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Trigonometric Functions



The Sine Function
The Cosine Function
The Tangent Function
The Cosecant Function
The Secant Function
The Cotangent Function
Reciprocal and Quotient Relationships
Domain and Range Summary
Evaluating Trigonometric Functions at Any Angle
Finding All Function Values from One Known Value



From Geometric Ratios to Functions of a Real Variable

In right triangle trigonometry, sine and cosine are ratios — static numbers attached to a specific angle in a specific triangle. On the unit circle, they become coordinates — values that change as the angle sweeps around the circle. The next conceptual step is to treat them as functions: machines that accept any real number as input and return a real number as output, subject to the same analysis applied to polynomial, rational, exponential, or any other class of function.

This shift in perspective opens up the full toolkit of function analysis. Each of the six trigonometric functions has a domain (where it is defined), a range (what values it can produce), intervals of increase and decrease, symmetry properties, and characteristic behavior near its points of discontinuity. Sine and cosine are defined everywhere and bounded between −1-1 and 11. Tangent and cotangent are defined everywhere except at regularly spaced asymptotes, and their ranges span all real numbers. Cosecant and secant, as reciprocals of sine and cosine, inherit the zeros of their counterparts as points of undefinedness and are bounded away from zero but unbounded overall. These domains and ranges are not arbitrary — they follow inevitably from the unit circle definitions and govern every computation involving graphs, equations, inequalities, and inverse functions.



The Sine Function

The sine function assigns to each real number θ\theta the yy-coordinate of the corresponding point on the unit circle:

sin⁡(θ)=y-coordinate of the point at angle θ\sin(\theta) = y\text{-coordinate of the point at angle }\theta


Its domain is the entire real line — every real number corresponds to a position on the unit circle, so sin⁡(θ)\sin(\theta) is defined for all θ∈(−∞,∞)\theta \in (-\infty, \infty). Its range is the closed interval [−1,1][-1, 1], since the yy-coordinate of a point on a unit circle can never exceed 1 or fall below −1-1.

The function equals zero whenever the terminal side of θ\theta lies on the xx-axis — that is, at θ=nπ\theta = n\pi for every integer nn: …,−2π,−π,0,π,2π,…\ldots, -2\pi, -\pi, 0, \pi, 2\pi, \ldots. It reaches its maximum value of 11 at θ=π2+2nπ\theta = \frac{\pi}{2} + 2n\pi (the top of the circle) and its minimum value of −1-1 at θ=3π2+2nπ\theta = \frac{3\pi}{2} + 2n\pi (the bottom of the circle).

Sine is periodic with period 2π2\pi: sin⁡(θ+2π)=sin⁡(θ)\sin(\theta + 2\pi) = \sin(\theta) for all θ\theta. It is an odd function: sin⁡(−θ)=−sin⁡(θ)\sin(-\theta) = -\sin(\theta), which means its graph is symmetric about the origin. On the interval [−π2,π2]\left[-\frac{\pi}{2}, \frac{\pi}{2}\right], sine is strictly increasing — a fact that becomes essential when defining the inverse sine function.

Because the sine function is continuous, bounded, periodic, and smooth, it serves as the prototype for modeling any oscillating quantity. The general sinusoidal model y=Asin⁡(Bx−C)+Dy = A\sin(Bx - C) + D modifies amplitude, period, phase, and baseline, but the underlying behavior is always that of the sine function on the unit circle.

The Cosine Function

The cosine function assigns to each real number θ\theta the xx-coordinate of the corresponding point on the unit circle:

cos⁡(θ)=x-coordinate of the point at angle θ\cos(\theta) = x\text{-coordinate of the point at angle }\theta


Its domain is the entire real line, and its range is [−1,1][-1, 1] — identical to sine in both respects. The function equals zero whenever the terminal side lies on the yy-axis: at θ=π2+nπ\theta = \frac{\pi}{2} + n\pi for every integer nn. It reaches its maximum of 11 at θ=2nπ\theta = 2n\pi (the rightmost point of the circle) and its minimum of −1-1 at θ=π+2nπ\theta = \pi + 2n\pi (the leftmost point).

Cosine is periodic with the same period as sine: cos⁡(θ+2π)=cos⁡(θ)\cos(\theta + 2\pi) = \cos(\theta). Unlike sine, cosine is an even function: cos⁡(−θ)=cos⁡(θ)\cos(-\theta) = \cos(\theta). Geometrically, this follows from the unit circle — reflecting an angle across the xx-axis negates the yy-coordinate but preserves the xx-coordinate. On the graph, evenness manifests as symmetry about the yy-axis.

