Trigonometric Functions, the Unit Circle and Graphs

Extending a triangle idea to every angle

Inside a right triangle, an acute angle has sine and cosine ratios formed from its sides. Those definitions do not directly describe a negative rotation or an angle beyond a full turn. A unit circle lets us keep the familiar acute-angle values while extending the functions to every real input.

Place a circle of radius one at the origin O of coordinate axes. Start at A = (1, 0) and rotate through x radians, anticlockwise for positive x and clockwise for negative x. Let the endpoint on the circle be P = (a, b). Define the cosine of x to be its horizontal coordinate a and the sine to be its vertical coordinate b.

For an acute angle in the first quadrant, drop a perpendicular from P to the horizontal axis. The right triangle has hypotenuse one, horizontal side a and vertical side b. Its adjacent/hypotenuse ratio is a/1 = a, and its opposite/hypotenuse ratio is b/1 = b. The circle definitions therefore agree with the acute-angle ratios already known.

In other quadrants a or b may be negative. Coordinates carry signs even though triangle side lengths do not. That is the crucial extension: the terminal point records both magnitude and direction relative to the axes. Every real x determines one terminal point, so sine and cosine each give exactly one output for every real input.

Quadrantal angles and the fundamental identity

The four axis points are especially useful because their coordinates are immediate. Angles that are integer multiples of π/2 land on these axes and are called quadrantal angles. The equation of the circle also supplies a relationship between sine and cosine at every input.

Since P lies on a unit circle, its coordinates satisfy a² + b² = 1. In the first quadrant this follows directly from the right triangle and the Pythagorean theorem. The circle equation holds in every quadrant, so replacing a and b by cos x and sin x gives an identity valid for every real x.

From the axis points, sin x = 0 precisely when x is an integer multiple of π. Cos x = 0 precisely when x is an odd multiple of π/2. We write these inputs using an integer n, where ℤ denotes the set of all integers, including negative integers and zero.

The other four functions and their restrictions

Sine and cosine are defined directly by coordinates. The other four functions are built from them using division. Consequently, their excluded inputs are not arbitrary: they occur exactly where a denominator becomes zero.

Thus cosec and cot exclude x = nπ, while sec and tan exclude x = π/2 + nπ. Cot is the reciprocal of tan wherever both expressions are defined and tan is nonzero. The coordinate definition cos x/sin x is more fundamental for cot: it still gives cot(π/2) = 0, although tan(π/2) is undefined.

Exact values at familiar angles

Standard angles give reference values for later calculations. The acute values come from simple triangles, while the axis values come from the circle. Keeping them exact makes identity calculations clearer and avoids unnecessary decimal approximations.

An isosceles right triangle with perpendicular sides 1 and 1 has hypotenuse √2. Its 45° ratios give sin(π/4) = cos(π/4) = 1/√2. Bisect an equilateral triangle of side 2 to get a right triangle with sides 1, √3 and 2; its 30° and 60° ratios give the remaining acute reference values.

The reciprocal and quotient rows follow from the first two. For example, sec(π/3) = 1/(1/2) = 2, while cosec(π/3) = 1/(√3/2) = 2/√3. Equivalent exact forms such as 1/√2 and √2/2 describe the same value.

Signs come from quadrants

The horizontal and vertical coordinates have known signs in each quadrant. Those signs determine the signs of all six functions through their definitions. This removes the need to guess a sign after taking a square root.

In quadrant I both coordinates are positive; in quadrant II horizontal is negative and vertical positive; in quadrant III both are negative; in quadrant IV horizontal is positive and vertical negative. A reciprocal keeps its denominator’s sign, while a quotient is positive for equal signs and negative for opposite signs.

The table refers to open quadrants: it does not assign a plus or minus sign to a zero value on an axis. At an axis, calculate directly and check denominators. For an angle outside [0, 2π), first remove complete turns to locate its terminal side.

Negative angles and complete-turn repetition

Reflection and repeated turning are two different ways to connect angles. Reversing a rotation reflects the endpoint across the horizontal axis. Adding a full turn returns to exactly the same endpoint. Both observations produce useful identities.

The definitions then give tan(−x) = −tan x and cot(−x) = −cot x on their respective domains. Sec keeps its sign under reflection, while cosec changes sign. These statements also preserve excluded inputs; they do not make a previously undefined function value valid.

Sine, cosine and their reciprocals cosec and sec repeat after 2π. Tangent and cotangent repeat after π: after a half turn both coordinates change sign, and their quotient stays the same. Their excluded-input patterns also repeat after π. These are the fundamental periods of the six functions, although larger integer multiples are periods too.

Domains and ranges explain the graph shapes

The domain lists allowable inputs; the range lists outputs that really occur. Sine and cosine accept every real input because every rotation has a terminal point. Their coordinate values lie between −1 and 1, including both endpoints.

For sec and cosec, nonzero sine or cosine has magnitude at most one. Its reciprocal therefore has magnitude at least one. Outputs between −1 and 1 are impossible, but ±1 occur and all larger magnitudes can occur. Tangent and cotangent can take any real value, as the ratios vary from arbitrarily large negative to arbitrarily large positive values along their branches.

In the domain table n can be any integer. Square brackets at −1 and 1 include those outputs. Infinity is never an attained output or an ordinary endpoint; the parentheses indicate that a branch can grow without a finite bound. Reciprocal functions never equal zero.

The sine graph starts at 0, reaches 1 at π/2, returns to 0 at π, reaches −1 at 3π/2 and returns to 0 at 2π. Cosine starts at 1, reaches 0 at π/2, −1 at π, 0 at 3π/2 and 1 at 2π. These landmarks reproduce the moving point’s vertical and horizontal coordinates.

A tangent branch passes through zero at nπ and is broken at π/2 + nπ. Cotangent passes through zero at π/2 + nπ and is broken at nπ. Near an excluded input, a denominator becomes very small while the numerator stays away from zero, so the quotient can have arbitrarily large magnitude. A dashed vertical line indicates this behaviour; it is not part of the function graph.

The positive secant branches reach a minimum output 1, and its negative branches reach a highest output −1. Cosecant has corresponding branches with those same boundary heights. Their gaps and signs follow the reciprocal definitions. In particular, neither graph crosses the horizontal axis or enters the strip −1 < y < 1.

Following changes across the four quadrants

A repeating graph is easier to sketch when you know how its values change within each quadrant. The unit-circle point provides the starting intuition. As it turns anticlockwise, its horizontal and vertical coordinates rise or fall in predictable directions.

The quadrant intervals exclude their axis endpoints. A phrase such as “toward +∞” says that outputs exceed any chosen positive bound sufficiently close to the excluded input; it does not assign infinity as a value there. For example, tan x increases without bound as x approaches π/2 from the left, while tan(π/2) remains undefined.

To see a reciprocal change without memorising the table, consider quadrant I: sin x increases from near 0 toward 1, so 1/sin x decreases from very large positive values toward 1. In quadrant II cosine becomes more negative, from near 0 toward −1; its reciprocal rises from very large negative magnitude toward −1. The sign and the reciprocal relationship together explain what the graph does.