CEUET Mathematics — TrigonometryCheat Sheet
A printable cheat sheet for Trigonometry, built for CEUET reviewers who want one go-to reference in the final stretch. Covers formulas, key definitions, common question types, and the Centro Escolar University-specific twists you will see on CEUET day.
Exam context
Centro Escolar University runs the Centro Escolar University Entrance Test on Q3–Q4 2026. Its Mathematics section sits under a "Core" weighting, and Trigonometry is the 7th chapter in the 9-chapter CEUET Mathematics rotation. The CEUET passing mark is Competitive overall score, and the most recent 2026 paper drew about a meaningful share of questions from Mathematics.
Trigonometry - Cheat sheet
Your last-minute revision companion for Trigonometry - all essential formulas, identities, and values for UPCAT and college entrance exams
Sections
Formulas
Formula
sin θ = opposite/hypotenuse
Meaning
θ = angle, opposite = side opposite to angle, hypotenuse = longest side
Watch Out
Make sure you identify the correct opposite side relative to the given angle
When To Use
When you have a right triangle and need to find sine
Formula
cos θ = adjacent/hypotenuse
Meaning
adjacent = side next to angle (not hypotenuse), hypotenuse = longest side
Watch Out
Don't confuse adjacent with opposite side
When To Use
When you have a right triangle and need to find cosine
Formula
tan θ = opposite/adjacent
Meaning
opposite = side opposite to angle, adjacent = side next to angle
Watch Out
Tangent is undefined when cos θ = 0 (at 90°, 270°, etc.)
When To Use
When you have two legs of a right triangle
Formula
csc θ = 1/sin θ = hypotenuse/opposite
Meaning
csc = cosecant, reciprocal of sine
Watch Out
Cosecant is undefined when sin θ = 0
When To Use
When you need the reciprocal of sine
Formula
sec θ = 1/cos θ = hypotenuse/adjacent
Meaning
sec = secant, reciprocal of cosine
Watch Out
Secant is undefined when cos θ = 0
When To Use
When you need the reciprocal of cosine
Formula
cot θ = 1/tan θ = adjacent/opposite
Meaning
cot = cotangent, reciprocal of tangent
Watch Out
Cotangent is undefined when tan θ = 0
When To Use
When you need the reciprocal of tangent
Section Title
Basic Trigonometric Functions
Important Facts
- SOH-CAH-TOA: Sine-Opposite/Hypotenuse, Cosine-Adjacent/Hypotenuse, Tangent-Opposite/Adjacent
- CHO-SHA-CAO: Cosecant-Hypotenuse/Opposite, Secant-Hypotenuse/Adjacent, Cotangent-Adjacent/Opposite
- Domain of sin θ and cos θ: all real numbers
- Range of sin θ and cos θ: [-1, 1]
- Domain of tan θ: all real numbers except (n + 1/2)π where n is integer
Key Definitions
Term
Unit Circle
Example
Used to define trig functions for all angles
Definition
Circle with radius 1 centered at origin, equation x² + y² = 1
Term
Reference Angle
Example
Reference angle of 240° is 60°
Definition
Acute angle between terminal side and x-axis (0° to 90°)
Term
Standard Position
Example
All unit circle angles are in standard position
Definition
Angle with vertex at origin, initial side on positive x-axis
Diagrams To Know
- Right triangle with labeled sides (opposite, adjacent, hypotenuse)
- Unit circle with coordinates
- 30-60-90 triangle
- 45-45-90 triangle
Common Values
Value
0
Symbol
sin 0°
Quantity
sin 0°
Value
1/2
Symbol
sin 30°
Quantity
sin 30°
Value
√2/2
Symbol
sin 45°
Quantity
sin 45°
Value
√3/2
Symbol
sin 60°
Quantity
sin 60°
Value
1
Symbol
sin 90°
Quantity
sin 90°
Value
1
Symbol
cos 0°
Quantity
cos 0°
Value
√3/2
Symbol
cos 30°
Quantity
cos 30°
Value
√2/2
Symbol
cos 45°
Quantity
cos 45°
Value
1/2
Symbol
cos 60°
Quantity
cos 60°
Value
0
Symbol
cos 90°
Quantity
cos 90°
Value
0
Symbol
tan 0°
Quantity
tan 0°
Value
√3/3
Symbol
tan 30°
Quantity
tan 30°
Value
1
Symbol
tan 45°
Quantity
tan 45°
Value
√3
Symbol
tan 60°
Quantity
tan 60°
Section Title
Special Angles and Values
Important Facts
- Memorize the special angle values - they appear in 80% of exam problems
- sin 30° = cos 60° = 1/2
- sin 45° = cos 45° = √2/2
- sin 60° = cos 30° = √3/2
- tan 30° = √3/3, tan 45° = 1, tan 60° = √3
Key Definitions
Term
30-60-90 Triangle
Example
If shortest side = 5, then other sides are 5√3 and 10
Definition
Special right triangle with sides in ratio 1 : √3 : 2
Term
45-45-90 Triangle
Example
If legs = 5, then hypotenuse = 5√2
Definition
Special right triangle with sides in ratio 1 : 1 : √2