On the interval [0,π][0, \pi], cosine is strictly decreasing — running from cos⁡(0)=1\cos(0) = 1 down to cos⁡(π)=−1\cos(\pi) = -1. This monotonic interval is the one used to define the inverse cosine function.

The relationship between sine and cosine is captured by a phase shift: cos⁡(θ)=sin⁡(θ+π2)\cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right). The cosine wave is the sine wave translated π2\frac{\pi}{2} units to the left. This is not a coincidence — it reflects the geometric fact that the xx-coordinate at angle θ\theta equals the yy-coordinate at angle θ+π2\theta + \frac{\pi}{2} (a quarter rotation ahead on the circle). Equivalently, the cofunction identity cos⁡(θ)=sin⁡(π2−θ)\cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right) from right triangle trigonometry extends to all angles through the unit circle.

The Tangent Function

Tangent is defined as the ratio of sine to cosine:

tan⁡(θ)=sin⁡(θ)cos⁡(θ)=yx\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{y}{x}


where (x,y)(x, y) is the point on the unit circle at angle θ\theta. This ratio is undefined whenever cos⁡(θ)=0\cos(\theta) = 0 — that is, at θ=π2+nπ\theta = \frac{\pi}{2} + n\pi for every integer nn. These are precisely the angles where the terminal side is vertical (lying along the yy-axis), and the xx-coordinate is zero.

The domain of tangent is all real numbers except the odd multiples of π2\frac{\pi}{2}:

θ≠π2+nπ,n∈Z\theta \neq \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}


The range, by contrast, is the entire real line (−∞,∞)(-\infty, \infty). As θ\theta approaches an excluded value, cos⁡(θ)\cos(\theta) approaches zero while sin⁡(θ)\sin(\theta) approaches ±1\pm 1, driving the ratio toward +∞+\infty or −∞-\infty. On the graph, these excluded points correspond to vertical asymptotes.

Tangent is periodic with period π\pi — half the period of sine and cosine. This shorter period arises because tan⁡(θ+π)=sin⁡(θ+π)cos⁡(θ+π)=−sin⁡θ−cos⁡θ=sin⁡θcos⁡θ=tan⁡θ\tan(\theta + \pi) = \frac{\sin(\theta + \pi)}{\cos(\theta + \pi)} = \frac{-\sin\theta}{-\cos\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta. A half rotation negates both coordinates, but their ratio is unchanged.

Tangent is an odd function: tan⁡(−θ)=−tan⁡(θ)\tan(-\theta) = -\tan(\theta). On the interval (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) — a single period — tangent is strictly increasing, running from −∞-\infty to +∞+\infty. This is the interval used to define the inverse tangent function.

Geometrically, tan⁡(θ)\tan(\theta) measures the slope of the terminal side of θ\theta. A line from the origin to (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta) has slope sin⁡θcos⁡θ=tan⁡θ\frac{\sin\theta}{\cos\theta} = \tan\theta. This interpretation connects trigonometry to coordinate geometry and explains why tangent is undefined for vertical lines (slope is undefined when the run is zero).

The Cosecant Function

Cosecant is the reciprocal of sine:

csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}


It is defined wherever sin⁡(θ)≠0\sin(\theta) \neq 0 — that is, for all θ\theta except integer multiples of π\pi: θ≠nπ\theta \neq n\pi, n∈Zn \in \mathbb{Z}. At these excluded points (where the terminal side lies on the xx-axis and sin⁡θ=0\sin\theta = 0), cosecant is undefined, and the graph has vertical asymptotes.

The range of cosecant is (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty). Since ∣sin⁡θ∣≤1|\sin\theta| \leq 1, the reciprocal ∣csc⁡θ∣=1∣sin⁡θ∣≥1|\csc\theta| = \frac{1}{|\sin\theta|} \geq 1. Cosecant can never take a value between −1-1 and 11. It equals 11 when sin⁡θ=1\sin\theta = 1 (at θ=π2+2nπ\theta = \frac{\pi}{2} + 2n\pi) and equals −1-1 when sin⁡θ=−1\sin\theta = -1 (at θ=3π2+2nπ\theta = \frac{3\pi}{2} + 2n\pi).

