CEUET Mathematics — TrigonometrySlides
Slide deck for CEUET Mathematics — Trigonometry. These slides are built for quick visual review, highlighting the key concepts, formulas, and question patterns from this chapter of the CEUET 2026 syllabus.
Exam context
The Centro Escolar University Entrance Test is conducted by Centro Escolar University and is scheduled for Q3–Q4 2026. The Mathematics subtest is marked as "Core" in the official pattern, and Trigonometry appears in position 7th of 9 in the CEUET Mathematics review rotation. Passing mark: Competitive overall score. Recent CEUET 2026 papers have drawn roughly a meaningful share of questions from this subject.
Trigonometry - Slides
Trigonometry is a fundamental branch of mathematics that studies relationships between angles and sides of triangles. This chapter will explore trigonometric functions, identities, and their real-world applications. You'll learn to solve problems using the unit circle, special angles, and various trigonometric laws. These concepts are essential for UPCAT and other college entrance examinations in the Philippines.
Slides
Introduction to Trigonometry
Trigonometry is the mathematical study of triangles and their properties. It helps us find unknown sides and angles when we know some information about a triangle.
Notes
Trigonometry is foundational for advanced mathematics and practical applications. Focus on understanding concepts rather than just memorizing formulas.
Topic
Introduction and Overview
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mermaid
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mindmap root((Trigonometry)) Applications Architecture Navigation Physics Engineering Core Concepts Angles Triangles Functions Identities Problem Solving SOH CAH TOA Unit Circle Special Angles Laws of Sines Cosines
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mermaid_mindmap
Description
Overview of trigonometry showing its applications, core concepts, and problem-solving tools
Angle Measurement: Degrees and Radians
Angles can be measured in degrees or radians. Understanding both systems is crucial for trigonometric calculations.
Notes
Master angle conversion early as it's used throughout trigonometry. Practice with common angles like 30°, 45°, 60°, 90°.
Topic
Angle Measurement
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S2
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mermaid
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flowchart TD A[Angle Conversion] --> B{Unit?} B -->|Degrees to Radians| C[Multiply by π/180] B -->|Radians to Degrees| D[Multiply by 180/π] C --> E[Result in Radians] D --> F[Result in Degrees] E --> G[Example: 45° × π/180 = π/4 rad] F --> H[Example: π/6 × 180/π = 30°]
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mermaid_flowchart
Description
Flowchart showing the process for converting between degrees and radians
The Unit Circle and Standard Position
The unit circle is fundamental to understanding trigonometric functions. It provides a geometric representation of angles and their corresponding coordinates.
Notes
The unit circle helps visualize angle positions and determine signs of trigonometric functions in different quadrants.
Topic
Unit Circle
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S3
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mermaid
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stateDiagram-v2 [*] --> Quadrant_I: 0° to 90° Quadrant_I --> Quadrant_II: 90° to 180° Quadrant_II --> Quadrant_III: 180° to 270° Quadrant_III --> Quadrant_IV: 270° to 360° Quadrant_IV --> [*]: Complete rotation state Quadrant_I { [*] --> Both_Positive Both_Positive: x > 0, y > 0 } state Quadrant_II { [*] --> X_Neg_Y_Pos X_Neg_Y_Pos: x < 0, y > 0 }
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mermaid_stateDiagram
Description
State diagram showing the four quadrants of the unit circle and the signs of coordinates in each quadrant
Right Triangle Trigonometry: SOH-CAH-TOA
SOH-CAH-TOA is a mnemonic to remember the basic trigonometric ratios. These ratios help find unknown sides and angles in right triangles.
Notes
Always draw and label the triangle first. Identify which side is opposite, adjacent, and hypotenuse relative to the given angle.
Topic
Right Triangle Trigonometry
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S4
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mermaid
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Mermaid Diagram
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flowchart TD A[Right Triangle Problem] --> B{What to find?} B -->|Opposite side| C[Use sine: sin θ = opp/hyp] B -->|Adjacent side| D[Use cosine: cos θ = adj/hyp] B -->|Angle| E[Use inverse functions] C --> F[Solve: opp = hyp × sin θ] D --> G[Solve: adj = hyp × cos θ] E --> H[θ = sin⁻¹ or cos⁻¹ or tan⁻¹] F --> I[Check answer using Pythagorean theorem] G --> I H --> I
Type
mermaid_flowchart
Description
Decision flowchart for solving right triangle problems using SOH-CAH-TOA
Reciprocal Trigonometric Functions
Reciprocal functions are the inverse ratios of the basic trigonometric functions. They provide alternative ways to express relationships in triangles.
