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8.5 Law of Sines, Law of Cosines, and Trigonometric Equations

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1 Chapter 1: ACCUPLACER Suite Overview & Testing Mechanics

1.1 The ACCUPLACER Testing System & Computer-Adaptive Mechanics 1.2 Placement Policies, Score Scales & Institutional Cut Scores 1.3 Test-Taking Strategies, Tool Mechanics & Pacing

2 Chapter 2: Next-Generation Reading

2.1 The Reading Test Blueprint: Set-Based and Discrete Items 2.2 Information and Ideas: Close Reading & Central Themes 2.3 Summarizing and Understanding Relationships 2.4 Rhetoric, Text Structure & Author's Purpose 2.5 Analyzing Point of View and Authorial Purpose 2.6 Analyzing Arguments: Claims, Counterclaims, and Evidence 2.7 Synthesis & Paired Text Comparative Analysis 2.8 Vocabulary in Context & Figurative Language 2.9 Passage Genre, Content Area, Mode, and Text Complexity 2.10 Answer-Choice Traps and Evidence Discipline

3 Chapter 3: Next-Generation Writing: Expression of Ideas

3.1 The Writing Test Blueprint: Passage Sets and Content Distribution 3.2 Development: Proposition and Supporting Evidence 3.3 Focus: Adding, Retaining, and Deleting for Relevance 3.4 Organization: Logical Sequence and Paragraph Order 3.5 Introductions, Conclusions, and Transitional Cohesion 3.6 Effective Language Use: Precision and Word Choice 3.7 Concision: Eliminating Wordiness and Redundancy 3.8 Style, Tone, and Voice Consistency 3.9 Syntax: Sentence Combining for Rhetorical Purpose

4 Chapter 4: Next-Generation Writing: Standard English Conventions

4.1 Sentence Boundaries: Fragments, Run-ons, and Comma Splices 4.2 Subordination and Coordination 4.3 Parallel Structure 4.4 Modifier Placement: Misplaced and Dangling Modifiers 4.5 Inappropriate Shifts in Verb Tense 4.6 Inappropriate Shifts in Verb Voice and Mood 4.7 Inappropriate Shifts in Pronoun Person and Number 4.8 Subject-Verb Agreement, Pronoun Case, and Verb Forms 4.9 Possessive Determiners, Contractions, and Confusable Adverbs 4.10 Pronoun Clarity and Pronoun-Antecedent Agreement 4.11 Frequently Confused Words, Noun Agreement, and Logical Comparison 4.12 Within-Sentence Punctuation: Semicolons, Colons, Dashes, and Apostrophes 4.13 Items in a Series and Nonrestrictive Elements 4.14 End-of-Sentence Punctuation, Hyphenation, and Unnecessary Punctuation

5 Chapter 5: Next-Generation Arithmetic

5.1 Whole Number Operations, Estimation & Multi-Step Applications 5.2 Fractions, Mixed Numbers & Operations 5.3 Decimal Operations & Place-Value Conversions 5.4 Percent Calculations, Increase, Decrease, and Applications 5.5 Number Comparisons and Equivalents Across Formats

6 Chapter 6: Next-Generation Quantitative Reasoning, Algebra, and Statistics (QAS)

6.1 Rational Numbers, Absolute Value, and Order of Operations 6.2 Exponents, Radicals, and Scientific Notation 6.3 Ratio, Proportional Relationships, and Unit Conversions 6.4 Creating and Evaluating Algebraic Expressions 6.5 Linear Equations, Inequalities, and Systems of Two Equations 6.6 Linear Applications, Functions, and Graphs 6.7 Probability and Sets 6.8 Descriptive Statistics: Center, Spread, and Graphical Displays 6.9 Geometry for Pre-Algebra and Algebra 1

7 Chapter 7: Next-Generation Advanced Algebra, Quadratics & Polynomials

7.1 Advanced Linear Equations, Systems, and Coordinate Geometry 7.2 Factoring Methods for Quadratics, Cubics, and Polynomials 7.3 Quadratic Equations, Vertex Form & Parabolic Functions 7.4 Polynomial Equations, Roots, and Graphs 7.5 Radical and Rational Equations and Functions 7.6 Geometry for Algebra 1 and Algebra 2

8 Chapter 8: Functions, Exponentials, Logarithms & Trigonometry

8.1 Function Notation, Transformations, Compositions & Inverses 8.2 Domain, Range, and Interpreting Functions in Context 8.3 Exponential & Logarithmic Functions, Laws & Modeling 8.4 Right-Triangle Trigonometry, the Unit Circle, and Radian Measure 8.5 Law of Sines, Law of Cosines, and Trigonometric Equations 8.6 Choosing Your Math Test: Arithmetic, QAS, or AAF

9 Chapter 9: WritePlacer & WritePlacer ESL Essays

9.1 WritePlacer Prompt Format, Planning, and the Writing Process 9.2 The WritePlacer Scoring Rubric & 6 Core Dimensions 9.3 Reading the WritePlacer Score Scale: What Each Band Demonstrates 9.4 Essay Architecture: Thesis, Evidence Integration & Structure 9.5 WritePlacer ESL Scoring Standards & Synthesis

