45° creates a perfect diagonal, bisecting a right angle. sin(45°) = cos(45°) = √2/2, tan(45°) = 1. This angle is fundamental in isosceles right triangles and creates equal x and y components. Full treatment of 45° · All special angles
First Quadrant (I)
In Quadrant I, both x and y coordinates are positive. All trigonometric functions (sin, cos, tan) are positive here. This quadrant contains angles from 0° to 90° and represents the 'northeast' section of the coordinate plane. Learn more about Quadrant I · All quadrants
Reference Angle
A reference angle is the acute angle between the terminal ray and the x-axis. It's always between 0° and 90° and helps determine the sign and magnitude of trigonometric functions. Reference angles make calculations easier by relating any angle to a familiar acute angle. Learn more about reference angles · Related angle pairs
Trigonometric Functions
The six trigonometric functions relate angles to ratios in right triangles and positions on the unit circle. Sine (sin) represents the y-coordinate, cosine (cos) the x-coordinate, and tangent (tan) the ratio y/x. The reciprocal functions are cosecant (csc), secant (sec), and cotangent (cot). Learn more about the six trig functions · Reading the values table
Type any value into the Angle Value input to update the diagram, properties panel, and trigonometric values in real time. The explorer accepts positive numbers, negative numbers, and values beyond a single rotation, so 720° or −45° are both valid inputs.
Tips for entering angles: • Use the up and down arrow keys for fine adjustments. • Type a decimal like 52.5 to explore non-special positions. • Press Reset to return to 0° at any time.
The terminal ray on the diagram rotates counterclockwise for positive values and clockwise for negative values, matching standard mathematical convention.
Choosing Degrees or Radians
Use the unit dropdown next to the input to switch between Degrees and Radians. Switching the unit reinterprets the current numeric value rather than converting it, so the diagram may jump.
When to pick each unit: • Degrees are intuitive for geometry, navigation, and common reference angles like 30°, 45°, and 60°. • Radians match the natural input for calculus, physics, and the unit circle, where π/2, π, and 2π are the key markers.
The properties panel always reports both the degree value and the radian value, making the tool useful for verifying degree–radian conversions in either direction.
Using Quick Preset Angles
The Quick Angles row provides one-click access to sixteen standard positions: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, and 330°.
Why these specific values: • They are the special angles of the unit circle, with exact sin, cos, and tan values. • They cover all four quadrants evenly, making them ideal for spotting symmetry. • Pairs like 30° and 150° or 45° and 135° show how reference angles repeat.
Each preset auto-converts to the active unit, so switching to radians and clicking 90° enters π/2.
Display Toggle Options
Four checkboxes in the Display Options group control what the diagram renders:
• Show Angle Arc draws the blue arc sweeping from the initial ray to the terminal ray. • Show Reference Lines adds the unit circle, axes, and Roman numeral quadrant labels (I, II, III, IV). • Show Complementary Angle overlays a green dashed arc to the vertical axis when the angle is between 0° and 90°. • Show Supplementary Angle overlays a red dashed arc to the negative x-axis when the angle is between 0° and 180°.
Toggle these to isolate a concept. Turning off the reference lines, for example, leaves only the rays and arc, useful for clean explanations.
Reading the Properties Panel
The first column of the panel summarizes everything the explorer derives from the input.
Fields you will see: • Current Angle in both degrees and radians, useful for unit conversion checks. • Type classifies the angle as acute, right, obtuse, straight, or reflex. • Quadrant identifies which of the four regions the terminal ray points into. • Reference Angle gives the acute angle between the terminal ray and the x-axis, always between 0° and 90°.
Watching these values change as you sweep through angles is one of the fastest ways to build intuition about how classification rules work.
Working with Related and Coterminal Angles
The middle and right columns of the panel report angles connected to the current value.
Related Angles column: • Complementary: 90°−θ, shown only when θ is between 0° and 90°. • Supplementary: 180°−θ, shown only when θ is between 0° and 180°. • Reflex: 360°−θ, the angle on the opposite side of the rotation.
Coterminal Angles column: • Positive adds 360° to the input. • Negative subtracts 360°. • General Formθ+360°n describes every coterminal value, where n is any integer.
Reading the Trigonometric Values Table
Below the diagram, two compact tables show the six trigonometric functions evaluated at the current angle.
How values are displayed: • At special angles (30°, 45°, 60°, etc.), the table shows the exact form, such as fractions and radical expressions. • At other angles, values are rounded decimals. • Undefined points like tan90° display the infinity symbol.
