01
The unit circle, piece by piece
Build the circle one idea at a time and see why every point on it is exactly (cos θ, sin θ).
Step 1 · See it
Step 2 · Try it
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All courses
MTH 162
Precalculus II
Presents trigonometry, trigonometric applications including Law of Sines and Cosines, and an introduction to conics.
How this course is taught: Video lessons, interactive demos, and the free OpenStax textbook.
How to use this page
- Watch the lesson for the section you are on, in the player below.
- Experiment with the matching interactive demo to build intuition.
- Read and practice that section in the free OpenStax book linked in the outline.
Practice problems: each OpenStax Precalculus 2e section has exercises with answers in the back. The demo checks are for understanding, not a grade.
Lessons
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Course outline
What we cover, and where to read it
Our lessons follow OpenStax Precalculus 2e, a free, peer-reviewed textbook. Each unit below lists its topics and links to the matching chapter so you can read alongside the videos.
Unit 1 · Chapter 5
Trigonometric Functions
- Angles and radian measure
- The unit circle, sine and cosine
- The other trigonometric functions
- Right triangle trigonometry
Unit 2 · Chapter 6
Periodic Functions
- Graphs of sine and cosine
- Graphs of the other trigonometric functions
- Inverse trigonometric functions
Unit 3 · Chapter 7
Trigonometric Identities & Equations
- Verifying trigonometric identities
- Sum, difference, double, and half-angle formulas
- Solving trigonometric equations
Unit 4 · Chapter 8
Further Applications of Trigonometry
- The Law of Sines
- The Law of Cosines
- Vectors
Official grades, submissions, and course announcements live in Canvas. This site hosts public course materials, explanations, examples, and study resources.
Interactive demos
Explore the ideas, hands on
Trigonometry finally clicks when you can turn the crank and watch it move. Each tool walks through the idea first, then hands you the controls. Nothing here is graded, so experiment freely. Not sure how to study this? See Study Strategies. Back to the lessons
02
Sine and cosine, unrolled
Turn the point around the circle and watch its height and width unroll into the sine and cosine waves.
Step 1 · See it
Step 2 · Try it
Step 3 · Quick check
03
The bridge: triangle meets circle
See how SOH-CAH-TOA from a right triangle and the coordinates on the unit circle are the same sine and cosine.
Step 1 · See it
Step 2 · Try it
Step 3 · Quick check
04
The sinusoid transformer
Stretch, squeeze, and slide a wave with y = A·sin(B(x − C)) + D, and watch amplitude, period, phase, and midline respond.
Step 1 · See it
Step 2 · Try it
Step 3 · Quick check
Quick reference
Notation, identities & formulas
The trig notation, the unit circle, and the identities and laws that run through Precalculus II, all in one place. Angles are in radians unless a degree symbol says otherwise.
Key notation
| Symbol | What it means |
|---|---|
| θ | Theta — the usual name for an angle, most often measured in radians. |
| radian | The angle whose arc length equals the radius. 180° = π rad, and a full turn is 2π. |
| sin, cos, tan | The three primary ratios, read off a right triangle or the unit circle. |
| csc, sec, cot | Reciprocals of sin, cos, and tan respectively. |
| (cos θ, sin θ) | The point on the unit circle at angle θ, the source of every trig value. |
| y = A sin(B(x−C))+D | The general sinusoid: |A| amplitude, period 2π/B, C phase shift, D midline. |
| sin−1, cos−1 | Inverse trig (arcsin, arccos…): give it a ratio, it returns the angle. Not a reciprocal. |
The unit circle, at a glance
The five first-quadrant angles every other value is built from. Coordinates on the unit circle are exactly (cos θ, sin θ).
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0 (0°) | 0 | 1 | 0 |
| π/6 (30°) | 1/2 | √3 / 2 | √3 / 3 |
| π/4 (45°) | √2 / 2 | √2 / 2 | 1 |
| π/3 (60°) | √3 / 2 | 1/2 | √3 |
| π/2 (90°) | 1 | 0 | undefined |
Formulas & identities
Right-triangle ratios (SOH‑CAH‑TOA)
sin θ = opphyp cos θ = adjhyp tan θ = oppadj
Reciprocal ratios
csc = 1sin sec = 1cos cot = 1tan
Degrees ↔ radians
radians = degrees · π180
Pythagorean identities
sin2θ + cos2θ = 1 · 1 + tan2θ = sec2θ
Even / odd
cos(−θ) = cos θ · sin(−θ) = −sin θ
Sum & difference
sin(A±B) = sinA cosB ± cosA sinB
Double angle
sin 2θ = 2 sin θ cos θ cos 2θ = cos2θ − sin2θ
Arc length & sector area
s = rθ · A = ½r2θ
Law of Sines
sin Aa = sin Bb = sin Cc
Law of Cosines
c2 = a2 + b2 − 2ab cos C
Triangle area (SAS)
Area = ½ab sin C
Polar ↔ rectangular
x = r cos θ, y = r sin θ r2 = x2+y2
Glossary
Radian
The natural unit for angles: the angle that cuts an arc equal to the radius. A full circle is 2π radians.
Amplitude
Half the distance between a wave’s highest and lowest points, the |A| in y = A sin(…).
Period
The horizontal length of one full cycle of a periodic function. For a sinusoid it is 2π/B.
Phase shift
A horizontal slide of a trig graph, the C in y = A sin(B(x − C)) + D.
Midline
The horizontal center line a wave oscillates around, raised or lowered by D.
Reference angle
The acute angle between the terminal side and the x-axis; it gives the size of every trig value, the quadrant gives the sign.
Coterminal angles
Angles that land in the same place, differing by full turns of 2π (or 360°).
Identity
An equation true for every value of the variable, used to rewrite trig expressions into friendlier forms.
Polar coordinates
Locating a point by distance r from the origin and angle θ, instead of (x, y).
All of this is developed in the free OpenStax Precalculus 2e textbook linked from the course outline above.
Turn it in
Your practice report
Every quick check above feeds this report: what you got right on the first try, and what you got right eventually. Download it and upload it to Canvas.
MTH practice report · bryantclass.com