My child knows SOH-CAH-TOA but is lost on the unit circle. Where does the gap start?

Almost always at the idea that cosine and sine are the x- and y-coordinates of a point, not just triangle ratios. Sessions rebuild that link by placing the familiar 30-60-90 triangle inside the circle in each quadrant, so the signs and the exact values come from one picture rather than from a memorised chart.

Do we work in degrees or radians?

Both, but Grade 11-12 work is mostly radians because graphing and later calculus require it. The tutor will ask which mode the calculator is in and will convert with you until π/3 reads as 60° without effort.

Are trig identity proofs actually covered, or just calculations?

Proofs are covered. You work one side at a time, out loud, choosing between converting everything to sine and cosine, using a Pythagorean identity, or multiplying by a conjugate — and the tutor asks you to justify the choice before writing the next line.

How is this different from the Pre-Calculus topic?

Pre-Calculus sessions cover functions broadly — polynomial, rational, exponential, sequences, limits at the edge. This topic stays inside trigonometry: the unit circle, periodic graphs, identities, trig equations, and oblique triangles.

Trigonometry

High School (11-12) · Mathematics

Trigonometry at the Grade 11-12 level moves past right-triangle SOH-CAH-TOA into the unit circle, radian measure, periodic graphs, identities, and equations that have infinitely many solutions. This is where most students first meet a function that repeats, and where an answer like "x = π/6 + 2πn or 5π/6 + 2πn" is expected instead of a single number. Sessions are spoken one-on-one with an AI tutor: you talk through where a solution came from, why a second angle exists, and which identity actually simplifies the expression in front of you.

Start a session on Trigonometry

What this covers

Radian measure and the unit circle: converting degrees to radians, exact values for multiples of π/6 and π/4, and reading sine, cosine and tangent as coordinates and slope rather than memorised table entries
Graphing y = a·sin(b(x − h)) + k and its cosine and tangent counterparts — amplitude, period 2π/b, phase shift, midline — and reading these back out of a graph or a word problem such as tides or Ferris wheels
Proving and applying identities: Pythagorean, reciprocal, quotient, sum and difference, double-angle and half-angle formulas, including simplifying one side of an identity to match the other
Solving trigonometric equations on a restricted domain such as [0, 2π) and writing general solutions with +2πn or +πn, including equations needing factoring or a quadratic substitution in sin x
Law of Sines and Law of Cosines for non-right triangles, area formulas, and recognising the SSA ambiguous case where zero, one or two triangles exist
Inverse trigonometric functions: restricted ranges of arcsin, arccos and arctan, and why the calculator returns only one of several valid angles

Where learners get stuck

Taking the calculator's inverse-sine output as the only answer and losing the second solution in the interval: Every other equation students have solved has had one answer per branch. Because arcsin is defined with a restricted range so it can be a function, the calculator is built to hide the other angle — students need the habit of asking which quadrants make the ratio positive or negative and reflecting the reference angle into each one.
Treating sin(A + B) as sin A + sin B, or sin(2x) as 2sin x: Function notation looks like multiplication, so the distributive law gets applied by reflex. The fix is checking a concrete case (sin 90° versus sin 30° + sin 60°) once, out loud, so the sum and difference formulas feel necessary rather than arbitrary.
Confusing the coefficient b in sin(bx) with the period, and mixing up degree and radian mode mid-problem: Students learn that b stretches the graph but expect the period to equal b rather than 2π/b — an inverse relationship that runs against intuition. Mode errors compound this because a graph that looks nothing like expected gets blamed on algebra instead of settings.
Applying the Law of Sines to SSA information and stopping at the first triangle: The supplement of an angle has the same sine, so the equation genuinely has two solutions; students trained to find one answer rarely check whether the second angle still leaves a positive third angle.

What a session looks like

A typical 25-40 minute session starts with a problem you are stuck on — a homework identity, a graph you have to sketch, or an equation with a domain restriction. You describe your working aloud and the tutor questions the step where it breaks: which quadrant, which identity, what the reference angle is. Expect to be asked to state the unit circle values you are using and to justify why a solution set is complete. Diagrams and equations can be shared and worked through together, and sessions usually end with one problem you solve unaided while narrating each step.

Helpful to know first

Right-triangle trigonometry: SOH-CAH-TOA, the Pythagorean theorem, and special right triangles (30-60-90, 45-45-90)
Algebra 2 skills: factoring quadratics, solving quadratic equations, and manipulating rational expressions with common denominators
Function transformations — vertical and horizontal shifts, stretches and reflections — applied to any parent function
Comfort with the coordinate plane, including signs of x and y in each of the four quadrants

Questions

My child knows SOH-CAH-TOA but is lost on the unit circle. Where does the gap start?: Almost always at the idea that cosine and sine are the x- and y-coordinates of a point, not just triangle ratios. Sessions rebuild that link by placing the familiar 30-60-90 triangle inside the circle in each quadrant, so the signs and the exact values come from one picture rather than from a memorised chart.
Do we work in degrees or radians?: Both, but Grade 11-12 work is mostly radians because graphing and later calculus require it. The tutor will ask which mode the calculator is in and will convert with you until π/3 reads as 60° without effort.
Are trig identity proofs actually covered, or just calculations?: Proofs are covered. You work one side at a time, out loud, choosing between converting everything to sine and cosine, using a Pythagorean identity, or multiplying by a conjugate — and the tutor asks you to justify the choice before writing the next line.
How is this different from the Pre-Calculus topic?: Pre-Calculus sessions cover functions broadly — polynomial, rational, exponential, sequences, limits at the edge. This topic stays inside trigonometry: the unit circle, periodic graphs, identities, trig equations, and oblique triangles.