An angle measures the amount of rotation between two rays meeting at a common vertex. In the explorer, the initial ray points along the positive x-axis and the terminal ray rotates to the input position.
Two units of measure dominate mathematics: • Degrees divide a full rotation into 360 equal parts. • Radians measure rotation by arc length on a unit circle, with 2π radians equal to 360°.
The explorer applies these rules automatically and updates the Type field as you change the angle. For full coverage of each type with examples, see the angle types page.
Complementary, Supplementary, and Reference Angles
Three derived angles appear repeatedly in trigonometry:
• Complementary: a pair summing to 90°. Used in cofunction identities like sinθ=cos(90°−θ). • Supplementary: a pair summing to 180°. Common in geometry and triangle angle sums. • Reference angle: the acute angle between the terminal ray and the x-axis. Used to evaluate trigonometric functions in any quadrant by relating them to first-quadrant values.
Continue exploring with these connected resources:
• Unit Circle — visualize how angle position determines sin, cos, and tan on a circle of radius 1. • Degrees and Radians Converter — quick numeric conversion without the diagram. • Trigonometric Functions — full theory of sine, cosine, tangent, and their reciprocals. • Special Angles Table — exact values at 0°, 30°, 45°, 60°, 90°, and beyond. • Coterminal Angles — practice problems and proofs for angles sharing a terminal side. • Reference Angle Calculator — focused tool for the reduction rules across quadrants.
Quadrants and Trigonometric Signs
The coordinate axes divide the plane into four quadrants, labeled counterclockwise with Roman numerals. An angle in standard position belongs to the quadrant containing its terminal ray: Quadrant I holds angles between 0° and 90°, Quadrant II between 90° and 180°, Quadrant III between 180° and 270°, and Quadrant IV between 270° and 360°. Angles landing exactly on an axis (0°, 90°, 180°, 270°) are called quadrantal angles and sit on the boundary rather than inside a quadrant.
The quadrant determines the sign of every trigonometric function, because cosθ is the x-coordinate and sinθ the y-coordinate of the point where the terminal ray meets the unit circle: • Quadrant I — all six functions positive. • Quadrant II — only sin (and csc) positive. • Quadrant III — only tan (and cot) positive. • Quadrant IV — only cos (and sec) positive.
A common mnemonic is All Students Take Calculus — one word per quadrant, naming the functions that stay positive there. Combine the quadrant's sign rule with the reference angle and you can evaluate any trigonometric function of any angle using first-quadrant values alone.
Special Angles and Their Exact Values
The special angles are the multiples of 30° and 45°: sixteen positions around the unit circle — 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330° — plus 360°, which completes the rotation. They matter because their trigonometric values are exact — expressible with fractions and square roots instead of rounded decimals.
The exact values come from two classical triangles: • The 45-45-90 triangle gives sin45°=cos45°=22 and tan45°=1. • The 30-60-90 triangle gives sin30°=21, cos30°=23 and sin60°=23, cos60°=21. • The axis angles 0°, 90°, 180°, 270°, and 360° take their values directly from the unit-circle coordinates (1,0), (0,1), (−1,0), and (0,−1).
Every other special angle is one of these first-quadrant angles reflected into another quadrant: its reference angle is 30°, 45°, or 60°, and only the sign changes. For example, 150° has reference angle 30°, so sin150°=21 while cos150°=−23. This is why the explorer can display exact forms at every preset: sixteen positions, but only three underlying triangles.
Acute Angles
An acute angle measures strictly between 0° and 90°: its terminal ray has left the initial ray but has not yet reached the vertical. Every acute angle lies in Quadrant I, where all six trigonometric values are positive.
Acute angle, frozen at 50°
The terminal ray inside the first quadrant, its arc stopping well short of the right-angle mark.
The diagram freezes the explorer at 35°, a typical acute angle: the blue arc has swept just over a third of the way to the vertical axis. Acute angles are the building blocks of trigonometry — both non-right angles of every right triangle are acute, and reference angles are acute by definition.
Familiar examples: each angle of an equilateral triangle (60°), the 30° and 45° angles of the classical special triangles, and most angles of elevation in applied problems. Any input with 0°<θ<90° reports Type: acute in the explorer's properties panel.
Right Angles
A right angle measures exactly 90°: the terminal ray points straight up the positive y-axis, perpendicular to the initial ray. It is the boundary between acute and obtuse, and is marked in diagrams with a small square at the vertex.
