FEUCAT Mathematics — TrigonometryRevision Notes
Final-week revision notes for Trigonometry. If you have already studied the full chapter, this page is your go-to refresher before sitting the FEUCAT. Compact, high-yield, and aligned with what Far Eastern University tests in the Mathematics subtest.
Exam context
Far Eastern University runs the Far Eastern University College Admission Test on Q3–Q4 2026. Its Mathematics section sits under a "Core section" weighting, and Trigonometry is the 7th chapter in the 9-chapter FEUCAT Mathematics rotation. The FEUCAT passing mark is Competitive overall score, and the most recent 2026 paper drew about a meaningful share of questions from Mathematics.
Trigonometry - Revision notes
Trigonometry is the study of triangles and the relationships between their sides and angles. This chapter covers fundamental trigonometric concepts essential for UPCAT and other college entrance exams, including right triangle trigonometry, unit circle concepts, trigonometric identities, and the laws of sines and cosines. Master these concepts through step-by-step problem solving and practical applications.
Sections
Formulas
Example
Convert 45° to radians: θ_rad = (π × 45°)/180° = π/4 radians
Formula
Degree to Radian: θ_rad = (π × θ_deg)/180°
Variables
θ_rad = angle in radians, θ_deg = angle in degrees
Application
Converting between angle measurements
Example
Convert π/3 to degrees: θ_deg = (180° × π/3)/π = 60°
Formula
Radian to Degree: θ_deg = (180° × θ_rad)/π
Variables
θ_deg = angle in degrees, θ_rad = angle in radians
Application
Converting radians back to degrees
Example
Find arc length with r = 6 and θ = π/3: s = 6 × π/3 = 2π units
Formula
Arc Length: s = rθ (θ in radians) or s = (πrθ)/180° (θ in degrees)
Variables
s = arc length, r = radius, θ = central angle
Application
Finding arc length on circles
Exam Tips
- Memorize key conversions: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2
- Draw unit circle diagrams to visualize angle positions
- Practice finding coterminal angles by adding/subtracting 360°
Key Points
- Angles can be measured in degrees (360° = full rotation) or radians (2π = full rotation)
- Standard position: vertex at origin, initial side on positive x-axis
- Coterminal angles differ by multiples of 360° or 2π
- Reference angles are acute angles (0° to 90°) used to find trigonometric values
- Unit circle has radius = 1 and equation x² + y² = 1
Definitions
Term
Standard Position
Definition
An angle positioned with vertex at origin and initial side along positive x-axis
Importance
Essential for defining trigonometric functions using coordinates
Term
Reference Angle
Definition
The acute angle between the terminal side and the x-axis
Importance
Used to find trigonometric values in any quadrant
Section Title
Angle Measurement and Unit Circle
Common Mistakes
- Forgetting to convert between degrees and radians
- Confusing coterminal angles with complementary angles
- Not identifying the correct reference angle for angles > 90°
Formulas
Example
In triangle with opposite = 3, hypotenuse = 5: sin θ = 3/5 = 0.6
Formula
sin θ = opposite/hypotenuse
Variables
θ = angle, opposite = side opposite to angle, hypotenuse = longest side
Application
Finding ratios in right triangles
Example
In triangle with adjacent = 4, hypotenuse = 5: cos θ = 4/5 = 0.8
Formula
cos θ = adjacent/hypotenuse
Variables
θ = angle, adjacent = side next to angle, hypotenuse = longest side
Application
Finding ratios in right triangles
Example
In triangle with opposite = 3, adjacent = 4: tan θ = 3/4 = 0.75
Formula
tan θ = opposite/adjacent
Variables
θ = angle, opposite = side opposite to angle, adjacent = side next to angle
Application
Finding ratios without hypotenuse
Exam Tips
- Draw and label triangles clearly before solving
- Memorize exact values: sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3
