The unit circle is a mathematical abstraction model invented to give us an intuitive understanding of how trigonometric functions behave in the coordinate system with respect to different angles. Rather than being just another geometric shape, it serves as a visual and computational tool that transforms abstract trigonometric relationships into concrete, observable patterns.
We create this circle by placing its center at the origin (0,0) of the coordinate plane and setting the radius to exactly 1. This positioning is crucial because when we take any point on the circle and drop perpendicular lines to the x and y axes, we automatically create a right triangle. The hypotenuse of this triangle is always the radius (which equals 1), while the legs are the x and y coordinates of the point.
This model allows us to see trigonometric functions as coordinates on a circle, making it possible to understand their behavior across all possible angles, not just the limited range we can explore with right triangles.
Why Radius = 1?
The choice of radius = 1 is not arbitrary—it's a deliberate simplification that eliminates unnecessary complexity from trigonometric calculations. Here's exactly why this matters: In any right triangle, we define trigonometric functions as ratios:
sinθ=hypotenuseopposite side
cosθ=hypotenuseadjacent side
When we replace the hypotenuse with 1, the trigonometric functions drastically simplify:
sinθ=1opposite side=opposite side cosθ=1adjacent side=adjacent side
This means sine becomes simply the opposite side, and cosine becomes simply the adjacent side. The unit circle allows us to visualize these values directly as coordinates, eliminating the need for division calculations.
Angle Measurements in the Unit Circle
The unit circle uses two primary systems for measuring angles: degrees and radians. Understanding both systems is essential because different applications favor different units.
Degrees: The most familiar system, where a complete rotation around the circle equals 360°. This system divides the circle into 360 equal parts, making it intuitive for everyday use. In the unit circle, we start measuring from the positive x-axis (0°) and move counterclockwise.
Radians: The mathematical standard, where a complete rotation equals 2π radians (approximately 6.28). One radian is defined as the angle created when the arc length equals the radius. Since our circle has radius 1, this means one radian corresponds to an arc length of 1 unit along the circle's circumference.
The conversion between these systems is straightforward:
radians=degrees×180π
degrees=radians×π180
Why radians matter: While degrees feel more natural, radians create cleaner mathematical relationships. In calculus and advanced mathematics, formulas involving trigonometric functions work more elegantly with radians. For example, the derivative of sin(x) is simply cos(x) when x is in radians, but requires additional conversion factors when x is in degrees.
Standard position: Regardless of the unit system, angles in the unit circle are measured from the positive x-axis in a counterclockwise direction. This creates a consistent reference point for all calculations and makes it possible to extend trigonometry beyond the first quadrant.
The Four Quadrants: A Sign Language
One of the most crucial concepts for mastering trigonometry is understanding how the signs of sine and cosine change as we move around the circle. The unit circle is divided into four quadrants, each with its own "personality":
Quadrant I (0° to 90° or 0 to π/2 radians): Both x and y coordinates are positive, so both cosine and sine are positive. This is the "happy quadrant" where everything is positive and straightforward.
Quadrant II (90° to 180° or π/2 to π radians): The x-coordinate becomes negative while y stays positive. This means cosine is negative but sine remains positive. Think "sine is still climbing, but cosine has crossed over to the negative side."
Quadrant III (180° to 270° or π to 3π/2 radians): Both coordinates are negative, making both cosine and sine negative. This is the "opposite quadrant" from Quadrant I.
Quadrant IV (270° to 360° or 3π/2 to 2π radians): The x-coordinate returns to positive while y becomes negative. Cosine is positive again, but sine has gone negative.
You can observe these sign changes by using the interactive tool above. Set the angle to 45° (π/4 radians), then 135° (3π/4 radians), then 225° (5π/4 radians), and finally 315° (7π/4 radians). Watch how the values in the trigonometric table change signs as you move through each quadrant.
This unit circle shows all four quadrants with degree markings every 30°. Notice how the signs of sine and cosine change in each quadrant - this is the foundation of trigonometric calculations.
Certain angles on the unit circle are particularly important because they produce exact, "clean" trigonometric values that appear frequently in mathematics and real-world applications. These special angles have trigonometric values that can be expressed as simple fractions involving square roots, rather than long decimal approximations.
