MathIsimple
Lesson 4-2

Solving Triangles – Sine/Cosine Law & Applications

Use sine and cosine laws to find unknown sides/angles, determine area, analyze ambiguous cases, and solve navigation/surveying problems.

Learning Objectives Back to Unit 4

Learning Objectives

  • Apply sine law to AAS/ASA/SSA configurations with domain checks.
  • Use cosine law for SAS/SSS and detect obtuse/acute via law of cosines.
  • Compute triangle area from side-angle pairs and transform between forms.
  • Resolve SSA ambiguous cases and enumerate possible solutions.
  • Model realistic distance/heading problems and validate results.

Sine/Cosine Laws & Area

Sine Law

asin⁡A=bsin⁡B=csin⁡C=2R\tfrac{a}{\sin A}=\tfrac{b}{\sin B}=\tfrac{c}{\sin C}=2R

Applies to AAS, ASA, and SSA (ambiguous) cases.

Cosine Law

a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A
b2=a2+c2−2accos⁡Bb^2=a^2+c^2-2ac\cos B
c2=a2+b2−2abcos⁡Cc^2=a^2+b^2-2ab\cos C

Use for SAS and SSS; reveals obtuse/acute via cosine sign.

Area

S=12bcsin⁡A=12acsin⁡B=12absin⁡CS=\tfrac{1}{2} b c \sin A=\tfrac{1}{2} a c \sin B=\tfrac{1}{2} a b \sin C

Combine with sine law to express area solely in terms of sides/angles.

General Area

S=12bcsin⁡A=12acsin⁡B=12absin⁡C S = \tfrac{1}{2} b c \sin A = \tfrac{1}{2} a c \sin B = \tfrac{1}{2} a b \sin C

Heron

S=s(s−a)(s−b)(s−c),  s=a+b+c2 S = \sqrt{s(s-a)(s-b)(s-c)},\; s=\tfrac{a+b+c}{2}

Right Triangle

S=12(leg1)(leg2) S = \tfrac{1}{2} (\text{leg}_1)(\text{leg}_2)

SSA Ambiguous Case

Given two sides and a non-included acute angle, there may be 0, 1, or 2 solutions depending on height h=bsin⁡Ah=b\sin A relative to side a.

  • if a<ha<h: no solution
  • if a=ha=h: one right triangle
  • if h<a<bh<a<b: two solutions
  • if a≥ba\ge b: one solution

Use sine law to compute possible angles, then check feasibility of remaining sides/angles.

Worked Examples

Example 1: Sine Law (AAS)

In △ABC, A=45°, B=60°, a=10. Find b, c, and C.

C=180∘−A−B=75∘C=180^\circ-A-B=75^\circ
asin⁡A=bsin⁡B=csin⁡C⇒b=sin⁡60∘sin⁡45∘⋅10,  c=sin⁡75∘sin⁡45∘⋅10\tfrac{a}{\sin A}=\tfrac{b}{\sin B}=\tfrac{c}{\sin C}\Rightarrow b=\tfrac{\sin 60^\circ}{\sin 45^\circ}\cdot 10,\; c=\tfrac{\sin 75^\circ}{\sin 45^\circ}\cdot 10

Example 2: Cosine Law (SSS)

Given a=7, b=8, c=9, find angle A and area.

cos⁡A=b2+c2−a22bc=82+92−722⋅8⋅9\cos A=\tfrac{b^2+c^2-a^2}{2bc}=\tfrac{8^2+9^2-7^2}{2\cdot 8\cdot 9}
S=s(s−a)(s−b)(s−c),  s=7+8+92=12S=\sqrt{s(s-a)(s-b)(s-c)},\; s=\tfrac{7+8+9}{2}=12

Example 3: SSA Ambiguity

Given A=30°, a=5, b=6, find possible triangles.

sin⁡B=bsin⁡Aa=6⋅0.55=0.6\sin B=\tfrac{b\sin A}{a}=\tfrac{6\cdot 0.5}{5}=0.6

Two solutions: B1=arcsin⁡0.6B_1=\arcsin 0.6, B2=180∘−arcsin⁡0.6B_2=180^\circ-\arcsin 0.6 if feasible, then compute C and side c.

Example 4: Navigation

A ship travels 60 nm NE (bearing 45°), then 30 nm SE (bearing 135° from north clockwise). Find displacement and distance from start.

Construct triangle and use cosine law or vector components; right-angle case yields d=305d=30\sqrt{5} nm if legs are perpendicular.

Example 5: Surveying

Two points on opposite banks observe a tower with angles of elevation α and β; baseline distance known. Solve for height using triangle decompositions.

Split into right triangles; use tangent relations or sine/cosine laws as needed.

Extended Practice Sets

Set 1: SAS Triangle

  1. Given a=8, b=12, C=60°. Find side c and angles A, B.
  2. Compute area using 12absin⁡C\tfrac{1}{2}ab\sin C.
  3. Verify angle sum equals 180°.
  4. Check if triangle is acute, right, or obtuse.

Set 2: SSS Triangle

  1. Given a=5, b=7, c=9. Find all three angles.
  2. Use cosine law to find the largest angle first.
  3. Calculate area using Heron's formula.
  4. Verify with side-angle area formula.

