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: (-∞, ∞)

Explore the basics of trigonometry, which studies relationships between angles and sides of triangles. Learn about trigonometric functions such as sine, cosine, and tangent, and their mathematical formulas. Understand the unit circle and its applications in trigonometry.