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Trigonometry tutoring online

Trigonometry is where many strong algebra students first feel lost. The ratios look simple in a triangle, then the angles grow past 90 degrees, switch to radians, wrap around a circle and turn into waves. The trouble is rarely the arithmetic. It is that the same idea is being shown in three different pictures, and nobody has joined them up. Our tutors teach trigonometry as one idea seen three ways, the ratio in a triangle, the point on a unit circle and the graph of a function, so each new unit builds on the last instead of starting again.

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In short

Trigonometry in US high schools is taught in two places: right-triangle ratios in Geometry, under the Common Core standards G-SRT, and trigonometric functions on the unit circle in Algebra 2 or Precalculus, under F-TF, which AP Precalculus covers in Unit 3. Modern Age Coders teaches trigonometry live online, in a group of 15 to 20 or one to one, through our High School Mathematics course, with AP Precalculus preparation for exam students. The first class is free.

Where to start

Three courses that teach trigonometry properly

Choose by goal: the school course, the AP exam or the SAT.

High School Mathematics course thumbnail

TT / 01

High school math, grades 9 to 12

Right triangles in Geometry, the unit circle in Algebra 2 and Precalculus.

Open the syllabus →
AP Precalculus course thumbnail

TT / 02

AP Precalculus

Units 1 to 3, including trigonometric and polar functions.

Open the syllabus →
SAT Math 800 Prep course thumbnail

TT / 03

SAT Math

Right triangles and trigonometry within the geometry domain.

Open the syllabus →

Where trig is taught

Which high school course teaches trigonometry?

Usually two of them. The Common Core high school standards put right-triangle trigonometry in the geometry category and the trigonometric functions in the functions category, so most students meet sine twice: once in Geometry and again in Algebra 2 or Precalculus.

Trigonometry in the Common Core high school standards (thecorestandards.org, read 31 October 2026; summaries ours, quoted words in quotation marks)
CodeStrandWhat the standard asksMarked (+)?
G-SRT.C.6GeometryBy similarity, "side ratios in right triangles are properties of the angles in the triangle", which defines the trig ratiosNo
G-SRT.C.7GeometryExplain and use the link between sine and cosine of complementary anglesNo
G-SRT.C.8GeometrySolve right triangles "in applied problems" with trig ratios and the Pythagorean TheoremNo
G-SRT.D.9 to D.11GeometryArea as 1/2 ab sin(C); prove and use the Laws of Sines and CosinesYes
F-TF.A.1 and A.2FunctionsRadian measure; the unit circle extends trig functions "to all real numbers"No
F-TF.A.3 and A.4FunctionsExact values for pi/3, pi/4 and pi/6; symmetry and periodicityYes
F-TF.B.5FunctionsModel periodic situations "with specified amplitude, frequency, and midline"No
F-TF.B.6 and B.7FunctionsInverse trig functions and solving trig equations in contextYes
F-TF.C.8Functions"Prove the Pythagorean identity" and use it to find one ratio from anotherNo
F-TF.C.9FunctionsProve and use the addition and subtraction formulasYes

What the (+) means

The standards document explains the mark on page 57: "Additional mathematics that students should learn in order to take advanced courses such as calculus, advanced statistics, or discrete mathematics is indicated by (+)". Standards without it are meant for every college-ready student.

In practice, a student heading for calculus needs the (+) items too: exact values from special triangles, inverse functions, the Laws of Sines and Cosines and the addition formulas. That is where most of our trigonometry tutoring hours go.

Why triangles come first

The geometry introduction says "The definitions of sine, cosine, and tangent for acute angles are founded on right triangles and similarity". Two similar right triangles share the same angle and the same ratios, whatever their size, and that is why sin 35° is a single number.

A student who has seen that argument does not need SOH CAH TOA as a spell. They know the ratio is fixed by the angle, and the memory aid becomes a label for something they already understand.

Sources: Similarity, Right Triangles, and Trigonometry, Trigonometric Functions and the high school geometry introduction at thecorestandards.org, and the standards PDF, read 31 October 2026. Some states use their own standards with different codes. We are not connected with the Common Core initiative, the College Board or any state or district.

Right triangles

How do you solve a right triangle with trigonometry?

Name the angle you know, label the sides from that angle (opposite, adjacent, hypotenuse), pick the one ratio that links what you have to what you want, and solve. Here are four of our own examples at Geometry level.