Cosecant is an odd function, inheriting this property from sine: csc⁡(−θ)=1sin⁡(−θ)=1−sin⁡θ=−csc⁡θ\csc(-\theta) = \frac{1}{\sin(-\theta)} = \frac{1}{-\sin\theta} = -\csc\theta. Its period is 2π2\pi, the same as sine.

In right triangle trigonometry, cosecant equals hypotenuse over opposite — a ratio that is always greater than 1 for acute angles, since the hypotenuse exceeds every leg. The general function preserves this: the absolute value of cosecant is always at least 1.

Cosecant appears less frequently in elementary problems than sine, cosine, or tangent, but it plays a significant role in the Pythagorean identity 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta, in certain identities and formula derivations, and in calculus (as the derivative of −cot⁡θ-\cot\theta and in various integrals).

The Secant Function

Secant is the reciprocal of cosine:

sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}


It is defined wherever cos⁡(θ)≠0\cos(\theta) \neq 0, which excludes θ=π2+nπ\theta = \frac{\pi}{2} + n\pi for every integer nn — the same points where tangent is undefined. At these angles, the terminal side is vertical, the xx-coordinate is zero, and both tan⁡θ\tan\theta and sec⁡θ\sec\theta have vertical asymptotes on their graphs.

The range of secant mirrors that of cosecant: (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty). Since ∣cos⁡θ∣≤1|\cos\theta| \leq 1, the reciprocal satisfies ∣sec⁡θ∣≥1|\sec\theta| \geq 1, and secant can never produce a value strictly between −1-1 and 11. It equals 11 at θ=2nπ\theta = 2n\pi and −1-1 at θ=π+2nπ\theta = \pi + 2n\pi.

Secant is an even function: sec⁡(−θ)=1cos⁡(−θ)=1cos⁡θ=sec⁡θ\sec(-\theta) = \frac{1}{\cos(-\theta)} = \frac{1}{\cos\theta} = \sec\theta, since cosine is even. Its period is 2π2\pi.

The Pythagorean identity 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta connects secant directly to tangent. This relationship is heavily used in the simplification of identities and in calculus, where sec⁡2θ\sec^2\theta appears as the derivative of tan⁡θ\tan\theta and in the integral of sec⁡θtan⁡θ\sec\theta\tan\theta.

In the right triangle context, secant equals hypotenuse over adjacent — the reciprocal of the cosine ratio. Like cosecant, secant always has absolute value at least 1, reflecting the fact that the hypotenuse always exceeds any leg in a right triangle.

The Cotangent Function

Cotangent is defined as the ratio of cosine to sine, or equivalently as the reciprocal of tangent:

cot⁡(θ)=cos⁡(θ)sin⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{1}{\tan(\theta)}


It is undefined wherever sin⁡(θ)=0\sin(\theta) = 0: at θ=nπ\theta = n\pi for every integer nn. These are the same points where cosecant is undefined — where the terminal side lies along the xx-axis.

The range of cotangent is the entire real line (−∞,∞)(-\infty, \infty), identical to tangent. As θ\theta approaches an excluded value, sin⁡θ\sin\theta approaches zero while cos⁡θ\cos\theta approaches ±1\pm 1, and the ratio diverges.

Cotangent is periodic with period π\pi, matching tangent. The derivation is analogous: cot⁡(θ+π)=cos⁡(θ+π)sin⁡(θ+π)=−cos⁡θ−sin⁡θ=cot⁡θ\cot(\theta + \pi) = \frac{\cos(\theta + \pi)}{\sin(\theta + \pi)} = \frac{-\cos\theta}{-\sin\theta} = \cot\theta. It is an odd function: cot⁡(−θ)=−cot⁡θ\cot(-\theta) = -\cot\theta.

On the interval (0,π)(0, \pi), cotangent is strictly decreasing — running from +∞+\infty down to −∞-\infty. This contrasts with tangent, which is increasing on its principal interval. The graph of cotangent is a decreasing curve between consecutive asymptotes, while the graph of tangent is an increasing curve.

Cotangent equals zero where cosine equals zero: at θ=π2+nπ\theta = \frac{\pi}{2} + n\pi. In the right triangle, it equals adjacent over opposite — the reciprocal of the tangent ratio. The cofunction relationship cot⁡θ=tan⁡(π2−θ)\cot\theta = \tan\left(\frac{\pi}{2} - \theta\right) ties cotangent to tangent through complementary angles, consistent with the cofunction pattern established for all six functions.