10 Chapter 10: ACCUPLACER ESL Placement Suite

10.1 The ESL Placement Suite: Structure, Scoring, and Score Interpretation 10.2 ESL Reading Skills & Passage Comprehension 10.3 ESL Sentence Meaning & Context Clues 10.4 ESL Language Use & Grammatical Mechanics 10.5 ESL Listening Comprehension & Spoken Contexts

8.5 Law of Sines, Law of Cosines, and Trigonometric Equations

Key Takeaways Explain This Topic AI

Trigonometry carries 1 to 3 of the 20 AAF questions and is the most advanced content area on the test.

The law of sines applies when you have an angle paired with its opposite side, in the ASA, AAS, or SSA cases.

The law of cosines applies in the SAS and SSS cases where no angle-side pair is complete.

The law of cosines reduces to the Pythagorean theorem when the included angle is 90 degrees.

Trigonometric equations usually have multiple solutions in a given interval because sine and cosine repeat their values.

Last updated: August 2026

The Most Advanced AAF Area

Trigonometry: Solving trigonometric equations, using right triangle trigonometry including special triangles, evaluating equivalent trigonometric functions, graphing trigonometric relationships, determining arc length and radian measures, and using the law of sines and the law of cosines .

At 1 to 3 of the 20 questions , trigonometry is a small share — but because these items sit at the top of the difficulty range, they matter disproportionately for students seeking placement into precalculus or calculus.

Beyond Right Triangles

Right-triangle ratios (SOH-CAH-TOA) require a 90° angle. For oblique triangles, two laws take over.

Standard notation: angles A A A , B B B , C C C with opposite sides a a a , b b b , c c c . Side a a a is always opposite angle A A A .

The Law of Sines a sin ⁡ A = b sin ⁡ B = c sin ⁡ C \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} sin A a ​ = sin B b ​ = sin C c ​

Use it when you have a complete angle-side pair — an angle and the side opposite it. That covers:

  • AAS — two angles and a non-included side
  • ASA — two angles and the included side (find the third angle first)
  • SSA — two sides and a non-included angle (the ambiguous case)

In triangle ABC, A = 40 ° A = 40° A = 40° , B = 75 ° B = 75° B = 75° , and a = 12 a = 12 a = 12 . Find b b b .

12 sin ⁡ 40 ° = b sin ⁡ 75 ° \dfrac{12}{\sin 40°} = \dfrac{b}{\sin 75°} sin 40° 12 ​ = sin 75° b ​ b = 12 sin ⁡ 75 ° sin ⁡ 40 ° ≈ 12 ( 0.966 ) 0.643 ≈ 18.0 b = \dfrac{12 \sin 75°}{\sin 40°} \approx \dfrac{12(0.966)}{0.643} \approx 18.0 b = sin 40° 12 sin 75° ​ ≈ 0.643 12 ( 0.966 ) ​ ≈ 18.0

The Ambiguous Case

SSA can yield two valid triangles, one, or none, because sin ⁡ θ = sin ⁡ ( 180 ° − θ ) \sin\theta = \sin(180° - \theta) sin θ = sin ( 180° − θ ) . If you solve for an angle and get 35°, then 145° may also satisfy the equation — check whether the three angles still sum to 180°. If they exceed it, discard the obtuse option.

The Law of Cosines c 2 = a 2 + b 2 − 2 a b cos ⁡ C c^2 = a^2 + b^2 - 2ab\cos C c 2 = a 2 + b 2 − 2 ab cos C

Use it when no complete angle-side pair exists:

  • SAS — two sides and the included angle
  • SSS — three sides (solving for an angle)

Sides a = 8 a = 8 a = 8 , b = 11 b = 11 b = 11 , included angle C = 40 ° C = 40° C = 40° . Find c c c .

c 2 = 64 + 121 − 2 ( 8 ) ( 11 ) cos ⁡ 40 ° c^2 = 64 + 121 - 2(8)(11)\cos 40° c 2 = 64 + 121 − 2 ( 8 ) ( 11 ) cos 40° c 2 = 185 − 176 ( 0.766 ) ≈ 185 − 134.8 = 50.2 c^2 = 185 - 176(0.766) \approx 185 - 134.8 = 50.2 c 2 = 185 − 176 ( 0.766 ) ≈ 185 − 134.8 = 50.2 c ≈ 7.1 c \approx 7.1 c ≈ 7.1

For SSS, rearrange: cos ⁡ C = a 2 + b 2 − c 2 2 a b \cos C = \frac{a^2 + b^2 - c^2}{2ab} cos C = 2 ab a 2 + b 2 − c 2 ​

The Pythagorean Connection

When C = 90 ° C = 90° C = 90° , cos ⁡ 90 ° = 0 \cos 90° = 0 cos 90° = 0 and the law of cosines becomes c 2 = a 2 + b 2 − 0 = a 2 + b 2 c^2 = a^2 + b^2 - 0 = a^2 + b^2 c 2 = a 2 + b 2 − 0 = a 2 + b 2

The Pythagorean theorem is a special case of the law of cosines. This is a useful sanity check: a right-triangle result computed with the law of cosines must agree with Pythagoras.

