Introduction to Trigonometry

Study Notes

Trigonometry is the study of the relationships between the sides and angles of triangles.
Trigonometry is foundational for navigation, surveying, engineering, and physics.

Trigonometric functions correlate the angles of a right triangle to the ratios of its sides.
Sine (sin), cosine (cos), and tangent (tan) are considered primary trigonometric functions.
For acute angles in a right triangle these functions are defined by:
- sin(θ) = Opposite / Hypotenuse
- cos(θ) = Adjacent / Hypotenuse
- tan(θ) = Opposite / Adjacent
Cosecant (csc), secant (sec), and cotangent (cot) as reciprocal trigonometric functions.
- csc(θ) = 1 / sin(θ) = Hypotenuse / Opposite
- sec(θ) = 1 / cos(θ) = Hypotenuse / Adjacent
- cot(θ) = 1 / tan(θ) = Adjacent / Opposite

A unit circle has a radius of 1 and is centered at the origin (0,0) on the Cartesian coordinate system.
The unit circle extends trigonometric function definitions to all real numbers.
For any angle θ, the point where the terminal side of the angle intersects the unit circle has coordinates (cos θ, sin θ).

Trigonometric identities include trigonometric functions and are true for all variable values where 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) = sin(θ) / (1 + cos(θ)) = (1 - cos(θ)) / sin(θ)

Angles can be measured using degrees or radians.
A full circle measures 360 degrees or 2π radians.
Equations converting between degrees and radians:
- radians = (degrees × π) / 180
- degrees = (radians × 180) / π

Trigonometric equations contain trigonometric functions.
Solving these equations requires finding variable values that satisfy the equation.
General solutions use trigonometric functions' periodicity to find all possible solutions.
The equation sin(x) = a has solutions x = arcsin(a) + 2πk or x = π - arcsin(a) + 2πk, where k is an integer.

Inverse trigonometric functions are the inverse functions of trigonometric functions.
Inverse trigonometric functions can also be called arcus functions.
The common inverse trigonometric functions:
- arcsin(x) or sin⁻¹(x): the inverse of sine
- arccos(x) or cos⁻¹(x): the inverse of cosine
- arctan(x) or tan⁻¹(x): the inverse of tangent
Domains and ranges of inverse trigonometric functions are restricted to pass the vertical line test.

The Law of Sines relates triangle side lengths to the sines of their opposite angles.
a / sin(A) = b / sin(B) = c / sin(C), where a, b, c are the sides and A, B, C are the opposite angles, according to the Law of Sines.
The Law of Cosines relates triangle side lengths to the cosine of one of its angles.
c² = a² + b² - 2ab cos(C), where a, b, c are the sides and C is the angle opposite side c, according to the Law of Cosines.
The Law of Sines is applicable when two angles and a side (AAS or ASA) are known or when two sides and an angle opposite one of them (SSA) are known; the SSA case may yield zero, one, or two triangles.
The Law of Cosines is applicable when three sides (SSS) are known or when two sides and the included angle (SAS) are known.

Sine and Cosine:
- sin(x) and cos(x) are periodic functions that have a period of 2π.
- Both sine and cosine functions have an amplitude of 1.
- Sine and cosine functions both range from [-1, 1].
Tangent:
- tan(x) is a periodic function, repeating every π.
- Vertical asymptotes exist on a tangent graph at x = (π/2) + kπ, where k is an integer.
- Tangent functions range from (-∞, ∞).
Cosecant, Secant, and Cotangent:
- Cosecant, secant, and cotangent are reciprocal functions of sine, cosine, and tangent, respectively.
- Graphs of these functions have vertical asymptotes where their corresponding reciprocal functions equal zero.
- These functions are also periodic.