Step 1 of 6
The two axes cut the plane into four quarters, numbered anticlockwise from the top right.
The x axis and the y axis cross at the origin and divide the plane into four regions. Each region is a , from the Latin for a quarter.
They are numbered anticlockwise, starting at the top right. Quadrant 1 is top right, quadrant 2 is top left, quadrant 3 is bottom left, quadrant 4 is bottom right. Anticlockwise, because that is the direction angles are measured in, so the numbering follows the rotation.
In terms of angles measured from the positive x axis: quadrant 1 covers 0 to 90 degrees, quadrant 2 covers 90 to 180, quadrant 3 covers 180 to 270, and quadrant 4 covers 270 to 360.
Quadrant 1 is the only one a right triangle can reach, which is why page 1 stopped there. The other three are what the unit circle bought.
Teal: the vertical drop, sin. Amber: the horizontal run, cos. All four teal segments are 0.500 long. All four amber segments are 0.866 long. Only the direction changes, and the direction is the sign.
An arrowhead points up or right for a positive value, down or left for a negative one. The + and − glyphs are plain text, never coloured. Click a circle to set the page angle.
What you are looking at
Four small circles laid out in their true quadrant positions, all showing the same reference angle. The vertical teal drop is the sine and the horizontal amber run is the cosine. Every teal segment is the same length as every other, and so is every amber one. Only the directions change, and the direction is the sign. Drag the reference angle slider to move all four together, or click any circle to send the rest of the page to that angle.
Step 2 of 6
The sign of each function comes from the signs of the x and y coordinates in that quadrant.
There is nothing to memorise here. The point on the circle is (cos θ, sin θ), so cosine carries the sign of x, sine carries the sign of y, and tangent is y over x so it carries the sign of the quotient.
Quadrant 1: x is positive and y is positive. So cosine is positive, sine is positive, and tangent is positive over positive, so positive. All three are positive.
Quadrant 2: x is negative and y is positive. Sine is positive. Cosine is negative. Tangent is positive over negative, so negative. Only sine is positive.
Quadrant 3: x is negative and y is negative. Sine is negative, cosine is negative, and tangent is negative over negative, which is positive. Only tangent is positive.
Quadrant 4: x is positive and y is negative. Cosine is positive, sine is negative, tangent is negative over positive, so negative. Only cosine is positive.
That is the whole rule, and it is a consequence of where the point sits, not a fact of its own. If you ever forget it, sketch the circle, put the point in the quadrant, and read the signs off the coordinates. That takes about five seconds and cannot be misremembered.
| Quadrant | these two decide everything to the right | the six functions that follow | ||||||
|---|---|---|---|---|---|---|---|---|
| sign of x | sign of y | sin teal, pairs with csc | cos amber, pairs with sec | tan purple, pairs with cot | csc teal, pairs with sin | sec amber, pairs with cos | cot purple, pairs with tan | |
| Q1 0 to 90° | + x is positive | + y is positive | ||||||
| Q2 90 to 180° | − x is negative | + y is positive | ||||||
| Q3 180 to 270° | − x is negative | − y is negative | ||||||
| Q4 270 to 360° | + x is positive | − y is negative | ||||||
Hover or focus any function cell to see the division its sign comes from.
A tick pointing up and a + mean positive; a tick pointing down and a − mean negative. Both glyphs are plain text. The colour under each heading says only which family the function belongs to: teal for sin and csc, amber for cos and sec, purple for tan and cot, solid for the function and dashed for its reciprocal.
150° is in quadrant 2, so that row is outlined. Every sign in the table is read from six(135°), not typed in. The magnitudes are elsewhere: this figure is only about sign. For example csc in Q2 is 1.155, positive, because sin is positive there.
What you are looking at
One row per quadrant, showing the sign of x, the sign of y, and the sign of all six functions that follow from them. The two coordinate columns are the only given; every column to their right is worked out from them. Hover or focus any function cell and the division it comes from is printed underneath, so quadrant 3 reads as a negative divided by a negative giving a positive, which is the one case that looks wrong at first. The row you are currently standing in is outlined.
Step 3 of 6
CAST is the four quadrant results as a word, read anticlockwise starting from the bottom right.
