Home ›
Do My Trigonometry Homework ›
Trig Functions Explained

Trig Functions Explained: All 6 Functions, Signs & Exact Values

Why Trig Functions Feel Like a Foreign Language

Most math up to this point has a simple contract: here’s a formula, plug in numbers, get an answer. Trig breaks that contract. Suddenly you’re not solving for x — you’re relating angles to ratios, using Greek symbols, and dealing with six different functions that all seem to do slightly different versions of the same thing.

The confusion usually has a specific source: students try to memorize trig functions as a list of disconnected facts rather than understanding that sin and cos are the foundation and everything else is built from them. Once you see that tan is just sin divided by cos, and that csc, sec, and cot are just reciprocals of the first three, the number of things to memorize drops from six separate concepts to two primary ones with four derived relationships.

That’s the frame for this entire guide.

The Six Functions at a Glance

Sin and Cos: The Foundation

Sin and cos are not arbitrary — they come directly from the geometry of a circle. Place a point P anywhere on a circle of radius 1 (the unit circle). Draw an angle θ from the positive x-axis to the line connecting the center to P. The x-coordinate of P is cos θ. The y-coordinate of P is sin θ. That’s the definition.

This means sin and cos are essentially a coordinate system for describing where you are on a circle. Every trig identity, every wave function in physics, every oscillation in engineering comes back to this: sin = vertical position, cos = horizontal position, on a circle of radius 1.

Key values to know cold: sin 0° = 0, cos 0° = 1. sin 90° = 1, cos 90° = 0. sin 180° = 0, cos 180° = −1. sin 270° = −1, cos 270° = 0. These are the axis angles — no triangle required, just the circle.

Tan and Its Quirks

Tan θ = sin θ / cos θ. That simple definition explains everything unusual about tangent.

Why tan is undefined at 90° and 270°: At those angles, cos θ = 0. Division by zero is undefined. This creates vertical asymptotes in the graph of tangent — walls where the function shoots toward ±infinity. You’ll see these appear when you get to graphing tangent later in the course.

Why tan can be any value: Sin and cos are locked between −1 and 1. But their ratio can be anything — a very large sin divided by a very small cos produces a huge result. Tan has no upper or lower bound, which is why its range is all real numbers (unlike sin and cos).

When tan is positive: Tan = sin/cos. It’s positive whenever sin and cos have the same sign — both positive (Q I) or both negative (Q III). It’s negative when they have opposite signs (Q II and Q IV). This is exactly what the ASTC sign rule captures.

The Reciprocal Functions: Csc, Sec, Cot

The three reciprocal functions are exactly what their name says — they flip the primary functions upside down:

Reciprocal	Equals	Memory hook	On calculator
csc θ	1 / sin θ	“cosecant pairs with sine” — the third letter matches	1 ÷ sin(θ)
sec θ	1 / cos θ	“secant pairs with cosine” — they don’t share a letter, but sec = 1/cos	1 ÷ cos(θ)
cot θ	1 / tan θ = cos θ / sin θ	“cotangent pairs with tangent” — both contain ‘tan’	1 ÷ tan(θ)

The most reliable memory trick: the pairs that share a third letter in common are NOT reciprocals of each other. Cosecant pairs with sine (not cosine), and secant pairs with cosine (not sine). If that sounds backwards, use the co- prefix as a warning sign: “co-secant” → watch out, this one pairs with the plain version (sine), not the co-version.

The Notation Trap: csc θ ≠ sin⁻¹ θ

This is the single most dangerous notational confusion in all of trig. It costs students real points on exams every semester.

The notation sin⁻¹ comes from function notation, where f⁻¹ means “inverse function.” But in trig, the −1 superscript specifically means arcsin — it does not mean 1/sin. This is unlike how x⁻¹ = 1/x in algebra, which is what makes it confusing.