Set 3: AAS Configuration

  1. Given A=40°, B=80°, a=6. Find b, c, and C.
  2. Use sine law for unknown sides.
  3. Calculate area and verify triangle inequality.
  4. Express all answers to 3 significant figures.

Set 4: SSA Ambiguous Case

  1. Given A=35°, a=4, b=7. Determine number of solutions.
  2. Calculate height h = b sin A and compare with a.
  3. Find both possible triangles if they exist.
  4. Compute areas of all valid triangles.

Set 5: Navigation Problem

  1. Ship travels 25 km on bearing 040°, then 30 km on bearing 130°.
  2. Find distance and bearing from starting point.
  3. Use cosine law to find displacement magnitude.
  4. Apply sine law to find final bearing.

Set 6: Surveying Application

  1. Two observers 500m apart see a tower at angles 35° and 42°.
  2. Find distances from each observer to the tower.
  3. Calculate tower height using trigonometry.
  4. Verify using alternative triangle approach.

Set 7: Mixed Practice

  1. Given any three elements, classify triangle type (SAS, SSS, AAS, SSA).
  2. Choose appropriate law (sine or cosine) and solve completely.
  3. Calculate area using most efficient method.
  4. Check solution validity and reasonableness.

Set 8: Obtuse Triangle Challenge

  1. Given a=15, b=12, c=20. Prove triangle has obtuse angle.
  2. Find the obtuse angle using cosine law.
  3. Calculate area and verify triangle exists.
  4. Determine if triangle is valid using triangle inequality.

Set 9: Altitude and Area

  1. In triangle with sides 6, 8, 10, find all three altitudes.
  2. Use area to derive altitude formulas.
  3. Verify using direct trigonometric calculation.
  4. Identify which altitude is shortest and explain why.

Set 10: Bearing and Distance

  1. Aircraft flies 200 km on bearing 060°, then 150 km on bearing 150°.
  2. Find final position relative to start.
  3. Calculate fuel needed for direct return flight.
  4. Determine heading for most efficient return route.

Set 11: Law Selection Strategy

  1. Given triangle data, identify best solution approach.
  2. Explain when to use sine law vs cosine law.
  3. Handle ambiguous cases systematically.
  4. Develop checking strategies for solution validity.

Set 12: Real-World Modeling

  1. Model triangular lot with given constraints.
  2. Calculate area for landscaping cost estimation.
  3. Find perimeter for fencing requirements.
  4. Optimize triangle shape for maximum area given perimeter.

Deep Drills & Mixed Review

Drill 1: SAS to Complete Triangle

  • Given a=6, b=9, C=45°. Find side c using cosine law.
  • Calculate area using 12absin⁡C\tfrac{1}{2}ab\sin C.
  • Find angles A and B using sine law.
  • Verify all angles sum to 180°.

Drill 2: AAS Configuration

  • Given A=50°, B=70°, c=15. Find angle C first.
  • Use sine law to find sides a and b.
  • Calculate area and verify triangle inequality.
  • Check that largest side opposes largest angle.

Drill 3: Obtuse Angle Detection

  • Given a=7, b=5, c=11. Identify which angle is obtuse.
  • Use cosine law to find cos C and check sign.
  • Calculate exact angle measure and verify > 90°.
  • Confirm using side relationships c2>a2+b2c^2 > a^2 + b^2.

Drill 4: Altitude Calculations

  • In triangle with sides 8, 15, 17, find altitude to each side.
  • Use area formula: ha=2Sah_a = \tfrac{2S}{a}.
  • Calculate area first using Heron's formula.
  • Verify: shortest altitude corresponds to longest side.

Drill 5: SSA Double Solution

  • Given A=45°, a=8, b=10. Find both possible triangles.
  • Calculate sin B and find two angle solutions.
  • Complete both triangles and find their areas.
  • Determine which solution is more practical in context.

Drill 6: Navigation Accuracy

  • Boat travels 12 km NE then 8 km SE. Find displacement.
  • Convert bearings to interior triangle angles.
  • Apply cosine law for distance from start.
  • Use sine law to find final bearing direction.

Drill 7: Area Method Comparison

  • Triangle with a=13, b=14, c=15. Calculate area three ways.
  • Use Heron's formula with semiperimeter.
  • Use 12absin⁡C\tfrac{1}{2}ab\sin C after finding angle C.
  • Compare results and explain any discrepancies.

Drill 8: Isosceles Special Cases

  • Isosceles triangle with equal sides a=b=10, angle C=30°.
  • Find base c and equal angles A=B.
  • Calculate area and altitude to the base.
  • Verify using special right triangle properties.

Drill 9: Surveying Precision

  • Baseline 100m, angles to target 65° and 48°.
  • Find distances to target from each baseline end.
  • Calculate target's perpendicular distance to baseline.
  • Estimate measurement error propagation.

Drill 10: Law Selection Mastery

  • Given mixed triangle data sets, choose optimal solution method.
  • Identify when sine law leads to ambiguity.
  • Know when cosine law is more direct than sine law.
  • Develop systematic checking procedures for all solutions.
Back to Lesson 4-1Next Lesson 4-3