Introduction to Trigonometric Ratios

Study Notes

Introduction to Trigonometric Ratios

Trigonometric ratios relate the angles and sides of a right-angled triangle.
These ratios are fundamental in trigonometry and have widespread applications in various fields including physics, engineering, and navigation.
The basic trigonometric ratios are sine, cosine, and tangent.

Sine, Cosine, and Tangent

Sine (sin): The ratio of the side opposite to an angle to the hypotenuse.
Cosine (cos): The ratio of the side adjacent to an angle to the hypotenuse.
Tangent (tan): The ratio of the side opposite to an angle to the side adjacent to the angle.

Definitions and Formulas

For a right-angled triangle with an angle θ:
- sin(θ) = opposite/hypotenuse
- cos(θ) = adjacent/hypotenuse
- tan(θ) = opposite/adjacent

Reciprocal Trigonometric Ratios

Cosecant (csc): The reciprocal of sine. csc(θ) = 1/sin(θ) = hypotenuse/opposite
Secant (sec): The reciprocal of cosine. sec(θ) = 1/cos(θ) = hypotenuse/adjacent
Cotangent (cot): The reciprocal of tangent. cot(θ) = 1/tan(θ) = adjacent/opposite

Relationship Between Trigonometric Ratios

The trigonometric ratios are interconnected through various identities.
These identities are essential for simplifying expressions and solving trigonometric equations.
For example: sin²(θ) + cos²(θ) = 1

Applications of Trigonometric Ratios

Calculating unknown sides and angles in right-angled triangles.
Solving real-world problems involving inclined planes, heights and distances, and angles of elevation and depression.
Used in navigation for determining directions and distances.
Critical in various branches of engineering for designing structures and calculating forces.

Trigonometric Ratios for Special Angles

Trigonometric ratios for certain angles (e.g., 0°, 30°, 45°, 60°, and 90°) have specific values that are often memorized.
These values are derived from the properties of 30-60-90 and 45-45-90 triangles.

Trigonometric Functions in Degrees

Angles in trigonometric functions are frequently expressed in degrees.

Trigonometric Functions in Radians

Angles can also be expressed in radians, which are a different unit of measurement for angles.

Solving for Unknown Sides and Angles

Given a right triangle and some known sides or angles, you can use trigonometric ratios to find unknown sides and angles.
Methods include drawing a diagram, labeling known and unknown components.

Description

This quiz explores the fundamental concepts of trigonometric ratios, specifically focusing on sine, cosine, and tangent in right-angled triangles. Gain a better understanding of these essential ratios, their definitions, and their applications in various fields such as physics and engineering.