These angles, along with their equivalents in other quadrants, form the backbone of trigonometric calculations. Notice the pattern: 30° and 60° are complementary (they add to 90°), and their sine and cosine values are swapped. The 45° angle creates an isosceles right triangle, giving equal sine and cosine values.
These values come from two fundamental right triangles: the 30-60-90 triangle and the 45-45-90 triangle. Because their side ratios involve simple square roots and fractions, calculations remain exact rather than requiring decimal approximations.
Extended Special Angles:
The unit circle extends these basic angles to all four quadrants:
120° (2π/3), 135° (3π/4), 150° (5π/6) in Quadrant II
210° (7π/6), 225° (5π/4), 240° (4π/3) in Quadrant III
300° (5π/3), 315° (7π/4), 330° (11π/6) in Quadrant IV
Try entering these angles in the interactive tool above. You'll notice that the absolute values of sine and cosine remain the same as their first-quadrant counterparts, but the signs change according to the quadrant rules we discussed earlier.
Zero degrees is where every measurement on the circle begins: the terminal ray lies along the positive x-axis and the blue point sits at (1,0).
0° (0), frozen with the hover box open
The point sits at (1, 0): the red leg is cos 0° = 1, the blue leg sin 0° = 0, and the white box repeats the exact values the tool shows on hover.
All six values.sin0°=0, cos0°=1, tan0°=0, csc0° undefined, sec0°=1 and cot0° undefined.
Being an axis angle, 0° sits on the boundary between two quadrants rather than inside one, so the usual quadrant sign rules do not apply; the coordinates (1,0) decide everything directly. This is also the state the reset button (↺) returns to, and where the angle measurement of every other point starts.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 30° (π/6)
At 30° the point is (23,21): the vertical sine leg is exactly half the radius, the first of the "clean" values the circle is famous for.
30° (π/6), frozen with the hover box open
The point sits at (√3/2, 1/2): the red leg is cos 30° = √3/2, the blue leg sin 30° = 1/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin30°=21, cos30°=23, tan30°=33, csc30°=2, sec30°=323 and cot30°=3.
The point lies in Quadrant I with reference angle 30°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 150°. The values come from the 30-60-90 triangle: hypotenuse 1, short leg 21, long leg 23. Its complement 60° has the same two values swapped.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 45° (π/4)
At 45° the terminal ray bisects the first quadrant and the point is (22,22): the sine and cosine legs are equal.
45° (π/4), frozen with the hover box open
The point sits at (√2/2, √2/2): the red leg is cos 45° = √2/2, the blue leg sin 45° = √2/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin45°=22, cos45°=22, tan45°=1, csc45°=2, sec45°=2 and cot45°=1.
The point lies in Quadrant I with reference angle 45°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 135°. The values come from the 45-45-90 triangle: two equal legs and hypotenuse 1 force each leg to be 21=22, which is why tan45°=1 exactly.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 60° (π/3)
At 60° the point is (21,23), the 30° coordinates swapped: the cosine leg is now the half-radius.
60° (π/3), frozen with the hover box open
The point sits at (1/2, √3/2): the red leg is cos 60° = 1/2, the blue leg sin 60° = √3/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin60°=23, cos60°=21, tan60°=3, csc60°=323, sec60°=2 and cot60°=33.
The point lies in Quadrant I with reference angle 60°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 120°. The same 30-60-90 triangle as at 30°, read from the other acute vertex, which is what the complementary relation cos60°=sin30° means geometrically.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 90° (π/2)
At 90° the terminal ray points straight up and the point is (0,1): the cosine leg has shrunk to nothing and the sine leg is the whole radius.
90° (π/2), frozen with the hover box open
The point sits at (0, 1): the red leg is cos 90° = 0, the blue leg sin 90° = 1, and the white box repeats the exact values the tool shows on hover.
All six values.sin90°=1, cos90°=0, tan90° undefined, csc90°=1, sec90° undefined and cot90°=0.
Being an axis angle, 90° sits on the boundary between two quadrants rather than inside one, so the usual quadrant sign rules do not apply; the coordinates (0,1) decide everything directly. Because the cosine leg is zero, tan90° and sec90° divide by zero and the table shows Undefined for both. The circle explains the vertical asymptotes of the tangent graph without any algebra.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 120° (2π/3)
At 120° the point is (−21,23): the first special angle past the y-axis, so the cosine leg now runs to the left.
