6
For equilateral triangle with side a:
Δ = (√3/4) a²
Height = (√3/2) a
📝 Solution of Triangles Quiz (50 MCQs)
Quiz Results
Answer Key
🎓 Study Guidelines for Students
1. Understand the Fundamentals
- Memorize the triangle angle sum: A + B + C = 180°
- Learn the six elements of a triangle (3 sides, 3 angles)
- Understand when a triangle can be solved (SAS, ASA, SSS cases)
2. Master the Three Laws
- Law of Sines: Best for AAS or ASA cases
- Law of Cosines: Best for SAS or SSS cases
- Law of Tangents: Useful for checking solutions
- Practice identifying which law to use for different problems
3. Circle Relationships
- Memorize formulas for circumradius (R) and inradius (r)
- Understand the geometric meaning of circumcenter, incenter, excenters
- Learn the special properties for equilateral triangles
4. Area Calculation Strategies
- Memorize all area formulas (Heron’s, trig formula, etc.)
- Practice choosing the right formula based on given information
- Work through examples with different given elements
5. Problem-Solving Approach
- Step 1: Identify what’s given and what’s needed
- Step 2: Choose the appropriate law/formula
- Step 3: Solve systematically
- Step 4: Check if solution is reasonable (angles sum to 180°, etc.)
Create a “cheat sheet” with all formulas organized by use case. Practice solving triangles with random values until you can quickly identify which method to use. Time yourself to improve speed for exams.