Right-triangle examples (ours; answers rounded to two decimal places)
ProblemSides from the angleRatio chosenWorking and answer
A 20-foot ladder leans on a wall at 70° to the ground. How high does it reach?Hypotenuse 20, want oppositeSineheight = 20 sin 70° = 18.79 feet
A tree's shadow is 30 feet long when the sun is 35° above the horizon. How tall is the tree?Adjacent 30, want oppositeTangentheight = 30 tan 35° = 21.01 feet
A right triangle has legs 8 and 15. Find the smaller acute angle.Opposite 8, adjacent 15Inverse tangentangle = tan-1(8/15) = 28.07°; hypotenuse 17
Show that sin 25° equals cos 65°.The same triangle, read from each acute angleComplementary angles (G-SRT.C.7)Both are 0.4226, because the side opposite 25° is adjacent to 65°

The step students skip

Labeling the sides from the chosen angle is the step most errors come from. "Opposite" and "adjacent" are not fixed names for sides; they change when you look from the other acute angle. In the 8-15-17 triangle, the side of length 8 is opposite one angle and adjacent to the other.

We have students mark the angle with a dot and write O, A and H on the sides every time, until they no longer need to. It takes ten seconds and removes most wrong-ratio mistakes.

Check with a second fact

Every right-triangle answer can be checked another way. For the ladder, the foot of the ladder is 20 cos 70° = 6.84 feet from the wall, and 6.84 squared plus 18.79 squared comes to about 400, which is 20 squared.

For the 8-15-17 triangle, the two acute angles must add to 90°, so the other angle is 61.93°. Small checks like these are what make a student confident on a test without anyone marking the work.

The unit circle

How do you learn the unit circle without memorizing it?

Learn three triangles and four sign rules, then rebuild any point. Standard F-TF.A.1 defines a radian as "the length of the arc on the unit circle subtended by the angle", so the circle and radians arrive together.

Radians as a length

On a circle of radius 1, an angle of 1 radian cuts off an arc of length 1. A full turn is the whole circumference, 2π, so 360° equals 2π radians and 180° equals π. Every conversion follows from that one line.

Our example: 225° is 225/180 of π, which is 5π/4. It is π/4 past the half turn, so it sits in quadrant III, with reference angle 45°.

Rebuilding a point

The 45° triangle on the unit circle has both legs √2/2. In quadrant III both coordinates are negative, so the point at 5π/4 is (−√2/2, −√2/2). Cosine is the x coordinate and sine is the y coordinate, so both equal −√2/2 and the tangent is 1.

The same three steps, reference angle, triangle, signs, give every special value. Students who learn the steps can answer for 7π/6 or 5π/3 just as quickly as for angles they drilled.

Signs of sine, cosine and tangent by quadrant (ours, from the coordinates of the unit circle)
QuadrantAngles in radianssin (y)cos (x)tan (y/x)
I0 to π/2PositivePositivePositive
IIπ/2 to πPositiveNegativeNegative
IIIπ to 3π/2NegativeNegativePositive
IV3π/2 to 2πNegativePositiveNegative

Our identity example for F-TF.C.8: if sin θ = −3/5 and θ is in quadrant III, the Pythagorean identity sin2θ + cos2θ = 1 gives cos2θ = 1 − 9/25 = 16/25, so cos θ = ±4/5. Quadrant III makes cosine negative, so cos θ = −4/5, and tan θ = (−3/5)/(−4/5) = 3/4. The identity gives the size; the quadrant gives the sign.

Graphs and models

How are trigonometric functions used to model real situations?

Anything that repeats can be modeled with a sine or cosine. Standard F-TF.B.5 asks students to choose a trig function for a situation with a given amplitude, frequency and midline. Our Ferris wheel example shows how each number in the equation comes from the story.

A Ferris wheel modeled with cosine (our example): diameter 100 feet, center 60 feet above the ground, one turn every 8 minutes, rider starts at the bottom
FeatureFrom the storyIn h(t) = 60 − 50 cos(πt/4)
MidlineHeight of the center60
AmplitudeRadius, half the diameter50
PeriodOne turn every 8 minutes2π divided by π/4 is 8
Starting pointRider begins at the lowest pointMinus cosine starts at its minimum: h(0) = 10
CheckTop of the wheel after half a turnh(4) = 60 + 50 = 110

Solving with the model

When is the rider 85 feet up? Set 60 − 50 cos(πt/4) = 85, so cos(πt/4) = −1/2. On one turn, πt/4 = 2π/3 or 4π/3, giving t = 8/3 and t = 16/3 minutes, about 2.67 and 5.33 minutes.

Two answers in one turn is the point. The rider passes 85 feet on the way up and again on the way down, and a student who stops at the first answer has missed half the question.

The same idea without a story

Our equation example: solve 2 sin x − 1 = 0 for 0 ≤ x < 2π. Then sin x = 1/2. The reference angle is π/6, and sine is positive in quadrants I and II, so x = π/6 or x = 5π/6.

