Trigonometry - Cheat sheet
Your last-minute revision companion for Trigonometry - all essential formulas, identities, and values for UPCAT and college entrance exams
Basic Trigonometric Functions
Formulas
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
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
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
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
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
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
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
Unit Circle
Example
Used to define trig functions for all angles
Circle with radius 1 centered at origin, equation x² + y² = 1
Reference Angle
Example
Reference angle of 240° is 60°
Acute angle between terminal side and x-axis (0° to 90°)
Standard Position
Example
All unit circle angles are in standard position
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
Special Angles and Values
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°
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
30-60-90 Triangle
Example
If shortest side = 5, then other sides are 5√3 and 10
Special right triangle with sides in ratio 1 : √3 : 2
45-45-90 Triangle
Example
If legs = 5, then hypotenuse = 5√2
Special right triangle with sides in ratio 1 : 1 : √2
Trigonometric Identities
Formulas
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
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
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
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
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
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
Pythagorean Identity
Example
If sin θ = 3/5, then cos²θ = 1 - 9/25 = 16/25
sin²θ + cos²θ = 1, the most fundamental trigonometric identity
Reciprocal Identity
Example
sin θ × csc θ = 1
Each trig function has a reciprocal (csc, sec, cot)
Reactions or equations
sin θ × csc θ = 1
Note
Reciprocal identity
Conditions
sin θ ≠ 0
cos θ × sec θ = 1
Note
Reciprocal identity
Conditions
cos θ ≠ 0
tan θ × cot θ = 1
Note
Reciprocal identity
Conditions
sin θ ≠ 0, cos θ ≠ 0
Angle Conversions
Formulas
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
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
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
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
Radian
Example
π radians = 180°, 2π radians = 360°
Angle that subtends an arc equal to the radius (≈57.3°)
Coterminal Angles
Example
45° and 405° are coterminal
Angles that differ by multiples of 360° (or 2π radians)
Quadrant Signs and Reference Angles
Formulas
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
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
Quadrant I
Example
sin, cos, tan all positive
All trig functions are positive (0° to 90°)
Quadrant II
Example
sin positive, cos and tan negative
Only sine (and csc) are positive (90° to 180°)
Quadrant III
Example
tan positive, sin and cos negative
Only tangent (and cot) are positive (180° to 270°)
Quadrant IV
Example
cos positive, sin and tan negative
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
Sum and Difference Identities
Formulas
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
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
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
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
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
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
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
Laws of Triangles
Formulas
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
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
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
Ambiguous Case (SSA)
Example
Check if given side opposite known angle is long enough
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
Trigonometric Function Signs by Quadrant
| Function | QI (0°-90°) | QII (90°-180°) | QIII (180°-270°) | QIV (270°-360°) |
|---|---|---|---|---|
| sin θ, csc θ | + | + | - | - |
| cos θ, sec θ | + | - | - | + |
| tan θ, cot θ | + | - | + | - |
Special Right Triangles
| Triangle | Angles | Side Ratios | Example (if shortest = 5) |
|---|---|---|---|
| 30-60-90 | 30°-60°-90° | 1 : √3 : 2 | 5 : 5√3 : 10 |
| 45-45-90 | 45°-45°-90° | 1 : 1 : √2 | 5 : 5 : 5√2 |
When to Use Triangle Laws
| Given Information | Use This Law | Formula |
|---|---|---|
| Two angles, one side (AAS/ASA) | Law of Sines | a/sin A = b/sin B = c/sin C |
| Two sides, non-included angle (SSA) | Law of Sines | a/sin A = b/sin B = c/sin C |
| Two sides, included angle (SAS) | Law of Cosines | c² = a² + b² - 2ab cos C |
| Three sides (SSS) | Law of Cosines | c² = a² + b² - 2ab cos C |
Diagrams
Trigonometry Concept Map
Complete overview of all trigonometry topics and their relationships
Problem-Solving Decision Tree
Step-by-step decision tree for choosing the right approach to any trigonometry problem
Common UPCAT Trigonometry Question Types
Typical distribution of trigonometry question types in UPCAT and similar entrance exams