∿ Trigonometry
Angles, the unit circle, identities, waves, inverse functions, complex form and applications.
98 topics
- Angle Measure: Degrees (Beginner) – A full turn is 360°; angles measure rotation.
- Radians (Intermediate) – Angle measured by arc length on a unit circle; 2π rad = 360°.
- Converting Degrees and Radians (Intermediate) – Multiply by π/180 or 180/π.
- Right Triangle Terminology (Beginner) – Hypotenuse, opposite and adjacent sides relative to an angle.
- Sine Ratio (Intermediate) – Opposite over hypotenuse.
- Cosine Ratio (Intermediate) – Adjacent over hypotenuse.
- Tangent Ratio (Intermediate) – Opposite over adjacent.
- SOH-CAH-TOA (Beginner) – Memory aid for the three basic ratios.
- Reciprocal Trig Functions (Intermediate) – Cosecant, secant and cotangent.
- Solving Right Triangles (Intermediate) – Find all sides and angles using ratios and the Pythagorean theorem.
- Angles of Elevation and Depression (Intermediate) – Angles measured up or down from the horizontal.
- Special Angle Values (Intermediate) – Exact values for 30°, 45°, 60°.
- Unit Circle (Intermediate) – Circle of radius 1; point at angle θ is (cosθ, sinθ).
- Reference Angles (Intermediate) – Acute angle to the x-axis used to evaluate trig functions.
- Quadrant Signs (ASTC) (Intermediate) – Which functions are positive in each quadrant.
- Coterminal Angles (Intermediate) – Angles differing by full turns share terminal side.
- Graph of Sine (Intermediate) – A smooth wave with amplitude 1 and period 2π.
- Graph of Cosine (Intermediate) – Sine shifted left by π/2.
- Graph of Tangent (Advanced) – Periodic with vertical asymptotes at odd multiples of π/2.
- Amplitude (Intermediate) – Half the distance from peak to trough.
- Period and Frequency (Intermediate) – Length of one cycle and cycles per unit.
- Phase Shift (Advanced) – Horizontal translation of a wave.
- Vertical Shift and Midline (Advanced) – Moves the wave up or down.
- General Sinusoid (Advanced) – Combine amplitude, period, phase and shift.
- Graphs of Reciprocal Trig Functions (Advanced) – Secant, cosecant and cotangent graphs.
- Pythagorean Identity (Advanced) – The key identity from the unit circle.
- Other Pythagorean Identities (Advanced) – Divide the identity by sin² or cos².
- Quotient Identities (Advanced) – Tangent and cotangent in terms of sine and cosine.
- Cofunction Identities (Advanced) – Sine of an angle equals cosine of its complement.
- Even-Odd Identities (Advanced) – Cosine is even; sine and tangent are odd.
- Sum and Difference Formulas (Advanced) – Trig of a sum in terms of single angles.
- Double Angle Formulas (Advanced) – Trig of 2θ.
- Half Angle Formulas (Expert) – Trig of θ/2.
- Power Reducing Formulas (Expert) – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas (Expert) – Convert products into sums.
- Sum-to-Product Formulas (Expert) – Convert sums into products.
- Triple Angle Formulas (Expert) – sin3θ and cos3θ in terms of θ.
- Proving Trig Identities (Advanced) – Transform one side of an equation into the other.
- Inverse Sine (Advanced) – Angle whose sine is x, principal range [−π/2, π/2].
- Inverse Cosine (Advanced) – Angle whose cosine is x, range [0, π].
- Inverse Tangent (Advanced) – Angle whose tangent is x, range (−π/2, π/2).
- Trig and Inverse Trig Compositions (Expert) – Evaluate expressions like sin(arccos x).
- Basic Trig Equations (Advanced) – Solve sinθ = k using reference angles and periodicity.
- Trig Equations with Identities (Expert) – Use identities to factor trig equations.
- Law of Sines (Advanced) – Relates sides to opposite angles in any triangle.
- Ambiguous Case of Law of Sines (Advanced) – SSA can give zero, one or two triangles.
- Law of Cosines (Advanced) – Generalised Pythagorean theorem for any triangle.
- Area of Triangle with Sine (Advanced) – Area from two sides and the included angle.
- Bearings and Navigation (Advanced) – Compass angles used to solve distance problems.
