Learn Trigonometry unit by unit

Reason with angles, periodic functions, identities, and triangles.

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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 12-week schedule.

Trigonometry course map: 6 units
#	Unit	What you can do after it	Weeks
01	Angle measure	You can switch between degrees and radians and find other angles that land in the same spot.	Weeks 1-2
02	Unit circle	You can give exact sine, cosine and tangent values for the common angles from memory.	Weeks 3-4
03	Graphs of sine and cosine	You can sketch a wave from its formula and pull the amplitude, period, phase shift and midline out of a picture.	Weeks 5-6
04	Identities	You can rewrite one trig expression as another using the Pythagorean, sum, difference and double-angle rules.	Weeks 7-8
05	Trig equations	You can solve an equation with sine, cosine or tangent and list every answer inside the interval you were given.	Weeks 9-10
06	Sine and cosine laws	You can solve triangles that have no right angle, whether you are given two sides and an angle or all three sides.	Weeks 11-12

Trigonometry course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

Angle measure

Weeks 1-2

You can switch between degrees and radians and find other angles that land in the same spot. You can also work out arc length and turning speed.

Convert 240 degrees into radians.
Find an angle between 0 and 360 degrees that matches -100 degrees.
Find the arc length cut by a 2 radian angle in a circle of radius 9.

Worked example

Convert 240 degrees into radians

Multiply by pi/180
240 pi / 180

Answer4 pi / 3

Most common mistakeMultiplying by 180/pi instead: reporting about 13,751, a number thousands of times bigger than a full turn.

Explain angle measure Practice angle measure

Unit circle

Weeks 3-4

You can give exact sine, cosine and tangent values for the common angles from memory. You can also work out the sign of each ratio from the quadrant alone.

State that cos(pi/3) = 1/2 and sin(pi/3) = root 3 over 2 without a calculator.
Find the reference angle for 210 degrees.
Say which ratios come out negative in the third quadrant.

Worked example

Find sin(210 degrees)

Reference angle: 210 - 180 = 30 degrees
210 degrees is in the third quadrant, where sine is negative

Answer-1/2

Most common mistakeIgnoring the quadrant sign: answering +1/2 because sin(30 degrees) = 1/2, which puts the point above the axis instead of below it.

Explain unit circle Practice unit circle

Graphs of sine and cosine

Weeks 5-6

You can sketch a wave from its formula and pull the amplitude, period, phase shift and midline out of a picture. You can also model repeating things like tides and wheels.

Find the midline of y = 3 sin(x) + 5.
Find the phase shift of y = cos(2x - pi).
Write a wave equation for a Ferris wheel that turns once every 12 minutes.

Worked example

Find the phase shift of y = cos(2x - pi)

Factor the inside: cos(2(x - pi/2))
The shift is whatever is subtracted from x

Answerpi/2 to the right

Most common mistakeReading the shift straight off the -pi: reporting pi to the right, twice the true amount, because the 2 was never factored out first.

Explain graphs of sine and cosine Practice graphs of sine and cosine

Identities

Weeks 7-8

You can rewrite one trig expression as another using the Pythagorean, sum, difference and double-angle rules. You can also prove two expressions are equal for every angle.

Replace 1 - sin squared x with cos squared x.
Expand sin(A + B) with the sum rule.
Turn sin(2x) into 2 sin x cos x to make an equation solvable.

Worked example

Simplify (1 - cos squared x) / sin x

1 - cos squared x is sin squared x
sin squared x divided by sin x

Answersin x

Most common mistakeRewriting 1 - cos squared x as (1 - cos x) squared: at x = 60 degrees that gives 0.25 instead of the correct 0.75.

Explain identities Practice identities

Trig equations

Weeks 9-10

You can solve an equation with sine, cosine or tangent and list every answer inside the interval you were given. You can also add the repeating pattern to describe all of them.

Solve 2 sin x = 1 for x between 0 and 2 pi.
Add 2 pi n to describe every solution, not just the first one.
Factor sin x cos x = sin x rather than dividing, so no roots get lost.

Worked example

Solve 2 sin x = 1 for x from 0 up to 2 pi

sin x = 1/2
Sine is positive in the first and second quadrants

Answerx = pi/6 and x = 5 pi/6

Most common mistakeStopping at the single value the calculator prints: reporting only x = pi/6 and losing the second-quadrant answer 5 pi/6.

Explain trig equations Practice trig equations

Sine and cosine laws

Weeks 11-12

You can solve triangles that have no right angle, whether you are given two sides and an angle or all three sides. You can also find a triangle's area without knowing its height.

Use the law of sines when you have a side paired with its opposite angle.
Use the law of cosines when all three sides are known.
Watch for the ambiguous case when two sides and a non-included angle are given.

Worked example

A triangle has sides 7 and 9 with a 40 degree angle between them. Find the third side.

c squared = 49 + 81 - 2(7)(9) cos 40
c squared = 130 - 126(0.766) = 130 - 96.5

Answerc is about 5.79

Most common mistakeUsing the Pythagorean theorem on a triangle with no right angle: computing the square root of 130 = 11.4 instead of 5.79.

Explain sine and cosine laws Practice sine and cosine laws

Prepare for Trigonometry practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Angle measure from a blank page. Then connect it to Unit circle. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The Trigonometry calculator can test calculations and representations used in Trigonometry. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Trigonometry practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesTrigonometry explainedConcepts, worked methods and checks behind this course.

PracticeMixed Precalculus questionsAnswer review after every attempt.

ReferenceFormulas for this courseConditions and a worked example for each one.

Assess

Take the complete Trigonometry test

Begin test →

Reference

Review essential formulas

Open library →

Calculate

Use the Trigonometry calculator

Open tool →

Plan

Prepare around your exam date

Build plan →

Questions about the Trigonometry course

Where should I start?

Start with Angle measure if you are following the full sequence. If that unit feels automatic, open the Trigonometry practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Trigonometry test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The Trigonometry calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.

Learn Trigonometry unit by unit

Units at a glance

Trigonometry course units

Angle measure

Worked example

Unit circle

Worked example

Graphs of sine and cosine

Worked example

Identities

Worked example

Trig equations

Worked example

Sine and cosine laws

Worked example

Prepare for Trigonometry practice

Start with the earliest uncertain prerequisite

Use tools to verify, not to choose the method for you

Know when to move to the full test

Learn the ideas, then practise them

Other high school courses

Probability & Statistics

Mathematical Modeling

Pre-Algebra

Take the complete Trigonometry test

Review essential formulas

Use the Trigonometry calculator

Prepare around your exam date

Questions about the Trigonometry course