Trigonometry
Trigonometry is the largest topic in precalculus — three chapters covering the unit circle, right triangle trig, and identities. Here's everything you need, organized to build from the ground up.
3
Chapters (5–7)
200+
Practice Problems
16
Topic Sections
Chapter 5
Unit Circle Trigonometry
“The unit circle is a circle of radius 1. At angle θ, the point on the circle is (cosθ, sinθ). This is the definition of cosine and sine.”
Topics covered:
- •Angles and radian measure
- •Unit circle — sin, cos, tan values
- •The six trig functions: sin, cos, tan, csc, sec, cot
- •Reference angles and symmetry
- •Graphs of sine and cosine
- •Amplitude, period, phase shift, vertical shift
- •Graphs of other trig functions
Chapter 6
Right Triangle Trigonometry
“Right triangle trig (SOH-CAH-TOA) and unit circle trig give the same values. They're two ways to define the same functions. The right triangle approach fails for angles > 90°; the unit circle approach handles all angles.”
Topics covered:
- •SOH-CAH-TOA: the right triangle definitions
- •Special right triangles: 30-60-90 and 45-45-90
- •Solving triangles (finding all sides and angles)
- •Angles of elevation and depression
- •Bearings and navigation problems
- •Law of Sines and the ambiguous case (SSA)
- •Law of Cosines
- •Heron's formula for area
Chapter 7
Analytic Trigonometry
“Everything in Chapter 7 can be derived from one identity: sin²θ + cos²θ = 1. Divide both sides by cos²θ to get tan²θ + 1 = sec²θ. Divide by sin²θ to get 1 + cot²θ = csc²θ.”
Topics covered:
- •Pythagorean identities and derivations
- •Sum and difference formulas
- •Double-angle formulas
- •Half-angle formulas
- •Product-to-sum and sum-to-product
- •Inverse trig functions: arcsin, arccos, arctan
- •Solving trigonometric equations
Unit Circle Reference Table
Memorize Q1 (0°–90°). Use sign rules for Q2–Q4: sin is negative in Q3/Q4, cos is negative in Q2/Q3.
| Degrees | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undef. |
| 180° | π | 0 | -1 | 0 |
| 270° | 3π/2 | -1 | 0 | undef. |
Essential Trig Identities
Pythagorean (primary)
sin²θ + cos²θ = 1
Derived forms: 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ
Quotient identities
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ
Sum formula (sin)
sin(A+B) = sinA cosB + cosA sinB
sin(A−B) = sinA cosB − cosA sinB
Sum formula (cos)
cos(A+B) = cosA cosB − sinA sinB
cos(A−B) = cosA cosB + sinA sinB
Double-angle (sin)
sin 2θ = 2 sinθ cosθ
Derived from sin(A+B) with A=B=θ
Double-angle (cos)
cos 2θ = cos²θ − sin²θ
Also: 1 − 2sin²θ = 2cos²θ − 1
Practice Trig Problems Now
NailTheTest has 200+ trigonometry practice problems across Chapters 5–7, with worked examples, visual diagrams, and an private tutor for every concept. Free to start.