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NOTEBOOK SHEET 1 (ANGLE CONVERSIONS SUMMARY)
Angle Conversions Core Reference
Fundamental Reference Formulas:
180° = π rad
1° = π / 180 rad 1 rad = 180° / π
Conversion Summary Table
No. Convert Radian into Degree Convert Degree into Radian (i) (^) π rad = 180° 30° = 30(π/180) = π/ (ii) (^) π/2 rad = 90° 60° = 60(π/180) = π/ (iii) (^) π/6 rad = 30° 90° = 90(π/180) = π/ (iv) (^) π/4 rad = 45° 45° = 45(π/180) = π/ (v) (^) π/8 rad = 22.5° 120° = 120(π/180) = 2π/ (vi) (^) 3π/5 rad = 108° 150° = 150(π/180) = 5π/ (vii) (^) 3π/2 rad = 270° 135° = 3π/4 rad
NOTEBOOK SHEET 2 (RADIAN TO DEGREE DERIVATIONS)
Detailed Radian to Degree Conversions
Demonstrating full fractional division cancellations:
Example (i):
π/2 × (180° / π) = 90°
Alternative Form:
1/2 × 180° = 90°
Example (ii):
π/6 × (180° / π) = 30°
Alternative Form:
1/6 × 180° = 30°
Example (iii):
π/4 × (180° / π) = 45°
Example (iv):
π/8 × (180° / π) = 22.5°
NOTEBOOK SHEET 4 (COMPLEMENTARY ANGLE TRANSFORMATIONS)
Complementary Angles Properties
In a standard right triangle, computing the secondary angle x:
θ + 90° + x = 180° x = 180° - 90° - θ x = 90° - θ
Co-Function Transformations
sin(θ) = a / c sin(90° - θ) = b / c cos(θ) = b / c cos(90° - θ) = a / c
This implies:
sin(θ) = cos(90° - θ) cos(θ) = sin(90° - θ) tan(θ) = cot(90° - θ) cosec(θ) = sec(90° - θ) sec(θ) = cosec(90° - θ)
Examples
sin(30°) = cos(90° - 30°) = cos(60°) tan(45°) = cot(90° - 45°) = cot(45°)
NOTEBOOK SHEET 5 (RATIOS & FUNCTION RANGES)
Trigonometric Function Ranges
Function Ranges Matrix
Function Analytical Boundaries / Ranges
Range of sin(θ) -1 ≤ sin(θ) ≤ +1sin(270°) = -1, sin(90°) = 1
Range of cos(θ) (^) -1 ≤ cos(θ) ≤ + Range of tan(θ) (^) -∞ to +∞
Proportional Note: As angle θ increases from 0° → 90°, the value of sin(θ) increases.
NOTEBOOK SHEET 7 (PYTHAGOREAN FUNDAMENTAL IDENTITIES)
Fundamental Pythagorean Identities
Derived systematically using the Pythagorean Theorem (c^2 = a^2 + b^2 ):
Identity I Derivation:
Divide the full system equation by c^2 :
c^2 /c^2 = a^2 /c^2 + b^2 /c^2 ⇒ 1 = sin^2 (θ) + cos^2 (θ)
Identity II Derivation:
Divide the full system equation by a^2 :
c^2 /a^2 = a^2 /a^2 + b^2 /a^2 ⇒ cosec^2 (θ) = 1 + cot^2 (θ)
Identity III Derivation:
Divide the full system equation by b^2 :
c^2 /b^2 = a^2 /b^2 + b^2 /b^2 ⇒ sec^2 (θ) = tan^2 (θ) + 1
Negative Angle Space Graph Derivation
sin(-θ) = -a / c = -sin(θ)
cos(-θ) = b / c = cos(θ)