Notes
Reciprocal functions are undefined when their corresponding basic function equals zero. For example, csc θ is undefined when sin θ = 0.
Topic
Reciprocal Functions
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S5
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mermaid
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Mermaid Diagram
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flowchart LR A[sin θ] -.reciprocal.- B[csc θ] C[cos θ] -.reciprocal.- D[sec θ] E[tan θ] -.reciprocal.- F[cot θ] A --> G[opp/hyp] B --> H[hyp/opp] C --> I[adj/hyp] D --> J[hyp/adj] E --> K[opp/adj] F --> L[adj/opp]
Type
mermaid_flowchart
Description
Diagram showing the relationship between basic trigonometric functions and their reciprocals
Special Angles: 30°, 45°, 60°
Special angles have exact trigonometric values that frequently appear in problems. These come from 30-60-90 and 45-45-90 triangles.
Notes
Create a reference table for special angles. Practice until you can recall these values instantly without calculation.
Topic
Special Angles
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S6
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mermaid
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6
Mermaid Diagram
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pie title Special Angle Values Distribution "sin 30° = 1/2" : 16.7 "cos 30° = √3/2" : 16.7 "tan 30° = √3/3" : 16.6 "sin 45° = √2/2" : 16.7 "cos 45° = √2/2" : 16.7 "tan 45° = 1" : 16.6
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mermaid_pie
Description
Visual representation of the distribution of special angle values commonly used in problems
Quadrants and Signs of Trigonometric Functions
The signs of trigonometric functions depend on which quadrant the angle terminates in. Use reference angles to find exact values.
Notes
Remember ASTC: All Students Take Calculus (or All Stations to Central) to recall which functions are positive in each quadrant.
Topic
Quadrants and Signs
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S7
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mermaid
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stateDiagram-v2 [*] --> QI: 0° to 90° QI --> QII: 90° to 180° QII --> QIII: 180° to 270° QIII --> QIV: 270° to 360° QIV --> [*]: Back to start state QI { [*] --> All_Positive All_Positive: sin > 0, cos > 0, tan > 0 } state QII { [*] --> Sine_Positive Sine_Positive: sin > 0, cos < 0, tan < 0 } state QIII { [*] --> Tangent_Positive Tangent_Positive: sin < 0, cos < 0, tan > 0 } state QIV { [*] --> Cosine_Positive Cosine_Positive: sin < 0, cos > 0, tan < 0 }
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mermaid_stateDiagram
Description
State diagram showing the four quadrants and which trigonometric functions are positive in each
Fundamental Trigonometric Identities
Trigonometric identities are equations that are true for all values of the variable where both sides are defined. They're essential for simplifying expressions and solving equations.
Notes
These identities are tools for solving complex trigonometric problems. Practice using them to transform one expression into another.
Topic
Trigonometric Identities
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mermaid
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8
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mindmap root((Trig Identities)) Pythagorean sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = csc²θ Ratio tan θ = sin θ / cos θ cot θ = cos θ / sin θ Reciprocal csc θ = 1 / sin θ sec θ = 1 / cos θ cot θ = 1 / tan θ
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mermaid_mindmap
Description
Mind map showing the three main categories of fundamental trigonometric identities
Solving Trigonometric Identities - Step by Step
Proving trigonometric identities requires systematic application of known identities. Work on one side at a time until both sides match.
Notes
Don't work on both sides simultaneously when proving identities. Transform one side until it matches the other.
Topic
Proving Identities
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mermaid
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9
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flowchart TD A[Identity to Prove] --> B[Choose complex side] B --> C[Apply fundamental identities] C --> D[Simplify algebraically] D --> E{Sides equal?} E -->|Yes| F[Identity proven] E -->|No| G[Try different approach] G --> C F --> H[Verify with specific values]
Type
mermaid_flowchart
Description
Flowchart showing the systematic approach to proving trigonometric identities
Reference Angles and Coterminal Angles
Reference angles help find trigonometric values for any angle by relating them to acute angles. Coterminal angles have the same trigonometric function values.
Notes
Always reduce angles to their reference angle first, then apply the correct sign based on the quadrant.