How to keep them straight: If you see the −1 as a superscript on the function name (sin⁻¹, cos⁻¹, tan⁻¹), it’s an inverse trig function — input a ratio, output an angle. If you see csc, sec, or cot written as a separate abbreviation, it’s a reciprocal — input an angle, output a ratio. The inverse trig functions are covered fully in our Inverse Trig Functions guide.

The Hand Trick for Exact Values

This is one of the most useful memory devices in trig. Hold your left hand out, palm facing you, fingers pointing up. Number your fingers 0 through 4 from left (pinky) to right (thumb): pinky = 0°, ring = 30°, middle = 45°, index = 60°, thumb = 90°.

To find sin of any standard angle:

Exact Values Table

These are the values your homework expects you to enter. Platforms like Knewton Alta, Hawkes, and ALEKS almost always require exact form — not 0.866 for cos 30°, but √3/2.

θ (deg)	θ (rad)	sin θ	cos θ	tan θ	csc θ	sec θ	cot θ
0°	0	0	1	0	undef	1	undef
30°	π/6	1/2	√3/2	1/√3	2	2/√3	√3
45°	π/4	√2/2	√2/2	1	√2	√2	1
60°	π/3	√3/2	1/2	√3	2/√3	2	1/√3
90°	π/2	1	0	undef	1	undef	0
180°	π	0	−1	0	undef	−1	undef
270°	3π/2	−1	0	undef	−1	undef	0
360°	2π	0	1	0	undef	1	undef

For the full table with all 16 unit circle angles and their (cos θ, sin θ) coordinates, see our Unit Circle Explained guide.

Signs Across Quadrants

Whether a trig function is positive or negative depends on which quadrant the angle falls in. The mnemonic “All Students Take Calculus” tells you which functions are positive in each quadrant: All (Q I), Students (Q II — sin positive), Take (Q III — tan positive), Calculus (Q IV — cos positive). Reciprocals always match the sign of their primary function.

The underlying reason is simple: sin = y/r and cos = x/r, where r is always positive. The sign of sin follows the sign of y (positive in Q I and Q II, negative in Q III and Q IV). The sign of cos follows the sign of x (positive in Q I and Q IV, negative in Q II and Q III). Tan = sin/cos, so its sign is positive when they match (Q I and Q III) and negative when they differ (Q II and Q IV).

Domain and Range

Function	Domain (what θ can be)	Range (what output can be)
sin θ	All real numbers	−1 ≤ sin θ ≤ 1
cos θ	All real numbers	−1 ≤ cos θ ≤ 1
tan θ	All real numbers except θ = 90° + 180°n	All real numbers (−∞ to ∞)
csc θ	All real numbers except θ = 0°, 180°, 360°…	csc θ ≤ −1 or csc θ ≥ 1
sec θ	All real numbers except θ = 90°, 270°, 450°…	sec θ ≤ −1 or sec θ ≥ 1
cot θ	All real numbers except θ = 0°, 180°, 360°…	All real numbers (−∞ to ∞)

Notice that csc and sec never produce values between −1 and 1. That’s because they’re reciprocals of sin and cos, which are locked between −1 and 1 — and the reciprocal of a fraction smaller than 1 is always larger than 1 (in absolute value).

Even and Odd Functions

Trig functions have a specific behavior when you plug in a negative angle — and knowing which ones are even and which are odd saves significant time on simplification problems.

An even function satisfies f(−θ) = f(θ): the negative angle gives the same output as the positive angle. An odd function satisfies f(−θ) = −f(θ): the negative angle flips the sign of the output.

