Learning Objectives
- Define trigonometric ratios for acute angles in a right triangle
- Find trigonometric ratios of specific angles (0°, 30°, 45°, 60°, 90°)
- Establish and use trigonometric identities
- Find trigonometric ratios for complementary angles
Key Concepts
Trigonometric Ratios
In a right triangle with angle θ, hypotenuse (H), side opposite to θ (P for perpendicular), and side adjacent to θ (B for base):
- sin θ = P/H (Opposite / Hypotenuse)
- cos θ = B/H (Adjacent / Hypotenuse)
- tan θ = P/B (Opposite / Adjacent) = sin θ / cos θ
- cosec θ = H/P = 1/sin θ
- sec θ = H/B = 1/cos θ
- cot θ = B/P = 1/tan θ = cos θ / sin θ
Trigonometric Ratios of Standard Angles
Important values to memorise:
- sin 0° = 0, sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1
- cos 0° = 1, cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2, cos 90° = 0
- tan 0° = 0, tan 30° = 1/√3, tan 45° = 1, tan 60° = √3, tan 90° = not defined
Complementary Angles
Two angles are complementary if their sum is 90°. The following relations hold:
- sin(90° - θ) = cos θ and cos(90° - θ) = sin θ
- tan(90° - θ) = cot θ and cot(90° - θ) = tan θ
- sec(90° - θ) = cosec θ and cosec(90° - θ) = sec θ
Trigonometric Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
Summary
Trigonometric ratios relate the angles and sides of a right triangle. There are six ratios and three fundamental identities. Standard angle values are essential for solving problems. Complementary angle relationships simplify many expressions. These concepts form the foundation for applications in measurement, physics, and engineering.
Important Terms
- Trigonometric Ratio
- A ratio of any two sides of a right triangle with respect to an acute angle
- Hypotenuse
- The longest side of a right triangle, opposite the right angle
- Identity
- An equation that is true for all values of the variable for which it is defined
Quick Revision
- sin θ = P/H, cos θ = B/H, tan θ = P/B
- sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
- sin 30° = cos 60° = 1/2; sin 45° = cos 45° = 1/√2
- sin(90°-θ) = cos θ; tan(90°-θ) = cot θ
- tan θ = sin θ / cos θ; cot θ = cos θ / sin θ