A calculator's inverse sine gives only π/6. Standard F-TF.B.6 is about exactly that: inverse functions exist only on a restricted domain, so the other solutions come from the circle, not the button.

check_trig.py (our example)Python 3
import math

# Ferris wheel: h(t) = 60 - 50 cos(pi t / 4)
h = lambda t: 60 - 50 * math.cos(math.pi * t / 4)
print([round(h(t), 2) for t in (0, 2, 4, 8/3, 16/3)])

# Law of Cosines: sides 7 and 9 with a 60 degree angle between them
c = math.sqrt(7**2 + 9**2 - 2 * 7 * 9 * math.cos(math.radians(60)))
area = 0.5 * 7 * 9 * math.sin(math.radians(60))
print(round(c, 2), round(area, 2))

Output: [10.0, 60.0, 110.0, 85.0, 85.0], then 8.19 27.28. The third side is √67 because 49 + 81 − 63 = 67, and the area uses 1/2 ab sin(C) from G-SRT.D.9. Checking a hand answer with three lines of code is a habit we teach, never a replacement for the working.

AP and SAT

How much trigonometry is on AP Precalculus and the SAT?

A lot on AP Precalculus, a little on the SAT. In the College Board's AP Precalculus Course and Exam Description, Unit 3, Trigonometric and Polar Functions, carries 30 to 35 percent of the multiple-choice section. On the SAT, trigonometry sits inside the Geometry and Trigonometry domain.

The AP Precalculus Exam, as listed on AP Central for the May 2027 exam (read 31 October 2026)
SectionQuestionsTimeShare of scoreCalculator
I, Part A: multiple choice2965 minutes43.75%Not permitted
I, Part B: multiple choice1340 minutes18.75%Graphing calculator required
II, Part A: free response235 minutes18.75%Graphing calculator required
II, Part B: free response235 minutes18.75%Not permitted

What Unit 3 contains

The Course and Exam Description lists 15 topics in Unit 3, from 3.1 Periodic Phenomena through sine, cosine and tangent, sinusoidal transformations and modeling, the tangent function, inverse trigonometric functions, equations and inequalities, secant, cosecant and cotangent, equivalent representations, to 3.13 to 3.15 on polar coordinates, polar graphs and rates of change in polar functions.

AP Central lists the 2027 exam for Tue, May 11, 2027, and names Free-Response Question 3 as Modeling a Periodic Context, which is the Ferris wheel kind of problem above. Unit 4 is "Not assessed on the AP Exam".

Trigonometry on the SAT

The College Board's SAT Suite page lists four topics in its Geometry and Trigonometry domain, one of which is "Right triangles and trigonometry". Expect right-triangle ratios, complementary angles and radian conversions rather than identities or graphs.

For SAT students we keep trigonometry tight: the right-triangle work in the second section of this page, plus degrees and radians, practiced in the digital format alongside the rest of the math section.

Sources: the AP Precalculus exam page and the AP Precalculus Course and Exam Description on AP Central, and the SAT Geometry and Trigonometry page, read 31 October 2026. AP and SAT are College Board trademarks; we are not affiliated with the College Board. For official practice, use the free questions the College Board publishes.

A plan that works

What is the fastest way to learn trigonometry for beginners?

Learn it in the order the ideas depend on each other, and do not move on until the previous step is automatic. This is the order our tutors use for a student starting from scratch.

  1. Similar triangles and the three ratios. Why a ratio depends only on the angle; labeling sides from an angle.
  2. Solving right triangles. Missing sides, then missing angles with inverse functions; angles of elevation and depression.
  3. Special triangles. The 30-60-90 and 45-45-90 triangles, built from a square and an equilateral triangle, not memorized.
  4. Radians and the unit circle. Arc length, conversions, reference angles and signs by quadrant.
  5. Graphs. Sine and cosine as the unit circle unrolled; amplitude, period, midline and phase shift.
  6. Equations and identities. The Pythagorean identity first, then solving on an interval, then addition formulas.
  7. Non-right triangles. The Laws of Sines and Cosines, including the ambiguous case.
The trigonometry mistakes we correct most often (ours)
MistakeWhat it looks likeThe fix we teach
Calculator in the wrong modesin 30 gives −0.99 instead of 0.5Check the mode before every test; know sin 30° = 1/2 as a sense check
Wrong ratioUses cosine for the opposite sideMark the angle, label O, A, H, then choose
Sign lostcos 5π/4 written as positiveFind the quadrant before the value
One solution onlyStops at x = π/6Ask which other quadrant shares the sign
Ambiguous case missedOne triangle found where two existSketch both possible triangles for side-side-angle

The last row is in the standards too. The geometry introduction notes that the Laws of Sines and Cosines "yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angle is not a congruence criterion." A quick sketch of both triangles is the habit that catches it.

Where the student is

Four levels of trigonometry understanding

Our way of placing a student at the first class, so lessons start at the right point.