Formulas
Formula
sin²θ + cos²θ = 1
Meaning
Pythagorean identity - fundamental relationship
Watch Out
Remember ± when taking square roots - check the quadrant
When To Use
When you know one trig function and need to find another
Formula
1 + tan²θ = sec²θ
Meaning
Pythagorean identity for tangent and secant
Watch Out
This identity fails when cos θ = 0
When To Use
When working with tangent and secant functions
Formula
1 + cot²θ = csc²θ
Meaning
Pythagorean identity for cotangent and cosecant
Watch Out
This identity fails when sin θ = 0
When To Use
When working with cotangent and cosecant functions
Formula
tan θ = sin θ/cos θ
Meaning
Ratio identity for tangent
Watch Out
Undefined when cos θ = 0
When To Use
When converting between trig functions
Formula
cot θ = cos θ/sin θ
Meaning
Ratio identity for cotangent
Watch Out
Undefined when sin θ = 0
When To Use
When converting between trig functions
Section Title
Trigonometric Identities
Important Facts
- All trig identities can be derived from the basic definitions
- sin(-θ) = -sin(θ) (sine is odd function)
- cos(-θ) = cos(θ) (cosine is even function)
- tan(-θ) = -tan(θ) (tangent is odd function)
Key Definitions
Term
Pythagorean Identity
Example
If sin θ = 3/5, then cos²θ = 1 - 9/25 = 16/25
Definition
sin²θ + cos²θ = 1, the most fundamental trigonometric identity
Term
Reciprocal Identity
Example
sin θ × csc θ = 1
Definition
Each trig function has a reciprocal (csc, sec, cot)
Reactions Or Equations
Note
Reciprocal identity
Equation
sin θ × csc θ = 1
Conditions
sin θ ≠ 0
Note
Reciprocal identity
Equation
cos θ × sec θ = 1
Conditions
cos θ ≠ 0
Note
Reciprocal identity
Equation
tan θ × cot θ = 1
Conditions
sin θ ≠ 0, cos θ ≠ 0
Formulas
Formula
radians = degrees × π/180
Meaning
Convert degrees to radians
Watch Out
Don't forget the π in the conversion
When To Use
When angle is given in degrees but need radians
Formula
degrees = radians × 180/π
Meaning
Convert radians to degrees
Watch Out
Make sure to divide by π, not multiply
When To Use
When angle is given in radians but need degrees
Formula
Arc length: s = rθ
Meaning
s = arc length, r = radius, θ = angle in radians
Watch Out
Angle must be in radians, not degrees
When To Use
When finding arc length on a circle
Common Values
Value
180°
Symbol
π
Quantity
π radians
Value
90°
Symbol
π/2
Quantity
π/2 radians
Value
60°
Symbol
π/3
Quantity
π/3 radians
Value
45°
Symbol
π/4
Quantity
π/4 radians
Value
30°
Symbol
π/6
Quantity
π/6 radians
Section Title
Angle Conversions
Important Facts
- π radians = 180°
- 2π radians = 360°
- π/2 radians = 90°
- π/4 radians = 45°
- π/6 radians = 30°
- π/3 radians = 60°
Key Definitions
Term
Radian
Example
π radians = 180°, 2π radians = 360°
Definition
Angle that subtends an arc equal to the radius (≈57.3°)
Term
Coterminal Angles
Example
45° and 405° are coterminal
Definition
Angles that differ by multiples of 360° (or 2π radians)
Formulas
Formula
Reference angle = |θ - nearest x-axis|
Meaning
Find acute angle between terminal side and x-axis
Watch Out
Reference angle is always between 0° and 90°
When To Use
When finding exact trig values for any angle
Section Title
Quadrant Signs and Reference Angles
Important Facts
- ASTC: All Students Take Calculus (signs in QI, QII, QIII, QIV)
- QI: All positive, QII: Sin positive, QIII: Tan positive, QIV: Cos positive
- Use reference angles to find exact values
- Reference angle for QII: 180° - θ
- Reference angle for QIII: θ - 180°
- Reference angle for QIV: 360° - θ
Key Definitions
Term
Quadrant I
Example
sin, cos, tan all positive
Definition
All trig functions are positive (0° to 90°)
Term
Quadrant II
Example
sin positive, cos and tan negative
Definition
Only sine (and csc) are positive (90° to 180°)
Term
Quadrant III
Example
tan positive, sin and cos negative
Definition
Only tangent (and cot) are positive (180° to 270°)
Term
Quadrant IV
Example
cos positive, sin and tan negative
Definition
Only cosine (and sec) are positive (270° to 360°)
Diagrams To Know
- Unit circle divided into four quadrants with signs labeled
- Reference angle illustrations for each quadrant
Formulas
Formula
sin(u + v) = sin u cos v + cos u sin v
Meaning
Sum formula for sine
Watch Out
Order matters - don't mix up the terms
When To Use
When finding sine of sum of two angles
Formula
