Quadrantal angles and undefined values

Explain it like I am five

Picture a clock hand stopping at 3, 12, 9 or 6. An angle whose terminal side lies on an axis is a quadrantal angle. Use the point one step out, (1, 0), (0, 1), (−1, 0) or (0, −1), where r = 1, so sin θ = y and cos θ = x.

Try 450°. Remove a full turn: 450° − 360° = 90°. 450° and 90° are coterminal: they end on one ray and share every value, straight up at (0, 1). So sin 450° = 1 and cos 450° = 0. But tan 450° = $\frac{y}{x}$ = $1 over 0$ is undefined: no value exists. Yet cot 450° = $0 over 1$ = 0 is fine.

Why: $8 over 2$ = 4 because 4 × 2 = 8, but no number times 0 makes 1. tan and sec (bottom x) fail on the y-axis, where x = 0: 90° + n·180°. cot and csc (bottom y) fail on the x-axis, where y = 0: n·180°. Here n counts half turns: positive, negative or zero. Such angles lie outside the function's domain, its allowed inputs.

In plain words

Picture a clock hand stopping exactly at the top, bottom, left or right. An angle whose ending ray lies on an axis is called a Quadrantal angle. Extra full turns leave that ending ray in the same place. Instead of squeezing an angle on an axis into a triangle, choose the point one unit from the center: (1, 0), (0, 1), (−1, 0) or (0, −1). Its distance r is 1. Sine reads the up or down coordinate y; cosine reads the right or left coordinate x. For the other four functions, inspect the bottom of the fraction. A zero bottom means Undefined, which means no value. A zero top over a nonzero bottom gives the number zero.

At the top, x = 0: tangent and secant divide by zero; cotangent is zero.

Reminder

Coterminal angles. A full turn changes the travel but not the ending ray: −450° + 720° = 270°.
Signed division. Zero divided by a nonzero negative number is still zero: $0 over −4$ = 0.
Radians. π radians = 180°. A full turn is 2π radians, so − $5 π over 2$ + 4π = $3 π over 2$ .

Why it works. Division asks for a number that undoes multiplication. = 4 because 4 × 2 = 8. To give a value, that value times 0 would have to make 1. Every number times 0 is 0, so none works. Tangent and secant divide by x and fail on the y-axis. Cotangent and cosecant divide by y and fail on the x-axis. Sine and cosine divide by r = 1, so they work at every axis point.

Rulesin θ = y and cos θ = x at the axis points with r = 1.
tan θ and sec θ are undefined at θ = 90° + n·180°; cot θ and csc θ are undefined at θ = n·180°, where n is any integer (positive, negative or zero). In radians these are + nπ and nπ.

The same idea, five ways

Say it

An axis angle ends exactly right, up, left or down. A zero denominator means undefined.

Write it

A Quadrantal angle has its terminal side on an axis; its sine and cosine still have values.

In math

θ = n·90°, with n any integer
x = 0: tan and sec are undefined
y = 0: cot and csc are undefined
$0 over 4$ = 0; $4 over 0$ is undefined

A clock hand can stop exactly at one of the four main directions.

See it

At the top, x = 0: tangent and secant divide by zero; cotangent is zero.

The same idea, other ways

As a clock

Start pointing right, as a clock hand points at 3. A quarter turn counterclockwise points up, a half turn left, and three quarters down. Complete turns do not change the stopping place.

450° = 360° + 90°, so the hand ends on the positive y-axis.

As a fraction check

At an axis point one coordinate is zero. Write the six fractions before deciding which are undefined. At (0, 1), tangent is $1 over 0$ and has no value, while cotangent is $0 over 1$ = 0.

Point (0, 1): x = 0, y = 1

tan = : undefined

cot = = 0

The location of zero, on top or on the bottom, makes the difference.

Rebuild the five sine and cosine reference columns

The special triangles supply the middle three columns, and the axis points supply the two endpoints. Read sine under each angle in the picture. The matching cosine table runs in reverse order because exchanging the two legs exchanges horizontal and vertical shares. This memory device covers only these five reference angles.