Right angle, frozen at 90°
The terminal ray stands exactly perpendicular to the initial side — one quarter of a full turn.
At 90° the trigonometric values hit their quarter-turn landmarks: sin90°=1, cos90°=0, and tan90° is undefined, because the ratio cosθsinθ divides by zero there. The explorer's values table shows the infinity symbol for this case.
Right angles are the backbone of coordinate geometry: perpendicular lines, the corners of squares and rectangles, and the defining angle of every right triangle. A 90° angle is also a quadrantal angle — it sits on the boundary between Quadrant I and Quadrant II rather than inside either.
Obtuse Angles
An obtuse angle measures strictly between 90° and 180°: past the vertical, but not yet a straight line. Its terminal ray points into Quadrant II, where sine stays positive while cosine and tangent turn negative.
Obtuse angle, frozen
A terminal ray leaning into the second quadrant: more than 90°, less than 180°.
The diagram freezes the explorer at 110°. Its reference angle is 180°−110°=70°, so sin110°=sin70° while cos110°=−cos70° — the standard Quadrant II sign pattern.
Obtuse angles appear in triangles (a triangle can contain at most one), in bearings and rotations, and as the supplements of acute angles: every obtuse angle pairs with the acute angle 180°−θ to form a straight line.
Straight Angles
A straight angle measures exactly 180°: the two rays point in opposite directions and together form a straight line through the vertex. It is half of a full rotation.
Straight angle, frozen at 180°
The two rays form one straight line — half of a complete rotation.
At 180° the terminal ray lies along the negative x-axis, giving sin180°=0, cos180°=−1, and tan180°=0. Like 90°, it is a quadrantal angle — the boundary between Quadrant II and Quadrant III.
Straight angles anchor the idea of supplementary pairs: two angles are supplementary precisely when they compose a straight angle. The straight angle also marks the halfway point of the unit circle, where cosine reaches its minimum value of −1.
Reflex Angles
A reflex angle measures strictly between 180° and 360°: more than half a rotation, but less than a full turn. Every terminal-ray position below the x-axis admits two readings — the reflex angle measured counterclockwise, and the smaller angle 360°−θ on the other side.
Reflex angle, frozen
Beyond 180°: the arc sweeps the long way around toward a full turn.
The diagram freezes the explorer at 250°, deep in Quadrant III. The blue arc bends back past the straight-angle line: that long sweep is what makes the angle reflex. Its non-reflex partner is 360°−250°=110°.
Reflex angles matter wherever full rotational position matters — compass bearings, phase angles, rotations in graphics — and the explorer's Reflex field in the Related Angles column always reports the partner angle on the opposite side of the rotation.
Quadrant I
Quadrant I spans 0° to 90°: the region where both coordinates are positive. Any angle whose terminal ray lands here has every trigonometric value positive — sine, cosine, tangent, and all three reciprocals.
Quadrant I, frozen
The shaded quarter holds angles between 0° and 90°; here terminal ray and reference angle coincide.
The diagram shades Quadrant I with the explorer frozen at 50°. In this quadrant an angle is its own reference angle: no reduction is needed, which is why first-quadrant values serve as the reference data for the whole circle.
Quadrant I is where the special-triangle values live in their pure positive form: 30°, 45°, and 60° with their exact sines and cosines. Every other quadrant's values are these same magnitudes with signs adjusted.
Quadrant II
Quadrant II spans 90° to 180°: x negative, y positive. Only sine (and its reciprocal, cosecant) stays positive here; cosine and tangent are negative.
Quadrant II, frozen
Angles between 90° and 180°; the marked reference angle leans on the negative x-axis.
The diagram shades Quadrant II with the explorer frozen at 140°; the orange arc marks its reference angle180°−140°=40°. The reduction rule for this quadrant is θref=180°−θ, so for example sin140°=sin40° and cos140°=−cos40°.
The supplementary pairs live across Quadrants I and II: 30°/150°, 45°/135°, 60°/120° — equal sines, opposite cosines. That symmetry is easiest to see by sweeping the explorer from 40° to 140° and watching the values table.
Quadrant III
Quadrant III spans 180° to 270°: both coordinates negative. Sine and cosine are both negative, which makes their ratio positive — tangent (and cotangent) are the functions that stay positive here.
Quadrant III, frozen
Angles between 180° and 270°; the reference angle is measured up from the negative x-axis.