Levels we use when placing a trigonometry student (ours)
LevelCan doNext step
1. RatiosSolves a right triangle with a calculatorSpecial triangles and exact values
2. CircleFinds exact values on the unit circleGraphs and transformations
3. FunctionsGraphs and models with sine and cosineEquations on an interval, identities
4. FluentProves identities, uses the Laws of Sines and CosinesPolar functions, AP free response, calculus

For parents

A student who is fine at level 1 can still fall apart at level 2, because the unit circle is where memorizing stops working. That is the most common time families contact us.

Ask which course and unit your teen is in. It tells us whether to start with triangles or with the circle.

For students

If you cannot rebuild a value from a triangle, you do not know it yet, even if you remember it.

Draw the picture first. Every trig question has one.

The catalogue

Nine courses around trigonometry

What comes before trig, what includes it, and where it leads.

I

Before trigonometry

Grades 8 to 10

TT / B / 02

Middle school math

Similarity and the Pythagorean Theorem.

Open the syllabus
II

With trigonometry

Grades 10 to 12
III

After trigonometry

Grade 11 and up

TT / A / 01

AP Calculus AB and BC

Derivatives of sine and cosine.

Open the syllabus

How lessons run

Live trigonometry lessons that join the pictures up

Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays.

After school

Lessons that track the current unit.

Weekends

Longer sessions before a test.

1-on-1

Paced to your course or AP timeline.

Triangle, circle, graph

Every topic shown in all three pictures.

Exact before decimal

Special values rebuilt, not looked up.

Both calculator modes

Degrees and radians checked every lesson.

Your own work

We teach the method; graded work stays yours.

Exam aware

AP and SAT formats when they apply.

Batch or private

A class of 15 to 20, or one to one.

Student work

What our students build

Some students animate sine waves or draw polar roses in code. See them on student labs.

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Fees

Fees

Trigonometry lessons are billed month by month with no long contract, so a student can add support for the unit circle unit and stop once it clicks.

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Questions from students and parents

Trigonometry tutoring: common questions

Is there trigonometry tutoring near me?

Our trigonometry tutoring is live online, so location does not matter. Students join by video from home and work with the same teacher each week, in a group of 15 to 20 or one to one.

What grade is trigonometry taught in?

Usually grade 10 for right-triangle trigonometry in Geometry, then grades 11 or 12 for trigonometric functions in Algebra 2 or Precalculus. Course order varies by district, and some states use their own standards.

Is trigonometry hard?

Right-triangle trigonometry is usually manageable. The unit circle, radians and graphs feel harder because the same idea appears in new forms, which is why we teach the triangle, the circle and the graph together.

How do I memorize the unit circle?

Learn the 30-60-90 and 45-45-90 triangles and the signs in each quadrant instead. With a reference angle, a triangle and a sign, any special value can be rebuilt in seconds, and nothing has to be memorized point by point.

Is trigonometry on the AP Precalculus exam?

Yes. Unit 3, Trigonometric and Polar Functions, makes up 30 to 35 percent of the multiple-choice section according to the College Board's Course and Exam Description, and one free-response question models a periodic context. AP Central lists the 2027 exam for May 11, 2027.

Is trigonometry on the SAT?

Yes, a small part. The College Board lists right triangles and trigonometry as one of four topics in the Geometry and Trigonometry domain of the SAT math section.

Should my calculator be in degrees or radians?

Whichever the question uses. If angles are written with a degree symbol use degrees; if they are written with pi or as plain real numbers use radians. Check the mode before every test.

Can you help with my trigonometry homework?

Yes. The teacher works through a similar problem with you and checks your reasoning, but you write your own homework and graded work. We never complete assignments for students.

When are your trigonometry classes?

Weekdays or weekends, flexible: choose your batch, or for a very personalised plan choose 1-on-1 at your convenience, weekend mornings or weekdays.

Are you connected with the College Board or my school?

No. Modern Age Coders is an independent online school, not affiliated with the College Board, the Common Core initiative, any state or any district. We quote their published documents and link to them.

Elsewhere on this site

Related math pages

The courses around trigonometry.

Geometry tutoring

Similarity and right triangles, where trig begins.

Algebra 2 tutoring

Functions, then the trig functions.

Precalculus tutoring

Functions in depth, with trigonometry inside.

AP Calculus tutoring

Where sine and cosine get derivatives.

SAT math tutoring

The digital SAT math section.

ACT math tutoring

Timed math, trigonometry included.

Tenth grade math tutoring

The Geometry year.

Eleventh grade math tutoring

Algebra 2 and Precalculus.

Related pages in this guide

Part of the guide: US math placement and college readiness tests

High school math courses

School placement and growth tests

PSAT and PreACT

College placement and credit

More guides: US state math tests by state · US gifted and selective school tests · Homeschool and school math curricula guide · US math and coding competitions guide

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