Math explainers · 8 pages
Trigonometry explainers
Clear, citable explainers for the foundational concepts of trigonometry — unit circle, trig identities, law of sines, law of cosines, degrees to radians, and sin/cos/tan.
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All 8 trigonometry explainers pages
Unit circle
A unit circle is a mathematical tool with a radius of one, centred at the coordinate origin (0,0), used to extend the definitions of sine, cosine, and tangent to all real number angles. By placing an angle in the standard position, the coordinates of the intersection point on the circle represent the cosine (x-coordinate) and sine (y-coordinate) of that angle.
DefinitionTrigonometry identities
Trigonometric identities are equations relating trigonometric functions that remain true for any valid angle. The most fundamental is the Pythagorean identity, sin²(θ) + cos²(θ) = 1, derived from the unit circle. These identities are essential in calculus and physics for simplifying complex algebraic expressions, resolving integrals, and modelling periodic wave behaviour.
DefinitionLaw of cosines
The law of cosines states that for any triangle with sides a, b, and c and opposite angles A, B, and C, the relationship c² = a² + b² - 2ab cos(C) holds true. This law generalises the Pythagorean theorem to non-right triangles, allowing the calculation of an unknown side when two sides and the included angle are known, or any angle when all three sides are known.
DefinitionLaw of sines
The law of sines states that in any triangle, the ratio of a side length to the sine of its opposite angle is constant: a / sin(A) = b / sin(B) = c / sin(C). This law is used to solve triangles when we know either two angles and one side (ASA or AAS) or two sides and a non-included angle (SSA).
DefinitionSin, Cos, and Tan
Sine (sin), cosine (cos), and tangent (tan) are fundamental trigonometric functions representing ratio relationships in a right-angled triangle. Sine is the opposite side divided by the hypotenuse, cosine is the adjacent side divided by the hypotenuse, and tangent is the opposite side divided by the adjacent side. Together, they form the basis of geometric trigonometry and wave analysis.
DefinitionDegrees to radians
To convert degrees to radians, multiply the angle in degrees by π and divide by 180 (Radians = Degrees × π / 180). This conversion is necessary because degrees are an arbitrary division of a circle into 360 parts, whereas radians measure angles based on the radius of the circle, which is the standard unit in calculus.
DefinitionPi
Pi (π) is an irrational mathematical constant defined as the ratio of any circle's circumference to its diameter. It has a value of approximately 3.14159, with an infinite, non-repeating decimal expansion. Pi is a transcendental number, meaning it is not the root of any non-zero polynomial with rational coefficients, and it appears in countless formulas across geometry, trigonometry, and physics.
DefinitionTrigonometry
Trigonometry, derived from the Greek words "trigonon" (triangle) and "metron" (measure), is the study of how the angles and sides of triangles relate to one another. It spans from basic right-angled triangle ratios (sine, cosine, tangent) to periodic wave functions on the unit circle, serving as an essential tool in navigation, physics, engineering, and acoustics.
