What are the trigonometric ratios of complementary angles?
The trigonometric ratios of complementary angles state that sin(90° - θ) = cos(θ), cos(90° - θ) = sin(θ), tan(90° - θ) = cot(θ), cot(90° - θ) = tan(θ), sec(90° - θ) = cosec(θ), and cosec(90° - θ) = sec(θ).
How are the trigonometric ratios of complementary angles derived?
The trigonometric ratios of complementary angles are derived from the fact that the sum of two complementary angles is 90 degrees. Therefore, for any angle θ, sin(90° - θ) = cos(θ) and cos(90° - θ) = sin(θ), likewise for the other ratios: tan(90° - θ) = cot(θ) and cot(90° - θ) = tan(θ).
Why are the trigonometric ratios of complementary angles significant in geometry?
The trigonometric ratios of complementary angles are significant in geometry because they reveal the relationships between sine, cosine, and other functions for angles summing to 90°, simplifying calculations and proofs. This property supports problem-solving in right-angled triangles and applications in various geometric contexts.
What is the relationship between the trigonometric ratios of complementary angles?
The trigonometric ratios of complementary angles are related as follows: the sine of an angle equals the cosine of its complement, the cosine of an angle equals the sine of its complement, the tangent of an angle equals the cotangent of its complement, and the secant of an angle equals the cosecant of its complement.
How do the trigonometric ratios of complementary angles apply in real-life scenarios?
Trigonometric ratios of complementary angles are used in real-life scenarios such as engineering, architecture, and physics to calculate heights, distances, and angles accurately. This is crucial for construction, navigation, and designing various mechanical systems. Additionally, they aid in solving problems related to wave behaviour and circuit analysis.