Introduction to Trigonometry

• Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles.
• It is fundamentally concerned with the relationships between angles and ratios of sides.
• Trigonometry is crucial in fields like physics, engineering, astronomy, and navigation.

Basic Trigonometric Functions

• Sine (sin), cosine (cos), and tangent (tan) are the primary trigonometric functions.
• These functions relate angles of a right triangle to the ratios of its sides.

Right Triangle Definitions

• Consider a right triangle with one angle labeled θ (theta).
• The side opposite to θ is called the "opposite" side.
• The side adjacent to θ that is not the hypotenuse is the "adjacent" side.
• The longest side, opposite the right angle, is the "hypotenuse."
• sin(θ) = Opposite / Hypotenuse
• cos(θ) = Adjacent / Hypotenuse
• tan(θ) = Opposite / Adjacent
• An acronym to remember these ratios is SOH-CAH-TOA:
  • SOH: Sine = Opposite / Hypotenuse
  • CAH: Cosine = Adjacent / Hypotenuse
  • TOA: Tangent = Opposite / Adjacent

Reciprocal Trigonometric Functions

• Cosecant (csc), secant (sec), and cotangent (cot) are reciprocal functions.
• They are defined as follows:
  • csc(θ) = 1 / sin(θ) = Hypotenuse / Opposite
  • sec(θ) = 1 / cos(θ) = Hypotenuse / Adjacent
  • cot(θ) = 1 / tan(θ) = Adjacent / Opposite

Trigonometric Identities

• Trigonometric identities are equations that are true for all values of the variables involved.
• They are useful for simplifying expressions and solving equations.

Pythagorean Identity

• The most fundamental identity: sin²(θ) + cos²(θ) = 1
• Variations:
  • sin²(θ) = 1 - cos²(θ)
  • cos²(θ) = 1 - sin²(θ)
• Derived Identities:
  • 1 + tan²(θ) = sec²(θ)
  • 1 + cot²(θ) = csc²(θ)

Angle Sum and Difference Identities

• sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
• cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
• tan(A ± B) = (tan(A) ± tan(B)) / (1 ∓ tan(A)tan(B))

Double Angle Identities

• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ)
• tan(2θ) = (2tan(θ)) / (1 - tan²(θ))

Half Angle Identities

• sin(θ/2) = ±√((1 - cos(θ)) / 2)
• cos(θ/2) = ±√((1 + cos(θ)) / 2)
• tan(θ/2) = ±√((1 - cos(θ)) / (1 + cos(θ))) = sin(θ) / (1 + cos(θ)) = (1 - cos(θ)) / sin(θ)
• The sign (±) depends on the quadrant in which θ/2 lies.

Common Angles and Their Trigonometric Values

• Certain angles appear frequently, so it's beneficial to know their trigonometric values.
• These angles are often multiples of 0°, 30°, 45°, 60°, and 90°.

Values at Key Angles

• 0° (0 radians):
  • sin(0°) = 0
  • cos(0°) = 1
  • tan(0°) = 0
• 30° (π/6 radians):
  • sin(30°) = 1/2
  • cos(30°) = √3/2
  • tan(30°) = 1/√3 = √3/3
• 45° (π/4 radians):
  • sin(45°) = √2/2
  • cos(45°) = √2/2
  • tan(45°) = 1
• 60° (π/3 radians):
  • sin(60°) = √3/2
  • cos(60°) = 1/2
  • tan(60°) = √3
• 90° (π/2 radians):
  • sin(90°) = 1
  • cos(90°) = 0
  • tan(90°) = undefined

Unit Circle

• The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system.
• It provides a way to visualize trigonometric functions for all angles from 0 to 360 degrees (0 to 2π radians).
• Any point (x, y) on the unit circle can be represented as (cos(θ), sin(θ)), where θ is the angle formed by the positive x-axis and the line segment connecting the origin to the point.
• The x-coordinate corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine of the angle.
• tan(θ) can be visualized as the slope of the line segment.

Quadrantal Angles

• Angles that lie on the axes (0°, 90°, 180°, 270°, 360°) are called quadrantal angles.
• Their coordinates on the unit circle are:
  • 0°: (1, 0)
  • 90°: (0, 1)
  • 180°: (-1, 0)
  • 270°: (0, -1)
  • 360°: (1, 0)
• Using these coordinates, one can easily determine the sine, cosine, and tangent of these angles.

Law of Sines

• In any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
• a/sin(A) = b/sin(B) = c/sin(C)
• Here, a, b, and c are the side lengths, and A, B, and C are the opposite angles.
• The Law of Sines is used when you know:
  • Two angles and one side (AAS or ASA)
  • Two sides and an angle opposite one of them (SSA), which may lead to ambiguous cases.

Law of Cosines

• Relates the lengths of the sides of a triangle to the cosine of one of its angles.
• c² = a² + b² - 2ab cos(C)
• Variations:
  • a² = b² + c² - 2bc cos(A)
  • b² = a² + c² - 2ac cos(B)
• The Law of Cosines is useful when you know:
  • Three sides (SSS)
  • Two sides and the included angle (SAS)

Applications of Trigonometry

• Navigation: determining positions and directions.
• Physics: analyzing wave motion, forces, and fields.
• Engineering: designing structures, circuits, and mechanical systems.
• Astronomy: measuring distances to stars and planets.
• Surveying: measuring land and creating maps.