120° (2π/3), frozen with the hover box open
The point sits at (−1/2, √3/2): the red leg is cos 120° = −1/2, the blue leg sin 120° = √3/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin120°=23, cos120°=−21, tan120°=−3, csc120°=323, sec120°=−2 and cot120°=−33.
The point lies in Quadrant II with reference angle 60°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 60°. Its reference angle is 60°, so every magnitude equals the 60° value; only the signs follow Quadrant II.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 135° (3π/4)
At 135° the point is (−22,22): the 45° picture mirrored across the y-axis.
135° (3π/4), frozen with the hover box open
The point sits at (−√2/2, √2/2): the red leg is cos 135° = −√2/2, the blue leg sin 135° = √2/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin135°=22, cos135°=−22, tan135°=−1, csc135°=2, sec135°=−2 and cot135°=−1.
The point lies in Quadrant II with reference angle 45°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 45°. Its reference angle is 45°, and mirroring across the y-axis flips exactly one coordinate, which is why cos135°=−cos45° while sin135°=sin45°.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 150° (5π/6)
At 150° the point is (−23,21): the sine leg is again half the radius, the cosine leg negative and long.
150° (5π/6), frozen with the hover box open
The point sits at (−√3/2, 1/2): the red leg is cos 150° = −√3/2, the blue leg sin 150° = 1/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin150°=21, cos150°=−23, tan150°=−33, csc150°=2, sec150°=−323 and cot150°=−3.
The point lies in Quadrant II with reference angle 30°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 30°. Its reference angle is 30°, so the magnitudes are the 30° values with the Quadrant II signs.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 180° (π)
At 180° the terminal ray points along the negative x-axis and the point is (−1,0): a straight angle, half a turn.
180° (π), frozen with the hover box open
The point sits at (−1, 0): the red leg is cos 180° = −1, the blue leg sin 180° = 0, and the white box repeats the exact values the tool shows on hover.
All six values.sin180°=0, cos180°=−1, tan180°=0, csc180° undefined, sec180°=−1 and cot180° undefined.
Being an axis angle, 180° sits on the boundary between two quadrants rather than inside one, so the usual quadrant sign rules do not apply; the coordinates (−1,0) decide everything directly. Like 90° it is an axis angle: the sine leg vanishes, so csc180° and cot180° are undefined, and it is the supplementary partner of 0°.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 210° (7π/6)
At 210° the point is (−23,−21): the 30° picture rotated half a turn, both legs now negative.
210° (7π/6), frozen with the hover box open
The point sits at (−√3/2, −1/2): the red leg is cos 210° = −√3/2, the blue leg sin 210° = −1/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin210°=−21, cos210°=−23, tan210°=33, csc210°=−2, sec210°=−323 and cot210°=3.
The point lies in Quadrant III with reference angle 30°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 330°. Its reference angle is 30°; rotating by half a turn negates both coordinates, so the tangent, a ratio of two negatives, is positive again.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 225° (5π/4)
At 225° the point is (−22,−22): the 45° point through the origin, both coordinates negative and equal.
225° (5π/4), frozen with the hover box open
The point sits at (−√2/2, −√2/2): the red leg is cos 225° = −√2/2, the blue leg sin 225° = −√2/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin225°=−22, cos225°=−22, tan225°=1, csc225°=−2, sec225°=−2 and cot225°=1.
The point lies in Quadrant III with reference angle 45°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 315°. Its reference angle is 45°. Because both legs are equal, tan225°=1 just as at 45°, the signature of Quadrant III.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 240° (4π/3)
At 240° the point is (−21,−23): the 60° point rotated by 180°.
240° (4π/3), frozen with the hover box open
The point sits at (−1/2, −√3/2): the red leg is cos 240° = −1/2, the blue leg sin 240° = −√3/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin240°=−23, cos240°=−21, tan240°=3, csc240°=−323, sec240°=−2 and cot240°=33.
The point lies in Quadrant III with reference angle 60°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 300°. Its reference angle is 60°, and the magnitudes are the 60° values with both signs negative.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 270° (3π/2)
At 270° the terminal ray points straight down and the point is (0,−1): three quarters of a turn.
270° (3π/2), frozen with the hover box open
The point sits at (0, −1): the red leg is cos 270° = 0, the blue leg sin 270° = −1, and the white box repeats the exact values the tool shows on hover.