The diagram shades Quadrant III with the explorer frozen at 230°; the orange arc marks the reference angle230°−180°=50°. The rule for this quadrant is θref=θ−180°, giving tan230°=tan50° while sin230°=−sin50° and cos230°=−cos50°.
Every Quadrant III angle is the 180°-shift of a Quadrant I angle — which is exactly why tangent repeats with period 180° rather than 360°.
Quadrant IV
Quadrant IV spans 270° to 360°: x positive again, y still negative. Cosine (and secant) are positive; sine and tangent are negative.
Quadrant IV, frozen
Angles between 270° and 360°; the reference angle closes the gap to the positive x-axis.
The diagram shades Quadrant IV with the explorer frozen at 320°; the orange arc marks the reference angle360°−320°=40°. The rule is θref=360°−θ, so cos320°=cos40° while sin320°=−sin40°.
Quadrant IV angles are the mirror images of Quadrant I angles across the x-axis — the same geometry that makes cosine an even function (cos(−θ)=cosθ) and sine an odd one (sin(−θ)=−sinθ), since −40° and 320° are coterminal.
Special Angle: 0°
0° is the starting position: the terminal ray coincides with the initial ray along the positive x-axis, and no rotation has occurred. It is a quadrantal angle, sitting on a boundary rather than inside any quadrant.
0°, frozen
Read from the terminal point: sin = 0, cos = 1, tan = 0 — the y-coordinate, the x-coordinate, and their ratio.
sin0°=0cos0°=1tan0°=0
On the unit circle the terminal point is (1,0) — cosine at its maximum, sine at zero. Every multiple of 360° is coterminal with 0° and shares these values, which is why the explorer treats 360° and 0° as the same position.
Special Angle: 30°
30° is the smallest angle of the 30-60-90 triangle family and one-twelfth of a full rotation. In radians it is π/6.
30°, frozen
Read from the terminal point: sin = 1/2, cos = √3/2, tan = 1/√3 — the y-coordinate, the x-coordinate, and their ratio.
sin30°=21cos30°=23tan30°=33
The value sin30°=21 comes straight from the 30-60-90 triangle, where the side opposite 30° is exactly half the hypotenuse. 30° is the reference angle for 150°, 210°, and 330° — their values differ from these only in sign.
Special Angle: 45°
45° bisects the right angle: its terminal ray is the diagonal y=x, and its sine and cosine are equal. In radians it is π/4.
45°, frozen
Read from the terminal point: sin = √2/2, cos = √2/2, tan = 1 — the y-coordinate, the x-coordinate, and their ratio.
sin45°=22cos45°=22tan45°=1
The values come from the 45-45-90 triangle — an isosceles right triangle with legs 1 and hypotenuse 2. Because the legs are equal, tan45°=1 exactly. 45° is the reference angle for 135°, 225°, and 315°.
Special Angle: 60°
60° is the larger acute angle of the 30-60-90 triangle and the interior angle of the equilateral triangle. In radians it is π/3.
60°, frozen
Read from the terminal point: sin = √3/2, cos = 1/2, tan = √3 — the y-coordinate, the x-coordinate, and their ratio.
sin60°=23cos60°=21tan60°=3
Note the swap with 30°: sine and cosine trade values, because 30° and 60° are complementary and sinθ=cos(90°−θ). 60° is the reference angle for 120°, 240°, and 300°.
Special Angle: 90°
90° is the quarter turn: the terminal ray points straight up the positive y-axis. It is a quadrantal angle on the boundary between Quadrant I and Quadrant II, and equals π/2 radians.
90°, frozen
Read from the terminal point: sin = 1, cos = 0, tan = undefined — the y-coordinate, the x-coordinate, and their ratio.
sin90°=1cos90°=0tan90° undefined
The terminal point on the unit circle is (0,1): sine peaks at its maximum while cosine crosses zero, and the tangent ratio divides by zero — the explorer's table shows ∞. Secant is likewise undefined here, while cotangent equals 0.
Special Angle: 120°
120° lies in Quadrant II, 30° past the vertical. It is the supplement of 60° and equals 2π/3 radians.
120°, frozen
Read from the terminal point: sin = √3/2, cos = −1/2, tan = −√3 — the y-coordinate, the x-coordinate, and their ratio.
sin120°=23cos120°=−21tan120°=−3
Its reference angle is 180°−120°=60°, so the magnitudes are exactly the 60° values with Quadrant II signs: sine positive, cosine and tangent negative. 120° is also the interior angle of a regular hexagon.