sin(u - v) = sin u cos v - cos u sin v
Meaning
Difference formula for sine
Watch Out
Note the minus sign in the second term
When To Use
When finding sine of difference of two angles
Formula
cos(u + v) = cos u cos v - sin u sin v
Meaning
Sum formula for cosine
Watch Out
Cosine sum has minus sign (opposite of sine)
When To Use
When finding cosine of sum of two angles
Formula
cos(u - v) = cos u cos v + sin u sin v
Meaning
Difference formula for cosine
Watch Out
Cosine difference has plus sign
When To Use
When finding cosine of difference of two angles
Formula
tan(u + v) = (tan u + tan v)/(1 - tan u tan v)
Meaning
Sum formula for tangent
Watch Out
Denominator uses minus sign and product of tangents
When To Use
When finding tangent of sum of two angles
Formula
tan(u - v) = (tan u - tan v)/(1 + tan u tan v)
Meaning
Difference formula for tangent
Watch Out
Denominator uses plus sign for difference formula
When To Use
When finding tangent of difference of two angles
Section Title
Sum and Difference Identities
Important Facts
- These formulas work for any angles u and v
- Use these to find exact values like sin 15° = sin(45° - 30°)
- Memorize the patterns: sine formulas have same signs, cosine formulas have opposite signs
- Tangent formulas have the tangent sum/difference in numerator
Formulas
Formula
Law of Sines: a/sin A = b/sin B = c/sin C
Meaning
a, b, c = sides; A, B, C = opposite angles
Watch Out
May have 0, 1, or 2 solutions (ambiguous case)
When To Use
When you know 2 angles and 1 side, or 2 sides and 1 non-included angle
Formula
Law of Cosines: c² = a² + b² - 2ab cos C
Meaning
Extension of Pythagorean theorem for any triangle
Watch Out
Reduces to Pythagorean theorem when C = 90°
When To Use
When you know 3 sides, or 2 sides and included angle
Section Title
Laws of Triangles
Important Facts
- Law of Sines: Use when you have AAS, ASA, or SSA
- Law of Cosines: Use when you have SAS or SSS
- Always check if triangle exists (triangle inequality)
- Sum of angles in any triangle = 180°
Key Definitions
Term
Ambiguous Case (SSA)
Example
Check if given side opposite known angle is long enough
Definition
When given 2 sides and non-included angle, may have 0, 1, or 2 solutions
Diagrams To Know
- Triangle with all sides and angles labeled
- Ambiguous case diagram showing two possible triangles
Must Remember
- SOH-CAH-TOA for basic trig ratios
- sin²θ + cos²θ = 1 (most important identity)
- Special angle values: sin 30°=1/2, sin 45°=√2/2, sin 60°=√3/2
- ASTC: All Students Take Calculus (quadrant signs)
- π radians = 180 degrees
- Reference angles are always acute (0° to 90°)
- Law of Sines: a/sin A = b/sin B = c/sin C
- Law of Cosines: c² = a² + b² - 2ab cos C
- 30-60-90 triangle sides: 1 : √3 : 2
- Arc length formula: s = rθ (θ in radians)
Last Minute Tips
- Draw a quick unit circle with quadrant signs marked - saves time on sign questions
- For special angles, draw the 30-60-90 and 45-45-90 triangles to remember exact values
- When stuck on identities, try converting everything to sine and cosine first
- For Law of Sines vs Cosines: count what you know - if you have all angles or the included angle, use Cosines
- Always check if your triangle answer makes sense - angles sum to 180°, longest side opposite largest angle
Comparison Tables
Rows
Values
- +
- +
- -
- -
Property
sin θ, csc θ
Values
- +
- -
- -
- +
Property
cos θ, sec θ
Values
- +
- -
- +
- -
Property
tan θ, cot θ
Columns
- Function
- QI (0°-90°)
- QII (90°-180°)
- QIII (180°-270°)
- QIV (270°-360°)
Table Title
Trigonometric Function Signs by Quadrant
Rows
Values
- 30°-60°-90°
- 1 : √3 : 2
- 5 : 5√3 : 10
Property
30-60-90
Values
- 45°-45°-90°
- 1 : 1 : √2
- 5 : 5 : 5√2
Property
45-45-90
Columns
- Triangle
- Angles
- Side Ratios
- Example (if shortest = 5)
Table Title
Special Right Triangles
Rows
Values
- Law of Sines
- a/sin A = b/sin B = c/sin C
Property
Two angles, one side (AAS/ASA)
Values
- Law of Sines
- a/sin A = b/sin B = c/sin C
Property
Two sides, non-included angle (SSA)
Values
- Law of Cosines
- c² = a² + b² - 2ab cos C
Property
Two sides, included angle (SAS)
Values
- Law of Cosines
- c² = a² + b² - 2ab cos C
Property
Three sides (SSS)
Columns
- Given Information
- Use This Law
- Formula
Table Title
When to Use Triangle Laws
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