Read the sine under each angle; the square-root number increases by one.

θ	0° (0)	90° ( $π over 2$ )	180° (π)	270° ( $3 π over 2$ )	360° (2π)
point	(1, 0)	(0, 1)	(−1, 0)	(0, −1)	(1, 0)
sin θ	0	1	0	−1	0
cos θ	1	0	−1	0	1
tan θ	0	undefined	0	undefined	0
cot θ	undefined	0	undefined	0	undefined
sec θ	1	undefined	−1	undefined	1
csc θ	undefined	1	undefined	−1	undefined

.1The x-axis: y is zero

A hand pointing right or left has no upward travel. Among the axis points one unit from the origin, the point is (1, 0) or (−1, 0). The y-coordinate is zero, so sine and tangent are zero. Cotangent and cosecant put that zero on the bottom and have no value. Cosine and secant record whether the hand points right or left.

Right: sin = 0, cos = 1, tan = 0, cot undefined, sec = 1, csc undefined.
Left: sin = 0, cos = −1, tan = 0, cot undefined, sec = −1, csc undefined.

Two full turns end on the positive x-axis.

Worked exampleTwo turns, then read the point

Find sin 720°, cos 720° and csc 720°. In words, find the stopping point after two full turns.

Two counterclockwise turns end pointing right.

720° − 2 × 360° = 0°, so P = (1, 0).Removing full turns preserves the terminal side.
sin 720° = $0 over 1$ = 0 and cos 720° = $1 over 1$ = 1.Sine uses y and cosine uses x.
csc 720° = $1 over 0$ is undefined.Cosecant divides by y = 0.

Answer

sin 720° = 0
cos 720° = 1
csc 720° = undefined

Check The point has + = + = 1.

.2The y-axis: x is zero

A hand pointing up or down has no sideways travel. The point is (0, 1) or (0, −1). The x-coordinate is zero, so cosine and cotangent are zero. Tangent and secant put that zero on the bottom and have no value. Sine and cosecant distinguish up from down.

Up: sin = 1, cos = 0, tan undefined, cot = 0, sec undefined, csc = 1.
Down: sin = −1, cos = 0, tan undefined, cot = 0, sec undefined, csc = −1.

A full turn and a quarter turn ends up.

Worked exampleA turn and a quarter

Find cos 450°, cot 450° and sec 450°. In words, use the point where the ending ray lands.

The zero x-coordinate is below in secant and above in cotangent.

450° − 360° = 90°, so P = (0, 1).One full turn does not change the ending direction.
cos 450° = x = 0 and cot 450° = $\frac{x}{y}$ = $0 over 1$ = 0.Both calculations give zero without a zero denominator.
sec 450° = $1 over x$ = $1 over 0$ is undefined.No number multiplied by zero can equal one.

Answer

cos 450° = 0
cot 450° = 0
sec 450° = undefined

Check Sine at the top is 1, so cot = cos ÷ sin = 0 ÷ 1 = 0.

.3Undefined is different from zero

Sharing zero cookies among four people gives everyone zero cookies. Trying to share four cookies among zero people does not give a portion size. Fractions behave the same way: zero on top can be a real answer, but zero on the bottom never gives a defined quotient.

$0 over a$ = 0 when a ≠ 0.
$a over 0$ is undefined, including when a = 0.
For $0 over 0$ , every number times 0 gives 0, so division cannot choose one unique answer.

Four objects cannot be divided among zero groups.

Worked exampleCheck the location of zero

Evaluate $0 over 4$ , $4 over 0$ and $0 over 0$ . In words, decide whether each fraction names one definite number.

Division needs a nonzero number of groups.

$0 over 4$ = 0.0 × 4 = 0, so zero is the quotient.
$4 over 0$ is undefined.There is no number q with q × 0 = 4.
$0 over 0$ is undefined too.Every q satisfies q × 0 = 0, so there is no unique quotient.

Answer

$0 over 4$ = 0
$4 over 0$ = undefined
$0 over 0$ = undefined

Check The first fraction passes its multiplication check with a unique answer; the other two do not.