Special Angle: 135°
135° is the Quadrant II diagonal — the supplement of 45°, lying along the line y=−x. In radians it is 3π/4.
135°, frozen
Read from the terminal point: sin = √2/2, cos = −√2/2, tan = −1 — the y-coordinate, the x-coordinate, and their ratio.
sin135°=22cos135°=−22tan135°=−1
With reference angle45°, its sine and cosine share the same magnitude 22 and differ only in sign, forcing tan135°=−1 exactly.
Read from the terminal point: sin = 1/2, cos = −√3/2, tan = −1/√3 — the y-coordinate, the x-coordinate, and their ratio.
sin150°=21cos150°=−23tan150°=−33
Its reference angle is 30°, so sin150°=sin30°=21 — the classic illustration that supplementary angles have equal sines while their cosines are opposite.
Special Angle: 180°
180° is the half turn: the terminal ray points along the negative x-axis, forming a straight line with the initial ray. A quadrantal angle, it equals π radians.
180°, frozen
Read from the terminal point: sin = 0, cos = −1, tan = 0 — the y-coordinate, the x-coordinate, and their ratio.
Read from the terminal point: sin = −1/2, cos = −√3/2, tan = 1/√3 — the y-coordinate, the x-coordinate, and their ratio.
sin210°=−21cos210°=−23tan210°=33
Its reference angle is 210°−180°=30°. Both sine and cosine go negative in Quadrant III, so their ratio — tangent — turns positive again, matching tan30° exactly.
Special Angle: 225°
225° is the Quadrant III diagonal, pointing opposite to 45° along the extension of the line y=x. In radians it is 5π/4.
225°, frozen
Read from the terminal point: sin = −√2/2, cos = −√2/2, tan = 1 — the y-coordinate, the x-coordinate, and their ratio.
sin225°=−22cos225°=−22tan225°=1
With reference angle45° and both coordinates negative, sine and cosine are equal — so tangent equals 1, exactly as at 45°. Tangent's period of 180° is visible here: 225°=45°+180°.
Special Angle: 240°
240° lies 60° past the straight angle in Quadrant III, diametrically opposite 60°. It equals 4π/3 radians.
240°, frozen
Read from the terminal point: sin = −√3/2, cos = −1/2, tan = √3 — the y-coordinate, the x-coordinate, and their ratio.
sin240°=−23cos240°=−21tan240°=3
Its reference angle is 60°; the 30-60-90 magnitudes carry over with both coordinate signs flipped, leaving the tangent at a positive 3.
Special Angle: 270°
270° is the three-quarter turn: the terminal ray points straight down the negative y-axis. A quadrantal angle between Quadrant III and Quadrant IV, it equals 3π/2 radians.
270°, frozen
Read from the terminal point: sin = −1, cos = 0, tan = undefined — the y-coordinate, the x-coordinate, and their ratio.
sin270°=−1cos270°=0tan270° undefined
The terminal point is (0,−1): sine bottoms out at its minimum −1 while cosine crosses zero again, making tangent undefined — the second and last such point in a full rotation.
Special Angle: 300°
300° lies in Quadrant IV, 60° short of a full turn. It equals 5π/3 radians and is coterminal with −60°.
300°, frozen
Read from the terminal point: sin = −√3/2, cos = 1/2, tan = −√3 — the y-coordinate, the x-coordinate, and their ratio.
sin300°=−23cos300°=21tan300°=−3
Its reference angle is 360°−300°=60°. Cosine turns positive again in Quadrant IV while sine stays negative — the mirror image of 60° across the x-axis.
Special Angle: 315°
315° is the Quadrant IV diagonal, 45° short of a full turn and coterminal with −45°. In radians it is 7π/4.
315°, frozen
Read from the terminal point: sin = −√2/2, cos = √2/2, tan = −1 — the y-coordinate, the x-coordinate, and their ratio.
sin315°=−22cos315°=22tan315°=−1
With reference angle45°, sine and cosine again share the magnitude 22 but now with opposite signs, giving tan315°=−1.
Special Angle: 330°
330° sits 30° short of completing the rotation, in Quadrant IV, coterminal with −30°. It equals 11π/6 radians.