All six values.sin270°=−1, cos270°=0, tan270° undefined, csc270°=−1, sec270° undefined and cot270°=0.
Being an axis angle, 270° sits on the boundary between two quadrants rather than inside one, so the usual quadrant sign rules do not apply; the coordinates (0,−1) decide everything directly. Cosine is zero here, so tan270° and sec270° are undefined, as at 90°; the sine reaches its minimum −1.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 300° (5π/3)
At 300° the point is (21,−23): the 60° picture reflected in the x-axis, so only the sine has changed sign.
300° (5π/3), frozen with the hover box open
The point sits at (1/2, −√3/2): the red leg is cos 300° = 1/2, the blue leg sin 300° = −√3/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin300°=−23, cos300°=21, tan300°=−3, csc300°=−323, sec300°=2 and cot300°=−33.
The point lies in Quadrant IV with reference angle 60°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 240°. Its reference angle is 60°; the 60° point reflected in the x-axis keeps the cosine and negates the sine, the Quadrant IV rule.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 315° (7π/4)
At 315° the point is (22,−22): the 45° point reflected in the x-axis.
315° (7π/4), frozen with the hover box open
The point sits at (√2/2, −√2/2): the red leg is cos 315° = √2/2, the blue leg sin 315° = −√2/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin315°=−22, cos315°=22, tan315°=−1, csc315°=−2, sec315°=2 and cot315°=−1.
The point lies in Quadrant IV with reference angle 45°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 225°. Its reference angle is 45°, so tan315°=−1: equal legs of opposite sign. The angle is also written −45°, the same terminal ray reached clockwise.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Special Angle 330° (11π/6)
At 330° the point is (23,−21): the last marked point before the circle closes back on 0°.
330° (11π/6), frozen with the hover box open
The point sits at (√3/2, −1/2): the red leg is cos 330° = √3/2, the blue leg sin 330° = −1/2, and the white box repeats the exact values the tool shows on hover.
All six values.sin330°=−21, cos330°=23, tan330°=−33, csc330°=−2, sec330°=323 and cot330°=−3.
The point lies in Quadrant IV with reference angle 30°, the acute angle between the terminal ray and the x-axis. Its mirror image across the y-axis is 210°. Its reference angle is 30°; the 30° values with a negative sine. Written as −30° it shows the even-odd rules: cos(−30°)=cos30°, sin(−30°)=−sin30°.
Hover the marked point in the tool to open the box shown in the frozen picture, or type the angle in the input field; the table then reads the decimal forms of the exact values above. The full list of marked points is in Special Angles.
Quadrant I: 0° to 90°
Between 0° and 90° the terminal ray stays in the upper-right quarter of the plane. Both coordinates of the blue point are positive, so every one of the six functions is positive: the only quadrant where nothing needs a sign check.
Quadrant I, frozen at 50°
The tool parked at 50°: both legs point into the positive half-axes, and every entry in the table row below the circle is positive.
Sign pattern.sin>0, cos>0, tan>0; the reciprocals csc, sec, cot share those signs. The frozen picture is the tool at 50°: a short red cosine leg, a longer blue sine leg, both pointing into the positive directions.
Reference angle. In this quadrant an angle is its own reference angle, which is why the special angles30°, 45° and 60° live here and every other special value is a reflected copy of one of them.
As the angle grows from 0° to 90° the sine leg lengthens from 0 to 1 while the cosine leg shrinks from 1 to 0; drag the point slowly across the quadrant and watch the two table entries trade places.
Quadrant II: 90° to 180°
Between 90° and 180° the point is in the upper-left quarter: x negative, y positive. Sine (and its reciprocal cosecant) stays positive; cosine, tangent, secant and cotangent turn negative.
Quadrant II, frozen at 140°
The tool at 140°: the red cosine leg runs left of the origin, so cos, tan, sec and cot read negative while sin and csc stay positive.
Sign pattern.sin>0, cos<0, tan<0, and the reciprocals follow: csc>0, sec<0, cot<0. The mnemonic "Students" for this quadrant means only sine survives.
Reference angle. For θ in this quadrant the reference angle is 180°−θ; the picture shows 140°, whose reference angle is 40°. The special points here are 120°, 135° and 150°, mirrors of 60°, 45° and 30°.
Notice that the red cosine leg now runs to the left of the origin: its length is unchanged from the mirrored first-quadrant angle, but the table reports it with a minus sign.
