What Is Taught in Trigonometry? Complete Curriculum Guide

What Trigonometry Actually Covers

Trigonometry is the study of triangles, specifically the relationships between angles and side lengths. Most students encounter it after algebra and before calculus. The curriculum builds from basic triangle properties to complex function analysis.

Here's what you'll actually learn:

The Foundation: Angles and Triangles

Before touching any trig functions, you need to understand angles measured in degrees and radians. Most courses start here because radians are the natural unit for higher math.

• Degree measure (360° in a circle)
• Radian measure (2π radians in a circle)
• Converting between the two
• Complementary and supplementary angles
• Acute, obtuse, and right triangles

Right triangle geometry introduces the Pythagorean theorem: a² + b² = c². You use this constantly in trig, so memorize it if you haven't already.

The Three Core Functions

These are sine, cosine, and tangent. Every trig course centers on them. Here's how to think about them in a right triangle:

• Sin(θ) = opposite side / hypotenuse
• Cos(θ) = adjacent side / hypotenuse
• Tan(θ) = opposite side / adjacent side

The reciprocals—cosecant, secant, and cotangent—appear later and matter for identities and equations.

The Unit Circle

The unit circle is a circle with radius 1 centered at the origin. It's the bridge between right triangles and angles beyond 90°. Once you understand it, trig becomes much easier.

Key points on the unit circle:

• Coordinates (cos θ, sin θ) for common angles like 0°, 30°, 45°, 60°, 90°
• Reference angles and how to find trig values in any quadrant
• Sign conventions (positive/negative values by quadrant)

Most instructors expect you to memorize the unit circle. Do it early—it saves headaches later.

Graphing Trigonometric Functions

After mastering the unit circle, you graph sin, cos, and tan functions. This section covers:

• Amplitude (height from midline)
• Period (wavelength of one complete cycle)
• Phase shift (horizontal movement)
• Vertical shift (movement up/down)

You'll transform basic functions like y = sin(x) into forms like y = 2sin(3x - π) + 1. Each parameter changes the graph in predictable ways.

Trigonometric Identities

Identities are equations that hold true for all values. You'll memorize a stack of them:

• Pythagorean identities: sin²θ + cos²θ = 1
• Double-angle formulas: sin(2θ) = 2sinθcosθ
• Half-angle formulas
• Sum and difference formulas
• Product-to-sum and sum-to-product

These are tools for simplifying expressions and solving equations. Practice using them until the manipulations feel automatic.

Solving Trigonometric Equations

You'll solve equations like sin(x) = 0.5 or tan(2x) = √3. This requires combining your knowledge of:

• Inverse trig functions (arcsin, arccos, arctan)
• General solutions (infinite solutions, not just one)
• Restricted domains for inverse functions

Law of Sines and Law of Cosines

These laws let you solve any triangle, not just right triangles. They show up constantly in applications.

• Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
• Law of Cosines: c² = a² + b² - 2ab·cos(C)

Use these when you have two angles and a side, or two sides and an angle, or all three sides.

Applications of Trigonometry

Real-world uses include:

• Physics: projectile motion, wave analysis, alternating current
• Engineering: forces, structural analysis, signal processing
• Navigation: surveying, GPS, astronomy
• Computer graphics: rotation matrices, lighting calculations

Tools and Resources Comparison

How to Get Started

If you're starting from scratch or struggling with trig, here's what actually works:

1. Memorize the unit circle first. Spend 20 minutes daily until you know it cold. Quiz yourself on coordinates for 0°, 30°, 45°, 60°, 90°, and their equivalents in all quadrants.
2. Practice angle conversions. Convert degrees to radians and back until converting feels trivial.
3. Learn the three main trig functions cold. Sine, cosine, tangent definitions in a right triangle should be automatic.
4. Solve 10-15 problems daily. Trigonometry is a skill. You learn it by doing, not by reading.
5. Don't memorize identities blindly. Understand where the Pythagorean identity comes from. Derive the others when needed, then memorize the ones you use most.

What Comes After Trigonometry

Precalculus typically wraps up trig and introduces limits. Calculus then uses trig functions extensively—in derivatives, integrals, series, and differential equations. If you're heading toward STEM fields, solid trig skills are non-negotiable.

Most of the curriculum is standard across high schools and colleges, though depth varies. AP Calculus AB/BC expects full mastery of everything listed above. College algebra courses may go lighter on identities and graphing.