Introduction to Trigonometry

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.