Function	Even or Odd?	Rule	Example
cos θ	Even	cos(−θ) = cos(θ)	cos(−60°) = cos(60°) = 1/2
sec θ	Even	sec(−θ) = sec(θ)	sec(−45°) = sec(45°) = √2
sin θ	Odd	sin(−θ) = −sin(θ)	sin(−30°) = −sin(30°) = −1/2
csc θ	Odd	csc(−θ) = −csc(θ)	csc(−30°) = −csc(30°) = −2
tan θ	Odd	tan(−θ) = −tan(θ)	tan(−45°) = −tan(45°) = −1
cot θ	Odd	cot(−θ) = −cot(θ)	cot(−45°) = −cot(45°) = −1

Memory shortcut: only cos and its reciprocal sec are even. Every other trig function is odd. When in doubt, flip the sign of the angle and ask whether the output flips — for cos and sec it doesn’t; for all others it does.

Periodicity

All six trig functions are periodic — they repeat the same output values on a regular cycle. This is what creates the wave shapes you’ll see when graphing sine and cosine.

Function	Period	Meaning
sin θ, cos θ	2π (360°)	sin(θ + 2π) = sin(θ) for all θ
csc θ, sec θ	2π (360°)	Same period as their primary functions
tan θ, cot θ	π (180°)	tan(θ + π) = tan(θ) for all θ

Tan and cot have half the period of sin and cos because they depend on the ratio sin/cos — and that ratio repeats every half-rotation since both sin and cos flip sign together in opposite quadrants, making their ratio identical every 180°.

Periodicity is why trig equations have infinitely many solutions. If sin(θ) = 1/2 at θ = 30°, it’s also true at θ = 30° + 360°, 30° + 720°, and so on. When solving trig equations, you find the solutions within one period and then add the period multiplied by any integer n to capture all of them. This is covered fully in the Solving Trig Equations guide.

Co-function Identities

Co-function identities describe the relationship between complementary angles — angles that sum to 90°. The pattern: any trig function of an angle equals the co-function of its complement.

Identity	In radians	Example
sin(90° − θ) = cos(θ)	sin(π/2 − θ) = cos(θ)	sin(60°) = cos(30°) = √3/2
cos(90° − θ) = sin(θ)	cos(π/2 − θ) = sin(θ)	cos(60°) = sin(30°) = 1/2
tan(90° − θ) = cot(θ)	tan(π/2 − θ) = cot(θ)	tan(30°) = cot(60°) = 1/√3
csc(90° − θ) = sec(θ)	csc(π/2 − θ) = sec(θ)	csc(30°) = sec(60°) = 2
sec(90° − θ) = csc(θ)	sec(π/2 − θ) = csc(θ)	sec(30°) = csc(60°) = 2/√3
cot(90° − θ) = tan(θ)	cot(π/2 − θ) = tan(θ)	cot(30°) = tan(60°) = √3

Why the “co-” prefix makes sense now: Cosine is the co-function of sine, cosecant is the co-function of secant, and cotangent is the co-function of tangent. The “co-” literally means “complement” — each co-function gives the same value as its pair at the complementary angle. This is where those names came from historically.

Pythagorean Identity Preview

There is one identity you need to know before tackling almost anything else in trig:

This identity unlocks two more: divide both sides by cos²θ to get tan²θ + 1 = sec²θ. Divide both sides by sin²θ to get 1 + cot²θ = csc²θ. All three appear constantly in simplification problems and proofs. They’re covered fully in the Pythagorean Identities guide.

Calculator Tips

Degree vs radian mode: sin(30) in degree mode = 0.5. sin(30) in radian mode = −0.988. Before evaluating any trig function, check your mode. On a TI-84, press MODE and confirm the DEGREE or RADIAN setting. On Desmos, click the wrench icon. For a quick sanity check: type sin(90) — if it returns 1, you’re in degree mode. See our Radian and Degree Conversion guide for the full walkthrough.

Entering reciprocal functions: Most calculators have no csc, sec, or cot button. Use these instead:

Why This Matters in Calculus

If you’re in pre-calculus or heading into Calculus I, trig functions are not a detour — they’re the foundation of a third of the course. Here’s what’s coming and why these definitions matter now.

