What Is Trigonometry and What Are Its Core Concepts?

Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles, especially right triangles. It gives you a set of functions—sine, cosine, tangent, and their reciprocals—that connect an angle to a ratio of two sides. Those functions extend beyond triangles through the unit circle, which lets you work with any angle, not just the acute angles inside a right triangle. If you are starting from scratch, the practical path is: learn the six functions, understand the unit circle, memorize the core identities, then practice solving for missing sides and angles.

The six trigonometric functions

Every right triangle has three sides relative to a chosen angle: the hypotenuse (opposite the right angle), the opposite side (across from the angle), and the adjacent side (next to the angle). The six functions are just different pairings of those sides.

The last three are reciprocals of the first three: csc is 1/sin, sec is 1/cos, and cot is 1/tan. That means if you know sine, cosine, and tangent, you can derive the other three without memorizing new ratios.

A common memory aid for the first three is SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

The unit circle: extending trig to any angle

Right-triangle ratios only work for angles between 0° and 90°. The unit circle—a circle of radius 1 centered at the origin—removes that limit.

On the unit circle, an angle is measured from the positive x-axis, and the point where the angle's ray hits the circle has coordinates (cos θ, sin θ). This definition works for any angle, including angles larger than 90° and negative angles, and it produces the signs you see in each quadrant:

• Quadrant I (0°–90°): sin, cos, tan all positive
• Quadrant II (90°–180°): sin positive, cos and tan negative
• Quadrant III (180°–270°): tan positive, sin and cos negative
• Quadrant IV (270°–360°): cos positive, sin and tan negative

The unit circle also gives you the exact values for common angles (30°, 45°, 60°, and their multiples), which is why it is usually the second thing you study after the basic ratios.

Core identities and formulas

Identities are equations that hold for every angle. The ones you use most:

• Pythagorean identity: sin²θ + cos²θ = 1. This follows directly from the unit circle and the Pythagorean theorem.
• Reciprocal identities: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
• Quotient identity: tan θ = sin θ / cos θ.
• Complementary angles: sin(90° − θ) = cos θ, and tan(90° − θ) = cot θ.

These are used to simplify expressions, prove other identities, and rewrite an equation into a form you can solve. For example, if an equation contains sin²θ, substituting 1 − cos²θ often turns it into something solvable for cos θ.

Solving for a missing side or angle

This is the skill most courses test first. The steps:

1. Label the triangle. Identify the right angle, then mark the side opposite your known or target angle, the adjacent side, and the hypotenuse.
2. Choose a function. Pick the one that relates the two sides you care about. If you know the opposite and hypotenuse, use sine. If you know the adjacent and hypotenuse, use cosine. If you know opposite and adjacent, use tangent.
3. Set up the equation. For a missing side, write the ratio equal to the known values and solve. For a missing angle, take the inverse function (sin⁻¹, cos⁻¹, or tan⁻¹) of the ratio.
4. Check the result. A side must be shorter than the hypotenuse, and an angle must fall in a range consistent with the triangle's shape.

Example: A right triangle has a 35° angle and a hypotenuse of 12. To find the side opposite the 35° angle, use sine: sin 35° = opposite / 12, so opposite = 12 × sin 35° ≈ 6.88. The answer is less than 12, which checks out.

For non-right triangles, you switch to the law of sines (a/sin A = b/sin B = c/sin C) or the law of cosines (c² = a² + b² − 2ab cos C). These come up after the right-triangle material.

How to learn it step by step

Trigonometry builds on itself, so the order matters more than the pace. A workable sequence:

1. Angle measurement in degrees and radians
2. The six functions and SOH-CAH-TOA
3. Special right triangles (30-60-90 and 45-45-90)
4. The unit circle and reference angles
5. Graphing sine, cosine, and tangent
6. Identities and how to verify them
7. Solving triangles with the law of sines and law of cosines

Video-Tutor.net lists trigonometry as a dedicated course alongside algebra, precalculus, and calculus, and organizes its material into course videos, video playlists, and formula sheets. That structure fits this sequence: use the formula sheets as a reference while working problems, and the playlists to move through topics in order. The site also separates test-prep and final-exam videos, which are useful once you have covered the core material and want to check retention.

If you are deciding where to start, the honest test is whether you can already solve a right triangle for a missing side using sine, cosine, or tangent. If not, begin there before touching the unit circle or identities—those depend on it.