Topic
Reference and Coterminal Angles
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S10
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mermaid
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10
Mermaid Diagram
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flowchart TD A[Given Angle θ] --> B{Which Quadrant?} B -->|QI: 0° to 90°| C[Reference = θ] B -->|QII: 90° to 180°| D[Reference = 180° - θ] B -->|QIII: 180° to 270°| E[Reference = θ - 180°] B -->|QIV: 270° to 360°| F[Reference = 360° - θ] C --> G[Apply QI signs: all positive] D --> H[Apply QII signs: sin positive] E --> I[Apply QIII signs: tan positive] F --> J[Apply QIV signs: cos positive]
Type
mermaid_flowchart
Description
Decision tree for finding reference angles and applying correct signs based on quadrant
Arc Length and Sector Area
Arc length is the distance along a circle's circumference, while sector area is the area of a 'slice' of the circle. These formulas connect angles to geometric measurements.
Notes
Always check if the angle is in degrees or radians before applying formulas. Convert if necessary for consistency.
Topic
Arc Length and Sector Area
Slide Id
S11
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mermaid
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11
Mermaid Diagram
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flowchart TD A[Circle Problem] --> B{Units?} B -->|Radians| C[Arc length: s = rθ] B -->|Degrees| D[Arc length: s = πrθ/180] C --> E[Area: A = ½r²θ] D --> F[Area: A = πr²θ/360] E --> G[Calculate with given values] F --> G G --> H[Check units and reasonableness]
Type
mermaid_flowchart
Description
Problem-solving flowchart for arc length and sector area calculations
Sum and Difference Formulas
Sum and difference formulas allow us to find trigonometric values of angles that can be expressed as sums or differences of special angles.
Notes
Notice the sign patterns: sine uses same signs, cosine alternates signs, tangent has complex denominators.
Topic
Sum and Difference Formulas
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12
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flowchart TD A[Angle to evaluate] --> B[Express as sum/difference of special angles] B --> C{Which function?} C -->|Sine| D[sin(A±B) = sinA cosB ± cosA sinB] C -->|Cosine| E[cos(A±B) = cosA cosB ∓ sinA sinB] C -->|Tangent| F[tan(A±B) = (tanA ± tanB)/(1 ∓ tanA tanB)] D --> G[Substitute known values] E --> G F --> G G --> H[Simplify to get exact answer]
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mermaid_flowchart
Description
Process for applying sum and difference formulas to find exact trigonometric values
Law of Sines
The Law of Sines relates the sides of a triangle to the sines of their opposite angles. It's essential for solving oblique (non-right) triangles.
Notes
The ambiguous case occurs with SSA when the given angle is acute. Always check if a second solution exists.
Topic
Law of Sines
Slide Id
S13
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mermaid
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13
Mermaid Diagram
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flowchart TD A[Triangle Problem] --> B{What is given?} B -->|2 angles, 1 side AAS/ASA| C[Use Law of Sines directly] B -->|2 sides, 1 opposite angle SSA| D[Check for ambiguous case] C --> E[Set up proportion: a/sin A = b/sin B] D --> F{Is given angle acute?} F -->|Yes| G[May have 0, 1, or 2 solutions] F -->|No| H[At most 1 solution] E --> I[Solve for unknown] G --> I H --> I I --> J[Find remaining angles: sum = 180°]
Type
mermaid_flowchart
Description
Decision flowchart for applying the Law of Sines in different triangle scenarios
Law of Cosines
The Law of Cosines relates all three sides of a triangle to one of its angles. It's the go-to method when the Law of Sines doesn't apply.
Notes
Law of Cosines reduces to Pythagorean theorem when angle C = 90° (cos 90° = 0). Use it when Law of Sines isn't applicable.
Topic
Law of Cosines
Slide Id
S14
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mermaid
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14
Mermaid Diagram
Code
flowchart TD A[Triangle Problem] --> B{What is given?} B -->|3 sides SSS| C[Find angle: cos A = (b²+c²-a²)/(2bc)] B -->|2 sides, included angle SAS| D[Find third side: c² = a²+b²-2ab cos C] B -->|Other cases| E[Use Law of Sines instead] C --> F[Use inverse cosine: A = cos⁻¹(value)] D --> G[Take square root: c = √(calculated value)] F --> H[Find remaining angles] G --> I[Use Law of Sines for remaining angles] H --> J[Check: sum of angles = 180°] I --> J
Type
mermaid_flowchart
Description
Decision tree for applying the Law of Cosines based on given information
Real-World Applications of Trigonometry
Trigonometry has numerous practical applications in various fields. Understanding these applications helps see the relevance of trigonometric concepts.