Quadrant III: 180° to 270°
Between 180° and 270° both coordinates are negative. Sine and cosine are both negative, but tangent and cotangent, ratios of two negatives, are positive again.
Quadrant III, frozen at 230°
The tool at 230°: both legs point into the negative half-axes; sin and cos are negative, and tan and cot, ratios of two negatives, are positive.
Sign pattern.sin<0, cos<0, tan>0; hence csc<0, sec<0, cot>0. Tangent is positive because it is the ratio of two negative legs.
Reference angle. Here the reference angle is θ−180°; the frozen 230° has reference angle 50°. The special points are 210°, 225° and 240°, each the first-quadrant point rotated by half a turn.
Rotating any point by 180° negates both coordinates, so sin(θ+180°)=−sinθ and cos(θ+180°)=−cosθ; the tool shows both legs flipping when you add 180 to the input.
Quadrant IV: 270° to 360°
Between 270° and 360° the point is in the lower-right quarter: x positive, y negative. Cosine and secant are positive; sine, cosecant, tangent and cotangent are negative.
Quadrant IV, frozen at 320°
The tool at 320°: the blue sine leg hangs below the axis, so sin, csc, tan and cot read negative while cos and sec are positive.
Sign pattern.sin<0, cos>0, tan<0; hence csc<0, sec>0, cot<0. Only cosine and secant keep their first-quadrant sign.
Reference angle. The reference angle is 360°−θ; the picture shows 320°, reference angle 40°. The special points are 300°, 315° and 330°, the first-quadrant points reflected in the x-axis.
Angles in this quadrant are often written as negatives: 320° and −40° share the terminal ray. Type −40 in the input to see the same point reached clockwise; this is the picture behind cos(−θ)=cosθ and sin(−θ)=−sinθ.
The Six Functions in the Table
Below the circle the tool keeps a one-row table of all six trigonometric functions of the current angle, each with a Copy button. Every entry is read straight off the picture: two legs, and four ratios built from them.
The two coordinates are the primary functions, sine (the blue leg) and cosine (the red leg). Their ratio is the tangent, the slope of the radius. The remaining three are reciprocals: cosecant of sine, secant of cosine, and cotangent of tangent.
The table prints Undefined whenever a denominator is zero: tangent and secant at 90° and 270°, cosecant and cotangent at 0° and 180°. Each function has its own section below, frozen at an angle where its value is easy to read.
Sine on the Unit Circle
On the unit circle the sine of an angle is the y-coordinate of the blue point: the length of the vertical blue leg, taken with a sign. The first column of the table reports exactly that number.
Sine column highlighted, frozen at 50°
The blue vertical leg from the point down to the x-axis is sin θ; the highlighted first column of the table reads its length, 0.76604 at 50°.
Definition.sinθ=y, the signed height of the point above the x-axis. Because the radius is 1, no division is needed: the leg is the value. At 50°, the frozen state, sin50°≈0.76604.
Range and sign. The leg can never be longer than the radius, so −1≤sinθ≤1; it is 1 at 90°, −1 at 270° and 0 at 0° and 180°. It is positive in Quadrant I and Quadrant II, negative below the axis.
Drag the point around the circle and watch the sine column rise and fall: that column traced against the angle is the sine wave, which the Trigonometric Functions Graphs tool draws.
Cosine on the Unit Circle
The cosine of an angle is the x-coordinate of the blue point: the length of the horizontal red leg along the x-axis, with a sign. The second column of the table reports it.
Cosine column highlighted, frozen at 50°
The red horizontal leg from the origin to the foot of the point is cos θ; the highlighted second column reads 0.64279 at 50°.
Definition.cosθ=x, the signed distance of the point to the right of the y-axis. At the frozen 50°, cos50°≈0.64279; together with the sine the two legs satisfy x2+y2=1, which is the Pythagorean identity read off the circle.
Range and sign.−1≤cosθ≤1, with 1 at 0°, −1 at 180° and 0 at 90° and 270°. It is positive to the right of the y-axis, in Quadrant I and Quadrant IV.
Cosine is the "co-sine", the sine of the complementary angle: the red leg at θ has the same length as the blue leg at 90°−θ. Compare the table at 30° and 60° to see the two columns swap.