Derivatives of trig functions are among the first rules you’ll memorize in Calc I: d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x. Every one of those rules depends on knowing exactly what these functions are and being in radian mode. If you’re shaky on the definitions, the derivative rules are just more symbols to memorize without understanding.

Integration reverses those derivatives: ∫cos x dx = sin x + C, ∫sin x dx = −cos x + C. Trig substitution — a technique used to integrate expressions involving √(a²−x²) — requires fluency with all six functions and their identities.

Trig identities in calculus are used to simplify expressions before differentiating or integrating. The Pythagorean identity sin²θ + cos²θ = 1 appears in nearly every trig substitution problem. Students who don’t know their identities cold spend three times as long on these problems.

Common Mistakes

Practice Problems

Platform Tips

Hawkes Learning — when entering reciprocal functions in Certify mode, use the fraction input: type 1 in the numerator, then the trig function in the denominator. Hawkes does not accept “csc(30)” typed as text — it wants the explicit 1/sin format through the equation editor.

ALEKS — problems involving the sign of trig functions often show a diagram with a point in a specific quadrant and ask you to determine signs. Use the ASTC chart above. ALEKS will sometimes accept either the positive or negative version as a preliminary answer and then ask you to apply the quadrant restriction — read the full question before answering.

MyMathLab and WebAssign — exact value answers require the radical symbol, not a decimal. Type sqrt(3)/2, not 0.866. Both platforms have a palette with the √ symbol — use it. Entering an unsimplified form like sqrt(12)/4 instead of sqrt(3)/2 may or may not be accepted depending on the problem settings.

Frequently Asked Questions

Why are there six trig functions instead of just three?

Historically, all six were needed for navigation and astronomy calculations before calculators existed — having the reciprocals pre-computed in tables saved time. Today csc, sec, and cot appear mainly in calculus (where their derivatives and integrals have their own forms) and in proofs/identities. In practice, most problems can be solved with just sin, cos, and tan.

What’s the difference between sin⁻¹ and csc?

sin⁻¹ (arcsin) is an inverse function — input a ratio, output an angle. csc is a reciprocal function — input an angle, output a ratio. sin⁻¹(0.5) = 30°. csc(30°) = 2. They do completely different things. The −1 notation in sin⁻¹ is borrowed from function inverse notation (f⁻¹), not from the exponent meaning 1/x.

How do I enter csc, sec, or cot on a calculator?

Most calculators don’t have dedicated buttons. Enter: csc(θ) as 1÷sin(θ), sec(θ) as 1÷cos(θ), cot(θ) as 1÷tan(θ). Always check your mode (degrees vs radians) before evaluating.

Why is tan undefined at 90°?

tan θ = sin θ / cos θ. At 90°, cos 90° = 0. Division by zero is undefined. The same applies at 270° and all odd multiples of 90°. This is not a calculator error — there is genuinely no real number value for tan at those angles.

Do I need to memorize all six functions for a trig exam?

You need to know sin and cos cold — definitions, exact values, and signs. For the other four: know the definitions (tan = sin/cos, and the three reciprocals) and you can always derive them. The exact values at 30°, 45°, and 60° are the ones that appear most on exams — use the hand trick if you blank.

How do trig functions connect to the unit circle?

On a unit circle (radius = 1), the coordinates of the point at angle θ are exactly (cos θ, sin θ). Every trig value you’ll ever need is encoded in those two coordinates. The unit circle is the most efficient reference you can have — bookmark our Unit Circle Explained guide for the full table.

Related Resources

In this cluster:

• Do My Trigonometry Homework
• Unit Circle Explained — all 16 angles with coordinates
• SOH-CAH-TOA Explained — right triangle trig
• Radian and Degree Conversion

What comes next:

• Pythagorean Identities — sin²θ + cos²θ = 1 and its two derived forms
• Inverse Trig Functions — arcsin, arccos, arctan explained
• Graphing Sine and Cosine — amplitude, period, phase shift