Strategy: step by step

If this is a quadrantal angle, add or subtract full turns until 0° ≤ θ < 360°; in radians use turns of 2π.
Match its terminal side to (1, 0), (0, 1), (−1, 0) or (0, −1).
Read sin θ = y and cos θ = x.
Write tan = $\frac{y}{x}$ , cot = $\frac{x}{y}$ , sec = $1 over x$ and csc = $1 over y$ .
Inspect every denominator first. A zero denominator gives undefined; otherwise do the division.

Strategy

Find all six functions on an axis

Does the terminal side lie on an axis?

YesUse its point one unit from the origin.

NoUse an ordinary terminal-side point instead.

↓

Is the denominator of this function zero?

YesWrite undefined.

NoDivide; a zero numerator gives zero.

If this is a quadrantal angle, add or subtract full turns until 0° ≤ θ < 360°; in radians use turns of 2π.
Match its terminal side to (1, 0), (0, 1), (−1, 0) or (0, −1).
Read sin θ = y and cos θ = x.
Write tan = $\frac{y}{x}$ , cot = $\frac{x}{y}$ , sec = $1 over x$ and csc = $1 over y$ .
Inspect every denominator first. A zero denominator gives undefined; otherwise do the division.

Worked exampleAxis values after clockwise turns

Find all six functions at −540° and −450°. In words, remove complete turns, find the axis point, then read the six ratios.

A clockwise turn of −450° ends at the bottom of the circle.

One and a half clockwise turns ends on the negative x-axis.

One and a quarter clockwise turns ends on the negative y-axis.

Reference table: read the point entry directly under its θ label.

Reference table: read the sin θ entry directly under its θ label.

Reference table: read the cos θ entry directly under its θ label.

Reference table: read the tan θ entry directly under its θ label.

Reference table: read the cot θ entry directly under its θ label.

Reference table: read the sec θ entry directly under its θ label.

Reference table: read the csc θ entry directly under its θ label.

The cosine reference columns reverse the sine values.

−540° + 2 × 360° = 180°, so use (−1, 0), with r = 1.Adding full turns preserves the terminal side.
sin(−540°) = $0 over 1$ = 0; cos(−540°) = $−1 over 1$ = −1; tan(−540°) = $0 over −1$ = 0.Substitute y, x and r in the coordinate definitions.
cot(−540°) is undefined; sec(−540°) = −1; csc(−540°) is undefined.Cotangent and cosecant divide by y = 0. Secant is $1 over −1$ .
−450° + 2 × 360° = 270°, so use (0, −1), with r = 1.The terminal side points down after the full turns are removed.
sin(−450°) = −1; cos(−450°) = 0; cot(−450°) = $0 over −1$ = 0; csc(−450°) = −1.Read the coordinates and substitute in the ratios.
tan(−450°) and sec(−450°) are undefined.Their denominator x is 0.

Answer

At −540°:
sin = 0
cos = −1
tan = 0
cot = undefined
sec = −1
csc = undefined
At −450°:
sin = −1
cos = 0
tan = undefined
cot = 0
sec = undefined
csc = −1

Check At both points + = 1. A zero numerator over −1 gives zero; a zero denominator gives no value.

Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.

Rung 1Rung 1: a full-turn multiple

Find sine and cosine at 720°. In words, read the two coordinates after two turns.

Two turns end at the starting point.

720° − 720° = 0°, so the point is (1, 0).Two full turns return to the starting point.
sin 720° = 0 and cos 720° = 1.Sine is y and cosine is x.

Answer

sin 720° = 0
cos 720° = 1

Check + = 1.

Rung 2Rung 2: a negative axis angle

Find tan(−270°) and cot(−270°). In words, identify which fraction has a zero bottom.

The negative rotation ends on the positive y-axis.

−270° + 360° = 90°, so P = (0, 1).A clockwise three-quarter turn ends where a counterclockwise quarter turn ends.
tan(−270°) = $1 over 0$ is undefined; cot(−270°) = $0 over 1$ = 0.Tangent divides by x; cotangent divides by y.

Answer

tan(−270°) = undefined
cot(−270°) = 0

Check The ray points up, so x is zero and y is not.

