Right Triangle Trigonometry

• Trigonometry studies the relationship between triangle side lengths and their angles, especially in right triangles

Trigonometric Functions

• Trigonometric functions establish a link between a triangle's angles and its side lengths
• Sine (sin), cosine (cos), and tangent (tan) form the primary trigonometric functions

Right Triangle Definitions

• Right triangles contain an angle θ, which is used to define the sides
• Opposite side is across from θ
• Adjacent side is next to θ
• The hypotenuse is opposite the right angle and is the triangle's longest side
• The sine of θ is the ratio of the opposite side to the hypotenuse: sin(θ) = Opposite / Hypotenuse
• The cosine of θ is the ratio of the adjacent side to the hypotenuse: cos(θ) = Adjacent / Hypotenuse
• The tangent of θ is the ratio of the opposite side to the adjacent side: tan(θ) = Opposite / Adjacent
• The size of the triangle does not affect these ratios, only the angle θ does

Reciprocal Trigonometric Functions

• Cosecant (csc), secant (sec), and cotangent (cot) are the respective reciprocals of sine, cosine, and tangent
• Cosecant of θ = csc(θ) = 1 / sin(θ) = Hypotenuse / Opposite
• Secant of θ = sec(θ) = 1 / cos(θ) = Hypotenuse / Adjacent
• Cotangent of θ = cot(θ) = 1 / tan(θ) = Adjacent / Opposite

Pythagorean Theorem

• For right triangles, the Pythagorean Theorem is a² + b² = c², with 'a' and 'b' as the legs, and 'c' the hypotenuse

Trigonometric Identities

• Trigonometric identities act as equations involving trigonometric functions, which hold true for all angles
• The core Pythagorean Identity: sin²(θ) + cos²(θ) = 1
• Derived from the Pythagorean identity: tan²(θ) + 1 = sec²(θ)
• Another identity derived from the Pythagorean identity: 1 + cot²(θ) = csc²(θ)
• Sine is an odd function: sin(-θ) = -sin(θ)
• Cosine is an even function: cos(-θ) = cos(θ)
• Tangent is an odd function: tan(-θ) = -tan(θ)

Angle Sum and Difference Identities

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
• sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
• cos(A + B) = cos(A)cos(B) - sin(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))
• 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 of the half-angle identities depends on the quadrant of θ/2

Law of Sines

• The ratio of side length to the sine of its opposite angle remains constant: a / sin(A) = b / sin(B) = c / sin(C)
• Useful for solving triangles with two angles and a side (AAS or ASA), or two sides and a non-included angle (SSA)

Law of Cosines

• Law of Cosines relates sides and angles in any triangle:
• a² = b² + c² - 2bc * cos(A)
• b² = a² + c² - 2ac * cos(B)
• c² = a² + b² - 2ab * cos(C)
• Used to solve triangles with three sides (SSS) or two sides and an included angle (SAS)

Unit Circle

• A circle centered at the origin (0,0) with a radius of 1
• Coordinates on the unit circle are (cos(θ), sin(θ)) for a given angle θ
• It helps to visualize trigonometric functions for angles, particularly past 90° (π/2 radians)
• Trigonometric functions' values are provided by the unit circle at key angles like 0, π/6, π/4, π/3, π/2, π, 3π/2, and 2π

Radians

• Radians serve as a unit for measuring angles
• One radian is the angle created by an arc length equal to the circle's radius
• π radians are equivalent to 180 degrees
• Conversion from degrees to radians: multiply by π/180
• Conversion from radians to degrees: multiply by 180/π

Trigonometric Equations

• Solving trigonometric equations involves finding angles that satisfy the equation
• Use trigonometric identities and algebraic methods to isolate functions when solving
• Solutions must account for the periodic nature of trigonometric functions

Inverse Trigonometric Functions

• Arcsin, arccos, and arctan yield angles from given sine, cosine, or tangent values
• arcsin(x) or sin⁻¹(x) outputs an angle with sine x, in the range [-π/2, π/2]
• arccos(x) or cos⁻¹(x) outputs an angle with cosine x, in the range [0, π]
• arctan(x) or tan⁻¹(x) outputs an angle with tangent x, in the range (-π/2, π/2)

Graphs of Trigonometric Functions

• sin(x): Period is 2π, amplitude is 1, range is [-1, 1]
• cos(x): Period is 2π, amplitude is 1, range is [-1, 1]
• tan(x): Period is π, has vertical asymptotes, range is (-∞, ∞)
• Graph transformations include amplitude and period alterations, along with phase and vertical shifts

Applications of Trigonometry

• Essential for navigation, helping determine position and direction
• Used in Physics to describe wave motion, projectile motion and oscillations
• Used in Engineering for structural design and force analysis
• Used in Surveying for measurements
• Used in Computer Graphics for animations and 3d modelling

Understand the basics of right triangle trigonometry. This includes trigonometric functions such as sine, cosine and tangent. Also learn about reciprocal trigonometric functions.