TRIGONOMETRY – PART 2: IDENTITIES AND
COMPLEMENTARY ANGLES
Trigonometric Identities:
The following are fundamental trigonometric identities that hold true for all angles A: 1. sin²A +
cos²A = 1 2. 1 + tan²A = sec²A 3. 1 + cot²A = cosec²A
Proof of sin²A + cos²A = 1:
In right triangle ABC, with ∠B = 90°, sin A = Perpendicular / Hypotenuse and cos A = Base /
Hypotenuse. Thus, sin²A + cos²A = (Perpendicular² + Base²) / Hypotenuse² = Hypotenuse² /
Hypotenuse² = 1.
Trigonometric Ratios of Complementary Angles:
Two angles are complementary if their sum is 90°. sin(90° - A) = cos A cos(90° - A) = sin A tan(90° -
A) = cot A cot(90° - A) = tan A sec(90° - A) = cosec A cosec(90° - A) = sec A
Examples:
Example 1: If sin A = 3/5, find cos A. Solution: sin²A + cos²A = 1 → (3/5)² + cos²A = 1 → cos²A =
16/25 → cos A = 4/5.
COMPLEMENTARY ANGLES
Trigonometric Identities:
The following are fundamental trigonometric identities that hold true for all angles A: 1. sin²A +
cos²A = 1 2. 1 + tan²A = sec²A 3. 1 + cot²A = cosec²A
Proof of sin²A + cos²A = 1:
In right triangle ABC, with ∠B = 90°, sin A = Perpendicular / Hypotenuse and cos A = Base /
Hypotenuse. Thus, sin²A + cos²A = (Perpendicular² + Base²) / Hypotenuse² = Hypotenuse² /
Hypotenuse² = 1.
Trigonometric Ratios of Complementary Angles:
Two angles are complementary if their sum is 90°. sin(90° - A) = cos A cos(90° - A) = sin A tan(90° -
A) = cot A cot(90° - A) = tan A sec(90° - A) = cosec A cosec(90° - A) = sec A
Examples:
Example 1: If sin A = 3/5, find cos A. Solution: sin²A + cos²A = 1 → (3/5)² + cos²A = 1 → cos²A =
16/25 → cos A = 4/5.
