Trigonometric Functions- Complete Guide

What Are Trigonometric Functions?

Trigonometric functions are relationships between the angles and sides of right triangles. They show up everywhere—in physics, engineering, architecture, computer graphics, and even music theory. If you're studying math past algebra, you'll need to master these.

There are six primary trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. Most textbooks focus on the first three because the last three are just reciprocals.

The Three Core Functions

Before memorizing formulas, understand what each function actually represents. The definitions depend on which angle you're analyzing in a right triangle.

Sine (sin)

The ratio of the opposite side to the hypotenuse.

sin(θ) = opposite ÷ hypotenuse

Think: soh from the memory trick "SOH CAH TOA."

Cosine (cos)

The ratio of the adjacent side to the hypotenuse.

cos(θ) = adjacent ÷ hypotenuse

Think: cah from "SOH CAH TOA."

Tangent (tan)

The ratio of the opposite side to the adjacent side.

tan(θ) = opposite ÷ adjacent

Think: toa from "SOH CAH TOA."

Quick note: tan also equals sin ÷ cos. This identity becomes useful later.

The Reciprocal Functions

These are just inverses of the main three. You'll use them less often, but they show up in calculus and when simplifying expressions.

• Cosecant (csc) = 1 ÷ sin = hypotenuse ÷ opposite
• Secant (sec) = 1 ÷ cos = hypotenuse ÷ adjacent
• Cotangent (cot) = 1 ÷ tan = adjacent ÷ opposite = cos ÷ sin

If you forget these, just flip the fraction of the original function.

The Unit Circle

The unit circle is a circle with radius 1 centered at the origin. It's the backbone of trigonometry because it lets you find sine and cosine values for any angle, not just those in a right triangle.

For any point on the unit circle at angle θ:

• The x-coordinate equals cos(θ)
• The y-coordinate equals sin(θ)

This means sin²(θ) + cos²(θ) = 1 always. That identity shows up constantly.

Quadrants and Signs

The unit circle splits into four quadrants. Each one determines whether sine, cosine, and tangent are positive or negative.

Remember: All Students Take Calculus—A in Quadrant I, S in Quadrant II, T in Quadrant III, C in Quadrant IV. Each letter indicates which function is positive in that quadrant.

Key Trigonometric Identities

Identities are equations that are always true. These are the ones you'll encounter most often.

Pythagorean Identities

• sin²(θ) + cos²(θ) = 1
• 1 + tan²(θ) = sec²(θ)
• 1 + cot²(θ) = csc²(θ)

The first one comes directly from the unit circle. The other two are derived from it.

Reciprocal Identities

• sin(θ) = 1 ÷ csc(θ)
• cos(θ) = 1 ÷ sec(θ)
• tan(θ) = 1 ÷ cot(θ)

Quotient Identities

• tan(θ) = sin(θ) ÷ cos(θ)
• cot(θ) = cos(θ) ÷ sin(θ)

Co-Function Identities

These relate functions of complementary angles (angles that add to 90° or π/2):

• sin(θ) = cos(90° - θ)
• cos(θ) = sin(90° - θ)
• tan(θ) = cot(90° - θ)

Even-Odd Identities

• sin(-θ) = -sin(θ) — odd function
• cos(-θ) = cos(θ) — even function
• tan(-θ) = -tan(θ) — odd function

Graphing Trigonometric Functions

Each trig function has a distinctive wave pattern. Understanding their graphs helps you visualize amplitude, period, and phase shifts.

Sine and Cosine Graphs

Both sine and cosine have:

• Amplitude = 1 (the height from center to peak)
• Period = 2π (360°) — one complete wave
• Domain = all real numbers
• Range = [-1, 1]

The difference: sine starts at 0, cosine starts at 1.

Tangent Graph

Tangent has no amplitude (it goes to infinity). Its period is π (180°). The range is all real numbers, and it has vertical asymptotes at odd multiples of π/2.

Transformations

You can modify trig graphs by changing the equation:

• y = A·sin(θ) — amplitude becomes |A|
• y = sin(B·θ) — period becomes 2π ÷ |B|
• y = sin(θ - C) — horizontal shift by C
• y = sin(θ) + D — vertical shift by D

Inverse Trigonometric Functions

Sometimes you need to find the angle when you know the ratio. That's what inverse trig functions do.

• arcsin or sin⁻¹ — returns an angle from -90° to 90°
• arccos or cos⁻¹ — returns an angle from 0° to 180°
• arctan or tan⁻¹ — returns an angle from -90° to 90°

⚠️ Don't confuse sin⁻¹(x) with (sin x)⁻¹. The first is arcsine, the second is cosecant.

Solving Trigonometric Equations

When solving trig equations, isolate the trig function first, then find all possible solutions within the given interval.

Example: Solve sin(x) = 0.5 for x between 0 and 2π.

sin(x) = 0.5 at x = π/6 and x = 5π/6 within [0, 2π].

General solutions add 2πn:

• x = π/6 + 2πn
• x = 5π/6 + 2πn

Common Values to Memorize

These angles appear constantly in problems. Know them in both degrees and radians.

Applications of Trigonometric Functions

Trig shows up in real-world problems more than most students expect.

Physics

• Projectile motion — calculating trajectory angles
• Wave mechanics — modeling sound, light, and electromagnetic waves
• Force vectors — breaking forces into components

Engineering

• Structural analysis — calculating forces on bridges and buildings
• Signal processing — analyzing alternating current
• Robotics — determining joint angles

Everyday Uses

• Navigation and GPS
• Architecture and construction
• Computer graphics and video game development

How to Get Started

Here's a practical approach to mastering trig:

1. Memorize SOH CAH TOA — this is your foundation
2. Drill the unit circle — draw it from memory until it's automatic
3. Memorize the common angle values in the table above
4. Practice identity manipulation — start with sin² + cos² = 1
5. Solve lots of equations — find all solutions, not just the obvious one

Use flash cards for identities. Work through problems daily, even if just 20 minutes. Trigonometry rewards consistency more than cramming.

Tools and Calculators

You'll need a graphing calculator for complex problems. Here's how the common options stack up.

For class, check what's allowed on exams before buying anything expensive.

Common Mistakes to Avoid

• Confusing radians with degrees — check your calculator mode
• Forgetting the sign in different quadrants
• Using degrees when the problem expects radians (or vice versa)
• Mixing up sin⁻¹(x) with csc(x)
• Not checking if your answer is in the correct interval

Most errors come from rushing. Write down what you're solving for before you start plugging numbers in.

Where to Go From Here

Once trig feels solid, you'll move into topics that build directly on it:

• Law of Sines and Law of Cosines
• Sum and difference formulas
• Double-angle and half-angle formulas
• Complex numbers in polar form
• Calculus derivatives of trig functions

Each of these uses the fundamentals covered here. If the basics aren't automatic, everything else becomes a struggle. Put in the work now.