21/08/2026
π TRIGONOMETRIC FUNCTIONS β COMPLETE FORMULA GUIDE FOR CNC PROGRAMMERS & MACHINISTS
Trigonometry is one of the most important mathematical tools used in CNC Programming, Machining, Milling, Turning, Tool Positioning, Coordinate Calculation, and Engineering Geometry.
Understanding SIN, COS, TAN, COT, SEC, and CSC helps CNC programmers calculate unknown dimensions, angles, coordinates, tapers, chamfers, and tool positions accurately.
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π· 1. RIGHT-ANGLE TRIANGLE
For a right-angle triangle:
A = Adjacent Side
B = Opposite Side
C = Hypotenuse
Ξ± = Angle
Pythagoras Theorem
CΒ² = AΒ² + BΒ²
This formula is used when two sides are known and the third side needs to be calculated.
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π· 2. BASIC TRIGONOMETRIC RATIOS
For angle Ξ±:
SIN Ξ± = B / C
Opposite Γ· Hypotenuse
COS Ξ± = A / C
Adjacent Γ· Hypotenuse
TAN Ξ± = B / A
Opposite Γ· Adjacent
Reciprocal Functions
COT Ξ± = A / B
SEC Ξ± = C / A
CSC Ξ± = C / B
Easy Memory Rule:
SIN = Opposite / Hypotenuse
COS = Adjacent / Hypotenuse
TAN = Opposite / Adjacent
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π· 3. IMPORTANT TRIGONOMETRIC IDENTITIES
SINΒ² Ξ± + COSΒ² Ξ± = 1
1 + TANΒ² Ξ± = SECΒ² Ξ±
1 + COTΒ² Ξ± = CSCΒ² Ξ±
Other Useful Formulas:
TAN Ξ± = SIN Ξ± / COS Ξ±
COT Ξ± = COS Ξ± / SIN Ξ±
SEC Ξ± = 1 / COS Ξ±
CSC Ξ± = 1 / SIN Ξ±
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π· 4. FORMULA CONVERSION
From Sine:
B = C Γ SIN Ξ±
C = B / SIN Ξ±
From Cosine:
A = C Γ COS Ξ±
C = A / COS Ξ±
From Tangent:
B = A Γ TAN Ξ±
A = B / TAN Ξ±
From Cotangent:
A = B Γ COT Ξ±
B = A / COT Ξ±
These rearranged formulas are especially useful when solving CNC drawing dimensions.
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π· 5. DEGREE TO RADIAN CONVERSION
Degrees β Radians
RADIANS = DEGREES Γ Ο / 180
Radians β Degrees
DEGREES = RADIANS Γ 180 / Ο
Where:
Ο β 3.141592653589793
Important conversions:
180Β° = Ο radians
90Β° = Ο/2 radians
45Β° = Ο/4 radians
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π· 6. COMMON ANGLE VALUES
0Β°
SIN = 0
COS = 1
TAN = 0
30Β°
SIN = 1/2
COS = β3/2
TAN = 1/β3
45Β°
SIN = β2/2
COS = β2/2
TAN = 1
60Β°
SIN = β3/2
COS = 1/2
TAN = β3
90Β°
SIN = 1
COS = 0
TAN = Undefined
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π₯ 7. PRACTICAL CNC CALCULATION EXAMPLES
π EXAMPLE 1
Given:
C = 100 mm
Ξ± = 30Β°
Find A and B.
A = C Γ COS 30Β°
A = 100 Γ 0.8660
A = 86.60 mm
B = C Γ SIN 30Β°
B = 100 Γ 0.5
B = 50.00 mm
β
Answer:
A = 86.60 mm
B = 50.00 mm
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π EXAMPLE 2
Given:
A = 60 mm
Ξ± = 45Β°
Find B and C.
B = A Γ TAN 45Β°
B = 60 Γ 1
B = 60.00 mm
C = A / COS 45Β°
C = 60 / 0.7071
C = 84.85 mm
β
Answer:
B = 60.00 mm
C = 84.85 mm
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π EXAMPLE 3
Given:
B = 75 mm
Ξ± = 60Β°
Find A and C.
A = B / TAN 60Β°
A = 75 / 1.732
A = 43.30 mm
C = B / SIN 60Β°
C = 75 / 0.8660
C = 86.60 mm
β
Answer:
A = 43.30 mm
C = 86.60 mm
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π EXAMPLE 4
Given:
A = 80 mm
B = 60 mm
Find C and Ξ±.
Using Pythagoras:
C = β(AΒ² + BΒ²)
C = β(80Β² + 60Β²)
C = β(6400 + 3600)
C = β10000
C = 100.00 mm
Now calculate the angle:
Ξ± = TANβ»ΒΉ(B/A)
Ξ± = TANβ»ΒΉ(60/80)
Ξ± = TANβ»ΒΉ(0.75)
Ξ± = 36.87Β°
β
Answer:
C = 100.00 mm
Ξ± = 36.87Β°
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π EXAMPLE 5
Given:
C = 120 mm
Ξ± = 25Β°
Find A and B.
A = C Γ COS 25Β°
A = 120 Γ 0.9063
A = 108.76 mm
B = C Γ SIN 25Β°
B = 120 Γ 0.4226
B = 50.71 mm
β
Answer:
A = 108.76 mm
B = 50.71 mm
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π― WHY IS TRIGONOMETRY IMPORTANT IN CNC?
Trigonometric calculations are commonly used for:
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Angular Machining
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Taper Calculations
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Coordinate Calculations
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Chamfer Calculations
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Hole Position Calculations
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Tool Positioning
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Milling Geometry
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Turning Geometry
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Toolpath Calculations
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Radius & Angle Calculations
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CNC Drawing Interpretation
For a CNC Programmer or Machinist, strong knowledge of Trigonometry means:
Better Accuracy + Faster Calculations + Fewer Programming Errors
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π‘ IMPORTANT TIP
Don't just memorize the formulas.
First identify:
Adjacent Side
Opposite Side
Hypotenuse
Then select the correct formula:
SIN = Opposite / Hypotenuse
COS = Adjacent / Hypotenuse
TAN = Opposite / Adjacent
π Understand the Geometry β Select the Formula β Calculate the Dimension β Apply It in CNC Programming
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π CNC TECH SKILLS
By Anup Kumar Singh
π§ Learn CNC Programming
π Master CNC Calculations
βοΈ Improve Machining Skills
π Practice Daily β’ Improve Skills β’ Master CNC Technology