Rung 3Rung 3: all six after extra turns

Find all six functions at 630°. In words, remove a turn and use the bottom point.

630° ends down after one full turn and three quarters.

630° − 360° = 270°, so P = (0, −1) and r = 1.Coterminal angles share the terminal side.
sin = −1, cos = 0, cot = $0 over −1$ = 0, csc = $1 over −1$ = −1.Substitute the point into formulas with nonzero denominators.
tan = $−1 over 0$ and sec = $1 over 0$ are undefined.Their denominator x is zero.

Answer

sin 630° = −1
cos 630° = 0
tan 630° = undefined
cot 630° = 0
sec 630° = undefined
csc 630° = −1

Check 630° + 630° = (−1 + = 1.

Rung 4Rung 4: radians with a negative turn

Find all six functions at − $5 π over 2$ . In words, remove full clockwise laps and locate the axis point.

The radian angle represents the same clockwise travel as −450°.

− $5 π over 2$ + 4π = $3 π over 2$ = 270°.4π adds two full turns and preserves the ending ray.
Use P = (0, −1): sin = −1, cos = 0, cot = 0, csc = −1.With r = 1, the coordinate formulas give these four values.
tan and sec are undefined.Both put x = 0 on the bottom.

Answer

sin = −1
cos = 0
tan = undefined
cot = 0
sec = undefined
csc = −1

Check − radians = −450°, matching the main example.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: tan(−450°) = 0 because one coordinate is zero.

The angle ends at (0, −1), so tangent is . Zero is on the bottom.

✓ Instead: tan(−450°) is undefined; cot(−450°) = = 0.

✗ Not this: Every function is undefined when an angle ends on an axis.

Only the fractions with a zero denominator fail. Sine and cosine divide by r = 1.

✓ Instead: At −540°, sine is 0 and cosine is −1; only cotangent and cosecant are undefined.

Tips and tricks

Write the point before the functions. It prevents guessing which ratio has a zero denominator.
Circle the denominator: zero below means no value; zero above a nonzero number means zero.
Rebuild the axis table by walking right, up, left, down. Keep the general excluded-angle formulas on the cheat sheet.
Read reference tables by columns: keep the input label and its output in the same vertical column.

Trap. Writing 0 when the answer is undefined, or the reverse. = 0 is a real value; has no value. Look at which coordinate sits on the bottom before you answer.

Keep in mind

Zero on top is a real answer, but zero on the bottom is not: $0 over −1$ = 0, while $−1 over 0$ is undefined.
sin and cos never fail, because they divide by r, which is never 0: −810° + 3 × 360° = 270° ends at (0, −1), so sin(−810°) = −1 and cos(−810°) = 0.
Undefined at an angle means that angle is not in the domain: 450° is outside the domain of tan and sec but inside the domain of csc and cot.
The whole quadrantal table fits on a notebook-paper cheat sheet as one small plus sign with the points (1, 0), (0, 1), (−1, 0) and (0, −1), since every entry rebuilds from them.

Memory hookDraw a plus sign with dots at (1, 0), (0, 1), (−1, 0), (0, −1): cos is x, sin is y, and a zero bottom means undefined.

Flash cards: say the answer out loud, then flip

What is a quadrantal angle?

An angle whose terminal side lies on an axis: 0°, 90°, 180°, 270°, or one of those plus or minus full turns.

What does undefined mean for ?

No value exists: no number times 0 makes 1.

Where are tan and sec undefined? Where are cot and csc undefined?

tan and sec: where x = 0 (the y-axis), 90° + n·180°
cot and csc: where y = 0 (the x-axis), n·180°

Find sin, cos and tan of 900°.

900° − 720° = 180°, the point (−1, 0)
sin 900° = 0
cos 900° = −1
tan 900° = $0 over −1$ = 0

Is −630° in the domain of tan?

No. −630° + 720° = 90°, the point (0, 1) has x = 0, and tan = is undefined.

cot 630° = ?

0. 630° − 360° = 270°, the point (0, −1), and cot = = = 0. The zero is on top, so it is not undefined.