Introduction to Trigonometry

• Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles
• Trigonometry is fundamental to fields like physics, engineering, astronomy, and navigation

Trigonometric Functions

• The primary trigonometric functions relate angles to ratios of sides of a right triangle
• The basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan)
• In a right triangle, the side opposite to the right angle is the hypotenuse
• For a given angle θ in a right triangle:
  • sin(θ) = (length of the side opposite to θ) / (length of the hypotenuse)
  • cos(θ) = (length of the side adjacent to θ) / (length of the hypotenuse)
  • tan(θ) = (length of the side opposite to θ) / (length of the side adjacent to θ)
• Cosecant (csc), secant (sec), and cotangent (cot) are reciprocal trigonometric functions:
  • csc(θ) = 1 / sin(θ)
  • sec(θ) = 1 / cos(θ)
  • cot(θ) = 1 / tan(θ)

Unit Circle

• The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system
• The unit circle provides a way to visualize and define trigonometric functions for all real numbers
• An angle θ is measured counterclockwise from the positive x-axis
• The coordinates of the point where the terminal side of angle θ intersects the unit circle are (cos(θ), sin(θ))

Trigonometric Identities

• Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined
• Pythagorean identities:
  • sin²(θ) + cos²(θ) = 1
  • 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)

Radians and Degrees

• Angles can be measured in degrees or radians
• A full circle is 360 degrees or 2π radians
• Conversion between degrees and radians:
  • radians = (degrees × π) / 180
  • degrees = (radians × 180) / π

Inverse Trigonometric Functions

• Inverse trigonometric functions (arcsin, arccos, arctan) find the angle given a trigonometric ratio
• arcsin(x) or sin⁻¹(x) is the inverse of sin(x), returning the angle whose sine is x, the range is [-π/2, π/2]
• arccos(x) or cos⁻¹(x) is the inverse of cos(x), returning the angle whose cosine is x, the range is [0, π]
• arctan(x) or tan⁻¹(x) is the inverse of tan(x), returning the angle whose tangent is x, the range is (-π/2, π/2)

Laws of Sines and Cosines

• These laws relate the sides and angles in any triangle (not just right triangles)
• Law of Sines:
  • a / sin(A) = b / sin(B) = c / sin(C), where a, b, c are side lengths and A, B, C are opposite angles
• Law of Cosines:
  • a² = b² + c² - 2bc × cos(A)
  • b² = a² + c² - 2ac × cos(B)
  • c² = a² + b² - 2ab × cos(C)

Applications

• Solving triangles requires finding unknown sides and angles given known information
• Navigation involves calculating distances and bearings
• Physics uses trigonometry to analyze projectile motion, wave phenomena, and oscillations
• Engineering applies trigonometry to design structures and mechanisms
• Astronomy uses trigonometry to determine distances to stars and planets

Graphs of Trigonometric Functions

• Sine function: y = sin(x)
  • Periodic with a period of 2π
  • Amplitude of 1
  • Range: [-1, 1]
• Cosine function: y = cos(x)
  • Periodic with a period of 2π
  • Amplitude of 1
  • Range: [-1, 1]
• Tangent function: y = tan(x)
  • Periodic with a period of π
  • Vertical asymptotes at x = (π/2) + nπ, where n is an integer
  • Range: (-∞, ∞)