Notes
Practice solving word problems that relate to real situations. This helps develop problem-solving skills and shows the practical value of trigonometry.
Topic
Applications
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mindmap root((Trigonometry Applications)) Navigation GPS Systems Ship Navigation Aviation Surveying Construction Bridge Design Building Heights Roof Angles Foundation Work Science Physics Waves Astronomy Electronics Seismology Everyday Life Photography Sports Analysis Art and Design Gaming Graphics
Type
mermaid_mindmap
Description
Mind map showing the wide range of real-world applications of trigonometry across different fields
Problem-Solving Strategy for Trigonometry
Systematic problem-solving approach ensures you tackle trigonometry problems efficiently and accurately. Following these steps reduces errors and builds confidence.
Notes
A well-drawn diagram often makes the solution path obvious. Always start with a clear, labeled sketch of the problem situation.
Topic
Problem-Solving Strategy
Slide Id
S16
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mermaid
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16
Mermaid Diagram
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flowchart TD A[Trigonometry Problem] --> B[Draw and label diagram] B --> C[List given information] C --> D[Identify what to find] D --> E{Type of triangle?} E -->|Right triangle| F[Use SOH-CAH-TOA] E -->|Oblique triangle| G{What is given?} G -->|AAS/ASA or SSA| H[Use Law of Sines] G -->|SAS or SSS| I[Use Law of Cosines] F --> J[Calculate and solve] H --> J I --> J J --> K[Check answer reasonableness] K --> L[State conclusion with units]
Type
mermaid_flowchart
Description
Comprehensive flowchart for systematic trigonometry problem solving
Common Mistakes and How to Avoid Them
Recognizing common errors helps you avoid them and improve problem-solving accuracy. Most mistakes stem from rushing or not carefully reading the problem.
Notes
Slow down and be methodical. Most errors come from carelessness rather than not knowing the concepts.
Topic
Common Mistakes
Slide Id
S17
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mermaid
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17
Mermaid Diagram
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flowchart TD A[Trigonometry Problem] --> B[Check calculator mode] B --> C[Verify angle measurement unit] C --> D[Draw accurate diagram] D --> E[Label all known values] E --> F[Double-check triangle type] F --> G{Right triangle?} G -->|Yes| H[Identify opp, adj, hyp correctly] G -->|No| I[Choose appropriate law] H --> J[Apply correct trig function] I --> K[Set up equation carefully] J --> L[Check final answer] K --> L L --> M[Verify answer makes sense]
Type
mermaid_flowchart
Description
Error prevention flowchart highlighting key checkpoints to avoid common trigonometry mistakes
Review and Key Takeaways
Success in trigonometry comes from understanding fundamental concepts and practicing their application. Focus on building strong foundations before tackling complex problems.
Notes
Regular practice and understanding of when to use each method are keys to success. Don't just memorize - understand the reasoning behind each approach.
Topic
Review and Summary
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mindmap root((Trigonometry Mastery)) Fundamentals Unit Circle Special Angles Basic Ratios Identities Problem Solving Right Triangles Oblique Triangles Word Problems Applications Test Preparation Formula Sheet Practice Problems Time Management Error Checking Study Tips Draw Diagrams Check Units Verify Answers Practice Daily
Type
mermaid_mindmap
Description
Complete study guide mind map covering all essential aspects of trigonometry mastery
References
- CET 2026 Comprehensive Lecture Notes — Mathematics
- BRAINBOX UPCAT and Other College Entrance — Mathematics Proficiency
- The UPCAT Champion CET — Quantitative Reasoning
- Philippine Educational Standards for Senior High School Mathematics
- UPCAT and College Entrance Examination Review Materials
In summary
Trigonometry is a powerful mathematical tool that connects angles and sides in triangles, with applications spanning from basic geometry to advanced physics and engineering. Success in trigonometry requires mastering fundamental ratios (SOH-CAH-TOA), understanding the unit circle, memorizing special angle values, and knowing when to apply the Laws of Sines and Cosines. Practice systematic problem-solving approaches, always draw diagrams, and verify your answers. With consistent practice and understanding of core concepts, you'll be well-prepared for UPCAT and other college entrance examinations. Remember: trigonometry is not just about memorizing formulas, but understanding relationships and applying them logically to solve real-world problems.
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