- Use SOH-CAH-TOA mnemonic to remember basic ratios
Key Points
- SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent
- Reciprocal functions: csc = 1/sin, sec = 1/cos, cot = 1/tan
- Special triangles: 30-60-90 and 45-45-90 triangles have specific ratios
- Pythagorean theorem: a² + b² = c² for right triangles
- Applications in real-world problems: heights, distances, angles of elevation/depression
Definitions
Term
30-60-90 Triangle
Definition
Special right triangle with sides in ratio 1 : √3 : 2
Importance
Provides exact values for trigonometric functions at 30° and 60°
Term
45-45-90 Triangle
Definition
Special right triangle with sides in ratio 1 : 1 : √2
Importance
Provides exact values for trigonometric functions at 45°
Section Title
Right Triangle Trigonometry
Common Mistakes
- Confusing opposite and adjacent sides relative to the angle
- Using wrong triangle ratios for special angles
- Forgetting to rationalize denominators in exact answers
Formulas
Example
If sin θ = 3/5, then cos²θ = 1 - (3/5)² = 1 - 9/25 = 16/25, so cos θ = ±4/5
Formula
sin²θ + cos²θ = 1
Variables
θ = any angle
Application
Pythagorean identity - fundamental relationship
Example
If tan θ = 2, then sec²θ = 2² + 1 = 5, so sec θ = ±√5
Formula
tan²θ + 1 = sec²θ
Variables
θ = any angle where cos θ ≠ 0
Application
Pythagorean identity involving tangent and secant
Example
If cot θ = 1/3, then csc²θ = (1/3)² + 1 = 1/9 + 1 = 10/9, so csc θ = ±√10/3
Formula
cot²θ + 1 = csc²θ
Variables
θ = any angle where sin θ ≠ 0
Application
Pythagorean identity involving cotangent and cosecant
Exam Tips
- Remember ASTC (All Students Take Calculus) for quadrant signs
- Use identities to solve for unknown trigonometric values
- Practice converting between different forms using identities
Key Points
- Six trigonometric functions: sin, cos, tan, csc, sec, cot
- Signs of functions depend on quadrant location
- Fundamental identities: sin²θ + cos²θ = 1, tan²θ + 1 = sec²θ, cot²θ + 1 = csc²θ
- Even/odd functions: cos(-θ) = cos(θ), sin(-θ) = -sin(θ)
- Domain and range restrictions for each function
Definitions
Term
Quadrant Signs
Definition
QI: all positive, QII: sin positive, QIII: tan positive, QIV: cos positive
Importance
Determines the sign of trigonometric function values
Term
Reciprocal Functions
Definition
csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
Importance
Alternative ways to express trigonometric relationships
Section Title
Trigonometric Functions and Identities
Common Mistakes
- Forgetting to check quadrant when determining signs
- Confusing reciprocal relationships
- Not using identities to simplify complex expressions
Formulas
Example
sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4
Formula
sin(A + B) = sin A cos B + cos A sin B
Variables
A, B = any angles
Application
Finding sine of angle sums
Example
cos 75° = cos(45° + 30°) = cos 45° cos 30° - sin 45° sin 30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4
Formula
cos(A + B) = cos A cos B - sin A sin B
Variables
A, B = any angles
Application
Finding cosine of angle sums
Example
tan 75° = tan(45° + 30°) = (tan 45° + tan 30°)/(1 - tan 45° tan 30°) = (1 + √3/3)/(1 - √3/3) = 2 + √3
Formula
tan(A + B) = (tan A + tan B)/(1 - tan A tan B)
Variables
A, B = angles where tangent is defined
Application
Finding tangent of angle sums
Exam Tips
- Practice deriving exact values for angles like 15°, 75°, 105°
- Remember that difference formulas follow from sum formulas
- Use these formulas to verify trigonometric identities
Key Points
- Addition formulas for sine, cosine, and tangent
- Subtraction formulas derived from addition formulas
- Applications in solving trigonometric equations
- Used to find exact values of non-special angles
- Foundation for double angle and half angle formulas
Definitions
Term
Sum Formula
Definition