330°, frozen
Read from the terminal point: sin = −1/2, cos = √3/2, tan = −1/√3 — the y-coordinate, the x-coordinate, and their ratio.
sin330°=−21cos330°=23tan330°=−33
Its reference angle is 30°: the familiar 21 and 23 reappear one last time before the rotation closes, with sine negative below the x-axis.
Special Angle: 360°
360° completes the full rotation: the terminal ray returns to the positive x-axis, coterminal with 0°. In radians it is 2π.
360°, frozen
Read from the terminal point: sin = 0, cos = 1, tan = 0 — the y-coordinate, the x-coordinate, and their ratio.
sin360°=0cos360°=1tan360°=0
Geometrically the position is identical to 0° — same terminal point (1,0), same values — but the full blue circle records that an entire turn has been swept. Every angle θ shares its values with θ+360°n: this is the coterminal relationship, and it is why the trigonometric functions are periodic.
Complementary Angles
Two angles are complementary when their measures sum to 90°. Each is the other's complement: the complement of θ is 90°−θ, defined for angles between 0° and 90°.
Complementary pair, frozen
A 35° angle with its green 55° complement stacked on top — together exactly one right angle.
The diagram freezes the explorer at 35° with the Show Complementary Angle overlay on: the green dashed arc measures the remaining 55° up to the vertical axis, and 35°+55°=90°.
Complements drive the cofunction identities: sinθ=cos(90°−θ) and tanθ=cot(90°−θ) — "cosine" literally means *sine of the complement*. That is why sine and cosine swap values between 30° and 60°, and coincide at 45°, which is its own complement. Together with supplementary and reference angles, complements make up the panel's related angles column.
Supplementary Angles
Two angles are supplementary when their measures sum to 180° — together they form a straight angle. The supplement of θ is 180°−θ, defined for angles between 0° and 180°.
Supplementary pair, frozen
A 110° angle and its red 70° supplement completing the straight line.
The diagram freezes the explorer at 110° with the Show Supplementary Angle overlay on: the red dashed arc measures the 70° remaining to the negative x-axis, and 110°+70°=180°.
Supplementary pairs have equal sines and opposite cosines: sin(180°−θ)=sinθ and cos(180°−θ)=−cosθ. Linear pairs formed by intersecting lines are always supplementary, and co-interior angles between parallel lines sum to 180° for the same reason.
Reference Angles
The reference angle of θ is the acute angle between its terminal ray and the x-axis — always between 0° and 90°, whichever quadrant θ lies in.
Reference angle, frozen at 140°
The acute gap between the terminal ray and the negative x-axis — 40° — is the reference angle.
The diagram freezes the explorer at 140°: the orange arc against the negative x-axis is the reference angle 180°−140°=40°. The rule depends on the quadrant:
Reference angles reduce every trigonometric evaluation to a first-quadrant one: take the value at θref, then apply the quadrant's sign. This single idea is what makes the sixteen special positions around the circle follow from just three triangles.
Coterminal Angles
Coterminal angles share the same terminal ray but differ by whole rotations: θ and θ+360°n land in exactly the same position for every integer n.
Coterminal pair, frozen
Two opposite sweeps, one terminal ray: 45° and −315° are the same position reached two ways.
The diagram shows 45° (solid blue arc) and −315° (dashed purple arc): one short counterclockwise sweep, one long clockwise sweep, the same terminal ray. The explorer's Coterminal column always lists θ+360° and θ−360°, plus the general form θ+360°n.
Because trigonometric functions see only the position on the circle, coterminal angles have identical values for all six functions: sin405°=sin45° and cos(−315°)=cos45°. Reducing an angle to its coterminal representative between 0° and 360° is the standard first step of every evaluation.
The Six Trigonometric Functions
On the unit circle, the terminal point of angle θ has coordinates (cosθ,sinθ) — cosine is the x-coordinate and sine is the y-coordinate. Tangent is their ratio cosθsinθ, the slope of the terminal ray.
The six functions, frozen
The terminal point’s coordinates ARE the functions: x is cosine, y is sine, and their ratio is tangent — the other three are reciprocals.
The diagram freezes the explorer at 50°: the green horizontal segment is cos50°, the red dashed vertical segment is sin50°, and the marked point is where the terminal ray crosses the unit circle.
The remaining three functions are reciprocals: cscθ=sinθ1, secθ=cosθ1, cotθ=tanθ1. Each is undefined wherever its partner is zero — that is why the explorer's table shows ∞ for tan90°. Together the six functions turn every question about angles into a question about coordinates.