Collect the four results. Quadrant 1: All positive. Quadrant 2: Sine positive. Quadrant 3: Tangent positive. Quadrant 4: Cosine positive.
Take the initials in quadrant order and you get A, S, T, C. Take them starting from quadrant 4 instead and you get C, A, S, T, which spells a word. That is the CAST rule.
The catch is the starting point. CAST is read anticlockwise beginning at quadrant 4, the bottom right, then up to quadrant 1, then round to 2 and 3. Reading it clockwise from the top right, or anticlockwise from the top right, both give the wrong assignment. This is where the mnemonic loses people, and it loses them because the word is memorable while the starting corner is not.
ASTC is equally common and avoids the problem. It starts at quadrant 1 and runs anticlockwise in numerical order, which is the natural direction. It is not a word, so it usually gets a sentence attached, such as "All Students Take Calculus".
Neither is a reason. Both are compressions of the previous step, which is where the actual argument lives.
S sits in quadrant 2 for sine, A in quadrant 1 for all, T in quadrant 3 for tangent, C in quadrant 4 for cosine. Reading CAST means starting at quadrant 4 and going anticlockwise, which spells CAST. The letters never move between the two readings; only the corner you start from does. No colour is used here at all: these are letters, not functions.
150° is in quadrant 2, so that square is shaded: sin and csc positive. Every list of positives here is read from the same computation as the table above, not typed in.
What you are looking at
The four quadrants again, in the same positions as before, each carrying its letter, the word the letter stands for, and the functions that are positive there. The arrow through the middle is the reading order, and it starts at the bottom right, which is the part that gets forgotten. Switch to ASTC and nothing moves except the numbers and where the arrow starts: the same four quadrants with the same four meanings, read from a different corner. The square you are standing in is shaded.
Step 4 of 6
Every angle has an acute partner, and the value comes from that partner plus a sign.
The of an angle is the acute angle it makes with the x axis. Not with the y axis, and not with the nearest axis. The x axis.
Finding it is a subtraction that depends on the quadrant. In quadrant 1, the reference angle is the angle itself. In quadrant 2, it is 180 degrees minus the angle. In quadrant 3, it is the angle minus 180 degrees. In quadrant 4, it is 360 degrees minus the angle.
The reason this works is symmetry. A point at 150 degrees and a point at 30 degrees are mirror images in the y axis. Mirror images have coordinates of the same size, differing at most in sign. So the function values have the same magnitude, and only the sign has to be worked out.
That gives a procedure for any angle at all. Find the reference angle. Look up or compute the function for that acute angle, which is what page 1’s triangle definitions cover. Then attach the sign the quadrant demands, from the previous two steps.
Two numbers and a sign, and it works for any angle including negative ones and ones past a full turn. For those, subtract or add whole turns of 360 degrees first until you land between 0 and 360, which changes nothing because the functions are periodic.
| function | at 150° | sign × value at 30° | agree |
|---|---|---|---|
| sin | +0.500 | +0.500 | yes |
| cos | −0.866 | −0.866 | yes |
| tan | −0.577 | −0.577 | yes |
at 150°: (−0.866, +0.500) at 30°: (+0.866, +0.500)
Teal: the vertical drop, sin. Amber: the horizontal run, cos. The paler set with the hollow point is the same picture at the reference angle, 30°. The two teal segments are both 0.500 long and point the same way; the two amber segments are both 0.866 long and point opposite ways. Only an arrowhead and a glyph carry the sign, never a colour.
Both positions are drawn from the start, so the still picture already carries the comparison; folding only moves the paler copy onto the current one. The right hand column takes its sign from quadrant 2 and its size from 30°, then the two columns are compared: they agree.
What you are looking at
The circle with the current angle drawn, and the acute angle between the radius and the nearest part of the x axis marked separately with its own arc and value. A mirror copy of the radius is drawn in quadrant 1 at the reference angle, so the two points are visibly reflections. The coordinate readouts for both points sit side by side: identical magnitudes, and signs that differ according to the quadrant.
Step 5 of 6
Work 150 degrees through the procedure end to end.
Take 150 degrees.
Which quadrant? 150 is between 90 and 180, so quadrant 2.
Reference angle? In quadrant 2 it is 180 minus the angle, so 180 minus 150, which is 30 degrees.
