Introduction to Trigonometry

• Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles

Basic Trigonometric Functions

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

Right Triangles

• A right triangle has one angle of 90 degrees
• The side opposite the right angle is the hypotenuse, the longest side of the triangle
• The other two sides are the adjacent and opposite sides, defined relative to a specific angle

Sine Function

• sin(θ) = Opposite / Hypotenuse
• Where θ is the angle, "Opposite" is the length of the side opposite to the angle θ, and "Hypotenuse" is the length of the hypotenuse

Cosine Function

• cos(θ) = Adjacent / Hypotenuse
• Where "Adjacent" is the length of the side adjacent to the angle θ

Tangent Function

• tan(θ) = Opposite / Adjacent
• tan(θ) can also be expressed as sin(θ) / cos(θ)

Reciprocal Trigonometric Functions

• Cosecant (csc), secant (sec), and cotangent (cot) are the reciprocals of sine, cosine, and tangent, respectively

Cosecant Function

• csc(θ) = 1 / sin(θ) = Hypotenuse / Opposite

Secant Function

• sec(θ) = 1 / cos(θ) = Hypotenuse / Adjacent

Cotangent Function

• cot(θ) = 1 / tan(θ) = Adjacent / Opposite
• cot(θ) can also be expressed as cos(θ) / sin(θ)

Trigonometric Identities

• These are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined
• Pythagorean identities, angle sum and difference identities, and double-angle identities are examples of trigonometric identities

Pythagorean Identity

• sin²(θ) + cos²(θ) = 1
• This identity is derived from the Pythagorean theorem

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))

Unit Circle

• The unit circle is a circle with a radius of 1 centered at the origin in the Cartesian coordinate system
• It is used to define trigonometric functions for all real numbers

Angles in Standard Position

• An angle in standard position has its vertex at the origin and its initial side along the positive x-axis
• Angles are measured counterclockwise from the initial side

Coordinates on the Unit Circle

• For any point (x, y) on the unit circle corresponding to an angle θ, x = cos(θ) and y = sin(θ)
• The value of tan(θ) is given by y/x

Quadrantal Angles

• Quadrantal angles are angles that are integer multiples of 90 degrees (0, 90, 180, 270, and 360 degrees)
• The trigonometric functions of quadrantal angles have values of 0, 1, or -1

Trigonometric Values for Common Angles

• Common angles include 0°, 30°, 45°, 60°, and 90°
• sin(0°) = 0, sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1
• cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, cos(90°) = 0
• tan(0°) = 0, tan(30°) = √3/3, tan(45°) = 1, tan(60°) = √3, tan(90°) = undefined

Inverse Trigonometric Functions

• These functions return the angle whose sine, cosine, or tangent is a given number
• Arcsine (asin or sin⁻¹), arccosine (acos or cos⁻¹), and arctangent (atan or tan⁻¹) are the inverse trigonometric functions

Arcsine Function

• asin(x) or sin⁻¹(x) returns the angle whose sine is x
• The domain of arcsine is [-1, 1], and the range is [-π/2, π/2]

Arccosine Function

• acos(x) or cos⁻¹(x) returns the angle whose cosine is x
• The domain of arccosine is [-1, 1], and the range is [0, π]

Arctangent Function

• atan(x) or tan⁻¹(x) returns the angle whose tangent is x
• The domain of arctangent is all real numbers, and the range is (-π/2, π/2)

Applications of Trigonometry

• Solving triangles involves finding unknown angles and sides of a triangle using trigonometric functions
• Navigation, surveying, and physics utilize trigonometric principles

Law of Sines

• a / sin(A) = b / sin(B) = c / sin(C)
• Relates the lengths of the sides of a triangle to the sines of its angles

Law of Cosines

• c² = a² + b² - 2ab cos(C)
• Relates the lengths of the sides of a triangle to the cosine of one of its angles

Bearings and Directions

• Bearings are used to specify directions, often measured clockwise from the north
• Trigonometry is used to calculate distances and directions in navigation

Simple Harmonic Motion

• Trigonometric functions are used to model simple harmonic motion, such as the motion of a pendulum or a mass on a spring
• The position of the object can be described as a function of time using sine and cosine functions

Wave Phenomena

• Trigonometry is used to analyze wave phenomena, such as sound waves and electromagnetic waves
• Sine and cosine functions describe the amplitude, frequency, and phase of waves