Introduction to Trigonometry

Study Notes

Introduction to Trigonometry

Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles.
It's crucial in various fields, from astronomy and navigation to surveying and engineering.
The fundamental trigonometric functions relate angles to ratios of sides in a right-angled triangle.

Basic Trigonometric Ratios

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 it.
These ratios are defined for acute angles (0° to 90°).

Trigonometric Identities

Key relationships between trigonometric functions.
Examples include:
sin²(θ) + cos²(θ) = 1
tan²(θ) = sin²(θ)/cos²(θ)
These identities are crucial for simplifying expressions and solving equations.

Trigonometric Functions of Any Angle

Trigonometric functions can be extended to all angles.
For angles in other quadrants, sine, cosine, and tangent can be positive or negative.
Knowledge of reference angles is important. Reference angles are acute angles formed between the terminal side of an angle and the x-axis.

Trigonometric Graphs

Graphing trigonometric functions reveals the periodic nature of these functions.
Sine and cosine functions have a period of 360° (2π radians) and exhibit a sinusoidal shape.
Tangent functions have a period of 180° (π radians) and have vertical asymptotes.

Applications of Trigonometry

Solving Right Triangles: Finding unknown sides and angles in right-angled triangles.
Navigation: Determining directions and distances.
Surveying: Measuring distances and angles in land surveying.
Engineering: Calculating forces and stresses in structures.
Astronomy: Calculating distances to celestial bodies and tracking their movements.
Sound waves and light waves: Describe oscillatory nature.

Inverse Trigonometric Functions

Inverse functions such as arcsin, arccos, and arctan are used to find the angle given the trigonometric ratio.
These are crucial in solving triangles and other trigonometric equations.

Special Triangles

30-60-90 Triangles and 45-45-90 Triangles: Specific ratios are known by rote learning
These triangles have predictable side ratios, simplifying calculations.

Radians

Angular measure in radians, often used in calculus.
One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.
2π radians = 360 degrees.

Law of Sines

Relates the sides and angles of any triangle.
Used for solving triangles with non-right angles.
The ratio of the sine of an angle to the length of its corresponding side is constant for all three angles.

Law of Cosines

Relates the sides and angles of any triangle
Used for solving triangles with non-right angles.
Relates a side to the other two sides and cosine of the angle between them.

Description

This quiz covers the fundamental concepts of trigonometry, including basic trigonometric ratios, identities, and functions of any angle. It is essential for understanding relationships in triangles and has applications in various fields such as engineering and astronomy.