Signs? Quadrant 2 is the S in CAST, so sine is positive there and cosine and tangent are negative.
Now assemble. sin(150 degrees) = +sin(30 degrees) = 0.5 exactly. cos(150 degrees) = minus cos(30 degrees), and cos(30 degrees) is √3/2, so the answer is minus √3/2, which is minus 0.866 to three decimal places. tan(150 degrees) = minus tan(30 degrees) = minus 1/√3, which is minus 0.577 to three decimal places.
Check it against the picture. At 150 degrees the point on the unit circle is up and to the left, at coordinates (minus √3/2, 1/2). The y coordinate is positive, which is the positive sine. The x coordinate is negative, which is the negative cosine. Nothing was memorised and nothing needed a calculator.
The same procedure handles 210 degrees (quadrant 3, reference angle 30, only tangent positive) and 330 degrees (quadrant 4, reference angle 30, only cosine positive). All four of those angles share the reference angle 30 degrees and therefore share the same three magnitudes, in different sign combinations.
Click a card. It sets the angle to 150°, marks the part of the picture that step is about, and scrolls to it.
What you are looking at
Four cards, one per step of the procedure: quadrant, signs, reference angle, result. Clicking one sets the page to 150 degrees and marks the part of the picture that step is about, in the four quadrants above and in the folded circle. Both coordinate pairs stay printed throughout. To see 210 or 330 instead, click their circle in the four-quadrant figure: every magnitude holds still and only the signs change, which is the point.
Step 6 of 6
Secant, cosecant and cotangent carry the same sign as the function they invert.
CAST covers three functions. The other three need no extra rule.
One divided by a positive number is positive. One divided by a negative number is negative. So a reciprocal always has the same sign as the original. That settles all three at once.
Secant has the sign of cosine, so it is positive in quadrants 1 and 4. Cosecant has the sign of sine, so it is positive in quadrants 1 and 2. Cotangent has the sign of tangent, so it is positive in quadrants 1 and 3.
Reading CAST as covering six functions rather than three: quadrant 1 has all six positive; quadrant 2 has sine and cosecant positive and the other four negative; quadrant 3 has tangent and cotangent positive; quadrant 4 has cosine and secant positive.
The one thing to keep separate is undefined against negative. At exactly 90 degrees cosine is 0, so secant and tangent are undefined there. Undefined is not a sign. It is the absence of a value, and it sits on the boundary between two quadrants rather than inside one.
| Quadrant | these two decide everything to the right | the six functions that follow | ||||||
|---|---|---|---|---|---|---|---|---|
| sign of x | sign of y | sin teal, pairs with csc | cos amber, pairs with sec | tan purple, pairs with cot | csc teal, pairs with sin | sec amber, pairs with cos | cot purple, pairs with tan | |
| Q1 0 to 90° | + x is positive | + y is positive | ||||||
| Q2 90 to 180° | − x is negative | + y is positive | ||||||
| Q3 180 to 270° | − x is negative | − y is negative | ||||||
| Q4 270 to 360° | + x is positive | − y is negative | ||||||
Hover or focus any function cell to see the division its sign comes from.
A tick pointing up and a + mean positive; a tick pointing down and a − mean negative. Both glyphs are plain text. The colour under each heading says only which family the function belongs to: teal for sin and csc, amber for cos and sec, purple for tan and cot, solid for the function and dashed for its reciprocal.
150° is in quadrant 2, so that row is outlined. Every sign in the table is read from six(135°), not typed in. The magnitudes are elsewhere: this figure is only about sign. For example csc in Q2 is 1.155, positive, because sin is positive there.
- sin 0.500
- csc 2.000
- cos −0.866
- sec −1.155
- tan −0.577
- cot −1.732
A solid swatch is the function, a dashed one its reciprocal. Teal is the sine family, amber the cosine family, purple the tangent family.
What you are looking at
The same table again, read across all six columns. Each header names the function it pairs with, and hovering a cell lights up its partner in the same row: the linked pairs always carry matching signs, which is the rule this step adds. Park the angle on an axis, at 90 or 180 or 270 degrees, and no row is outlined at all; the note underneath names the functions that are undefined there rather than giving them a sign.