Formulas for trigonometric functions of angle sums (A + B)
Importance
Allows calculation of trigonometric values for non-standard angles
Term
Difference Formula
Definition
Formulas for trigonometric functions of angle differences (A - B)
Importance
Derived from sum formulas by substituting -B for B
Section Title
Sum and Difference Formulas
Common Mistakes
- Mixing up addition and subtraction formulas
- Sign errors when applying formulas
- Not simplifying final answers completely
Formulas
Example
Triangle with A = 60°, B = 45°, a = 10. Find b: b/sin 45° = 10/sin 60°, so b = 10 sin 45°/sin 60° = 10(√2/2)/(√3/2) = 10√6/3
Formula
Law of Sines: a/sin A = b/sin B = c/sin C
Variables
a, b, c = sides opposite to angles A, B, C respectively
Application
Solving triangles with known angles and sides
Example
Triangle with a = 5, b = 7, C = 60°. Find c: c² = 5² + 7² - 2(5)(7)cos 60° = 25 + 49 - 70(1/2) = 74 - 35 = 39, so c = √39
Formula
Law of Cosines: c² = a² + b² - 2ab cos C
Variables
a, b = two known sides, C = included angle, c = opposite side
Application
Finding third side when two sides and included angle are known
Exam Tips
- Identify which law to use based on given information
- Draw and label triangles clearly
- Always check if your answer makes sense (angles sum to 180°)
Key Points
- Law of Sines relates sides and angles in any triangle
- Law of Cosines generalizes Pythagorean theorem for any triangle
- Use Law of Sines when given two angles and one side, or two sides and an angle opposite one of them
- Use Law of Cosines when given three sides, or two sides and included angle
- Applications in solving oblique triangles and real-world problems
Definitions
Term
Oblique Triangle
Definition
Any triangle that is not a right triangle
Importance
Requires Laws of Sines and Cosines since SOH-CAH-TOA doesn't apply
Term
Ambiguous Case
Definition
When Law of Sines may yield two possible triangles
Importance
Occurs when given two sides and an angle opposite the shorter side
Section Title
Laws of Sines and Cosines
Common Mistakes
- Using wrong law for given information
- Not checking for ambiguous cases in Law of Sines
- Forgetting that angles in triangle must sum to 180°
Connections
- Trigonometry connects to Geometry through triangle relationships and circle properties
- Applications in Physics for wave motion, oscillations, and vector analysis
- Foundation for Calculus concepts including derivatives and integrals of trigonometric functions
- Real-world applications in engineering, navigation, architecture, and astronomy
- Connects to Complex Numbers through Euler's formula and polar coordinates
Exam Strategy
Focus on memorizing special angle values and key identities. Practice identifying which law or formula to use based on given information. Draw clear diagrams for every problem. Master the step-by-step approach: identify given information, choose appropriate method, substitute values carefully, and verify answers. For UPCAT preparation, emphasize exact values over decimal approximations, and practice both computational problems and conceptual questions about domains, ranges, and properties of trigonometric functions.
Quick Review Questions
Convert 150° to radians.
Use formula: θ_rad = (π × θ_deg)/180° = (π × 150°)/180° = 5π/6
In a right triangle, if sin θ = 4/5, find cos θ.
Using sin²θ + cos²θ = 1: cos²θ = 1 - (4/5)² = 1 - 16/25 = 9/25, so cos θ = ±3/5
Find the exact value of sin 15°.
sin 15° = sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4
In triangle ABC, a = 8, b = 6, C = 120°. Find c using Law of Cosines.
c² = a² + b² - 2ab cos C = 8² + 6² - 2(8)(6)cos 120° = 64 + 36 - 96(-1/2) = 100 + 48 = 148, so c = √148 = 2√37
What is the reference angle for 240°?
240° is in Quadrant III. Reference angle = 240° - 180° = 60°
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