- Vectors and Trigonometry (Advanced) – Components of a vector via sine and cosine.
- Resultant Forces (Advanced) – Add vectors using trigonometry.
- Polar Coordinates (Advanced) – Locate points with radius and angle.
- Polar–Rectangular Conversion (Advanced) – Switch between coordinate systems.
- Polar Graphs: Circles and Roses (Expert) – Curves like r = a cos nθ.
- Limaçons and Cardioids (Expert) – Curves like r = a + b cosθ.
- Complex Numbers in Polar Form (Expert) – Magnitude and argument.
- De Moivre’s Theorem (Expert) – Powers of complex numbers.
- Roots of Unity (Expert) – Solutions to zⁿ = 1 are equally spaced on the unit circle.
- Euler’s Formula (Expert) – Links exponentials and trigonometry.
- Euler’s Identity (Expert) – A famous special case.
- Hyperbolic Functions (Expert) – Analogues of trig functions defined with exponentials.
- Hyperbolic Identities (Expert) – cosh²x − sinh²x = 1.
- Inverse Hyperbolic Functions (Expert) – Logarithmic expressions for arsinh and arcosh.
- Fourier Series Introduction (Expert) – Any periodic function is a sum of sines and cosines.
- Simple Harmonic Motion (Advanced) – Oscillation modelled by sine or cosine.
- Sound Waves and Beats (Expert) – Sum of two nearby frequencies produces beats.
- Tides and Seasonal Modelling (Advanced) – Sinusoidal models of periodic data.
- Ferris Wheel Problems (Advanced) – Height as a sine function of time.
- Spherical Trigonometry Intro (Expert) – Triangles on a sphere and the spherical law of cosines.
- Trig Substitution Preview (Expert) – Using x = a sinθ to simplify radicals.
- Small Angle Approximation (Expert) – For small θ, sinθ ≈ θ and tanθ ≈ θ.
- Chebyshev Polynomials (Expert) – Polynomials such that Tₙ(cosθ) = cos(nθ).
- Trig Values via Pythagoras Triples (Intermediate) – Use 3-4-5 triangles to find exact trig ratios.
- Sine and Cosine Series (Expert) – Infinite series for sin and cos.
- Law of Tangents (Expert) – Relates differences and sums of sides and angles.
- Mollweide and Napier Formulas (Expert) – Alternative triangle-solving identities.
- Heron from Trigonometry (Expert) – Derive Heron’s formula using the law of cosines.
- Circumradius and Inradius (Expert) – Radii of triangle circles from trig.
- Trig Substitution in Algebra (Expert) – Use trig to solve algebraic inequalities.
- Sinusoidal Regression (Expert) – Fit a sine curve to periodic data.
- Phasors (Expert) – Represent sinusoids as rotating complex numbers in electrical engineering.
- Lissajous Curves (Expert) – Curves traced by two perpendicular oscillations.
- Parametric Unit Circle (Advanced) – x = cos t, y = sin t traces the unit circle.
- Trig Graph Transformations (Advanced) – Stretch, compress, reflect and shift trigonometric graphs.
- Solving Triangles in Surveying (Advanced) – Triangulation to measure inaccessible distances.
- Angle Between Vectors (Trig) (Advanced) – Using dot product and arccos.
- Rotation Matrices (Expert) – Rotate vectors using sine and cosine.
- Trigonometry in Physics (Advanced) – Projectile motion, inclined planes and optics.
- Snell’s Law (Expert) – Refraction of light uses sines of angles.
- Range of a Projectile (Advanced) – Horizontal distance depends on launch angle.
- Product of Trig Values (Expert) – Special products such as cos36° cos72° = 1/4.
- Exact Value of sin18° (Expert) – Derivation using pentagon properties.
- Haversine Formula (Expert) – Great-circle distance between points on Earth.
- Gudermannian Function (Expert) – Links trig and hyperbolic functions.
- Dirichlet Kernel (Preview) (Expert) – Sum of cosines used in Fourier analysis.
- Trig Sums and Telescoping (Expert) – Evaluate sums such as Σ sin kθ via telescoping.
- Contest Trigonometry (Expert) – Olympiad techniques using identities and complex numbers.
- Trig Review and Applications (Intermediate) – Mixed problems combining all key ideas.