Choosing Between the Laws

  • Given Use
  • AAS, ASA Law of sines
  • SSA Law of sines (check ambiguity)
  • SAS Law of cosines
  • SSS Law of cosines

The deciding question: do I have an angle together with the side opposite it? Yes → sines. No → cosines.

Special Triangles

Exact values worth knowing, since the test rewards recognizing them without a calculator.

45°-45°-90°: sides 1 : 1 : 2 1 : 1 : \sqrt{2} 1 : 1 : 2 ​ 30°-60°-90°: sides 1 : 3 : 2 1 : \sqrt{3} : 2 1 : 3 ​ : 2 (opposite 30°, 60°, 90°)

  • θ \theta θ sin ⁡ θ \sin\theta sin θ cos ⁡ θ \cos\theta cos θ tan ⁡ θ \tan\theta tan θ
  • 0° 0 1 0
  • 30° 1 2 \tfrac{1}{2} 2 1 ​ 3 2 \tfrac{\sqrt{3}}{2} 2 3 ​ ​ 3 3 \tfrac{\sqrt{3}}{3} 3 3 ​ ​
  • 45° 2 2 \tfrac{\sqrt{2}}{2} 2 2 ​ ​ 2 2 \tfrac{\sqrt{2}}{2} 2 2 ​ ​ 1
  • 60° 3 2 \tfrac{\sqrt{3}}{2} 2 3 ​ ​ 1 2 \tfrac{1}{2} 2 1 ​ 3 \sqrt{3} 3 ​
  • 90° 1 0 undefined
  • Solving Trigonometric Equations

Because sine and cosine are periodic, equations typically have multiple solutions in an interval. Missing the second one is the characteristic error.

Procedure

Isolate the trigonometric function.

Find the reference angle.

Identify all quadrants where the function has the required sign.

Keep the solutions inside the stated interval.

Signs by Quadrant

Remembered as ASTC — All, Sine, Tangent, Cosine positive in quadrants I, II, III, IV respectively.

  • Quadrant Positive functions
  • I all
  • II sine (and cosecant)
  • III tangent (and cotangent)
  • IV cosine (and secant)
  • Worked Example

Solve 2 sin ⁡ θ − 1 = 0 2\sin\theta - 1 = 0 2 sin θ − 1 = 0 for 0 ° ≤ θ < 360 ° 0° \le \theta < 360° 0° ≤ θ < 360° .

sin ⁡ θ = 1 2 \sin\theta = \tfrac{1}{2} sin θ = 2 1 ​

Reference angle: 30°. Sine is positive in quadrants I and II :

  • Quadrant I: θ = 30 ° \theta = 30° θ = 30°
  • Quadrant II: θ = 180 ° − 30 ° = 150 ° \theta = 180° - 30° = 150° θ = 180° − 30° = 150°

Solutions: 30° and 150°.

Solve cos ⁡ θ = − 2 2 \cos\theta = -\tfrac{\sqrt{2}}{2} cos θ = − 2 2 ​ ​ for 0 ° ≤ θ < 360 ° 0° \le \theta < 360° 0° ≤ θ < 360° .

Reference angle 45°; cosine is negative in quadrants II and III :

  • Quadrant II: 180 ° − 45 ° = 135 ° 180° - 45° = 135° 180° − 45° = 135°
  • Quadrant III: 180 ° + 45 ° = 225 ° 180° + 45° = 225° 180° + 45° = 225°

Solutions: 135° and 225°.

Reference Angle Formulas

  • Quadrant Solution from reference angle α \alpha α
  • I α \alpha α
  • II 180 ° − α 180° - \alpha 180° − α
  • III 180 ° + α 180° + \alpha 180° + α
  • IV 360 ° − α 360° - \alpha 360° − α
  • Radian Measure and Arc Length 180 ° = π radians 180° = \pi \text{ radians} 180° = π radians

Convert by multiplying by π 180 ° \dfrac{\pi}{180°} 180° π ​ or 180 ° π \dfrac{180°}{\pi} π 180° ​ .

60 ° = 60 ⋅ π 180 = π 3 60° = 60 \cdot \dfrac{\pi}{180} = \dfrac{\pi}{3} 60° = 60 ⋅ 180 π ​ = 3 π ​

Arc length: s = r θ s = r\theta s = r θ , with θ \theta θ in radians .

A circle of radius 9 subtends a central angle of 2 π 3 \tfrac{2\pi}{3} 3 2 π ​ . Arc length? s = 9 ⋅ 2 π 3 = 6 π s = 9 \cdot \tfrac{2\pi}{3} = 6\pi s = 9 ⋅ 3 2 π ​ = 6 π

Using degrees directly in s = r θ s = r\theta s = r θ is the standard error; convert first.