Tangent on the Unit Circle
The tangent is the ratio of the two legs, tanθ=cosθsinθ=xy: the slope of the blue radius line. The third column of the table reports it, and shows Undefined whenever the radius is vertical.
Tangent column highlighted, frozen at 50°
The blue radius line has slope y/x; the highlighted tangent column reads that slope, 1.19175 at 50°.
Slope reading. At 50° the radius rises 0.76604 for a run of 0.64279, so tan50°≈1.19175, the value highlighted in the frozen table. Steeper radius, larger tangent; at 45° the legs are equal and the tangent is exactly 1.
Where it breaks. At 90° and 270° the cosine leg is zero, the slope is infinite and the table prints Undefined: these are the vertical asymptotes of the tangent graph.
Sign. Tangent is positive where the legs have the same sign, Quadrant I and Quadrant III, and negative in Quadrant II and Quadrant IV. Adding 180° negates both legs and leaves the ratio unchanged, so tangent repeats every half turn.
Cosecant on the Unit Circle
The cosecant is the reciprocal of the sine, cscθ=sinθ1=y1. The tool freezes here at 30°, where the sine leg is exactly half the radius and the cosecant is exactly 2.
Cosecant column highlighted, frozen at 30°
At 30° the blue sine leg is half the radius, so its reciprocal, the highlighted csc column, reads exactly 2.00000.
Reading it. Because the radius is 1, cscθ is "how many sine legs fit into the radius": at 30° the blue leg is 21, so two of them make one radius and csc30°=2, the highlighted table entry.
Range. A reciprocal of a number between −1 and 1 is never between −1 and 1: cscθ≤−1 or cscθ≥1. It equals 1 at 90° and −1 at 270°.
Undefined wherever the sine is zero, at 0° and 180°; near those angles the table value grows without bound. Its sign is the sign of the sine: positive above the x-axis, negative below.
Secant on the Unit Circle
The secant is the reciprocal of the cosine, secθ=cosθ1=x1. The tool freezes at 60°, where the red cosine leg is half the radius and the secant is exactly 2.
Secant column highlighted, frozen at 60°
At 60° the red cosine leg is half the radius, so the highlighted sec column reads its reciprocal, exactly 2.00000.
Reading it. Two copies of the 60° cosine leg span the radius, so sec60°=2, as the highlighted table cell shows. Compare with the cosecant at 30°: the same picture with the roles of the legs exchanged.
Range.secθ≤−1 or secθ≥1, with sec0°=1 at 0° and sec180°=−1 at 180°.
Undefined where the cosine is zero, at 90° and 270°, the same angles that break the tangent. Its sign follows the cosine: positive to the right of the y-axis, negative to the left.
Cotangent on the Unit Circle
The cotangent is the reciprocal of the tangent, cotθ=sinθcosθ=yx: run over rise. The tool freezes at 45°, where the legs are equal and the cotangent, like the tangent, is exactly 1.
Cotangent column highlighted, frozen at 45°
At 45° the two legs are equal, so x/y is 1; the highlighted cot column reads 1.00000, the same as the tangent column beside it.
Reading it. At 45° the red and blue legs have the same length 22, so their ratio is 1 either way round; the highlighted table cell reads 1.00000. Away from 45° the two columns are reciprocals: at 30°tan=33 and cot=3.
Undefined where the sine leg is zero, at 0° and 180°, exactly where the tangent is 0; and cot=0 at 90° and 270°, where the tangent is undefined.
The input field accepts any number of degrees. Type 390 and the point lands where 30° put it, but the display reads "1 rounds + 30.0°": the tool counts the complete turns and shows what is left.
Input 390°, frozen: one full round plus 30°
Input 390°: the point, legs and table are those of 30°, while the angle display above the table reads "1 rounds + 30.0°".
Coterminal angles. Angles that differ by a whole number of turns, θ and θ+360°k, share a terminal ray and therefore every trigonometric value: sin390°=sin30°=21, and the table row is identical to the one at 30°. This is the periodicity of the trigonometric functions seen directly.
Negative input works the same way: −30 is one clockwise turn short of 330°, and the display reduces it to a value between 0° and 360°. The green arc always shows the reduced angle, since a full turn would draw a closed circle over itself.
In radians the same statement reads sin(θ+2πk)=sinθ; see Angle Measurements for the conversion and the Angle Explorer for coterminal pairs drawn side by side.