Reciprocal and Quotient Relationships

The six trigonometric functions are not independent — they are connected by a network of algebraic relationships that reduce all six to just two: sine and cosine. Every other function is expressible in terms of these two.

The reciprocal relationships:

csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}


The quotient relationships:

tan⁡θ=sin⁡θcos⁡θ,cot⁡θ=cos⁡θsin⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}


Combining reciprocal and quotient relationships, any expression involving the six functions can be rewritten purely in terms of sin⁡θ\sin\theta and cos⁡θ\cos\theta. For example, sec⁡θtan⁡θ=1cos⁡θ⋅sin⁡θcos⁡θ=sin⁡θcos⁡2θ\sec\theta\tan\theta = \frac{1}{\cos\theta} \cdot \frac{\sin\theta}{\cos\theta} = \frac{\sin\theta}{\cos^2\theta}. This conversion strategy — "write everything in sine and cosine" — is one of the most reliable first steps when simplifying trigonometric expressions or proving identities.

The relationships also imply that the six functions naturally pair off. Sine and cosecant are reciprocals. Cosine and secant are reciprocals. Tangent and cotangent are reciprocals. Within each pair, the product is always 1:

sin⁡θ⋅csc⁡θ=1,cos⁡θ⋅sec⁡θ=1,tan⁡θ⋅cot⁡θ=1\sin\theta \cdot \csc\theta = 1, \quad \cos\theta \cdot \sec\theta = 1, \quad \tan\theta \cdot \cot\theta = 1


These hold at every point where both functions in the pair are defined. They are among the simplest of the trigonometric identities, but their utility is pervasive.

Domain and Range Summary

The domains and ranges of all six functions follow from their definitions and from the geometry of the unit circle. Sine and cosine, being coordinates of a point on the unit circle, are defined for all real inputs and confined to [−1,1][-1, 1]. The remaining four functions involve division by sine or cosine (or both), so they are undefined wherever the relevant denominator is zero.

Sine: domain (−∞,∞)(-\infty, \infty), range [−1,1][-1, 1]

Cosine: domain (−∞,∞)(-\infty, \infty), range [−1,1][-1, 1]

Tangent: domain {θ:θ≠π2+nπ, n∈Z}\{\theta : \theta \neq \frac{\pi}{2} + n\pi,\, n \in \mathbb{Z}\}, range (−∞,∞)(-\infty, \infty)

Cotangent: domain {θ:θ≠nπ, n∈Z}\{\theta : \theta \neq n\pi,\, n \in \mathbb{Z}\}, range (−∞,∞)(-\infty, \infty)

Secant: domain {θ:θ≠π2+nπ, n∈Z}\{\theta : \theta \neq \frac{\pi}{2} + n\pi,\, n \in \mathbb{Z}\}, range (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty)

Cosecant: domain {θ:θ≠nπ, n∈Z}\{\theta : \theta \neq n\pi,\, n \in \mathbb{Z}\}, range (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty)

Several patterns are worth noting. Tangent and secant share the same excluded points — both are undefined where cos⁡θ=0\cos\theta = 0. Cotangent and cosecant share their excluded points — both are undefined where sin⁡θ=0\sin\theta = 0. Tangent and cotangent have range (−∞,∞)(-\infty, \infty), meaning any real number is a valid output. Secant and cosecant cannot produce values between −1-1 and 11 — their outputs are always at least 1 in absolute value. These constraints on range directly affect what equations have solutions: sin⁡(x)=2\sin(x) = 2 has none (since 2∉[−1,1]2 \notin [-1, 1]), and csc⁡(x)=0.5\csc(x) = 0.5 has none (since 0.5∉(−∞,−1]∪[1,∞)0.5 \notin (-\infty, -1] \cup [1, \infty)).

Evaluating Trigonometric Functions at Any Angle

The procedure for evaluating a trigonometric function at any angle combines three components: coterminal reduction, reference angle computation, and quadrant-based sign assignment.

Step 1: Reduce to a standard range. If the angle exceeds 360°360° (or 2π2\pi), or if it is negative, find a coterminal angle in [0°,360°)[0°, 360°) or [0,2π)[0, 2\pi) by adding or subtracting full rotations. For example, sin⁡(750°)=sin⁡(750°−2×360°)=sin⁡(30°)\sin(750°) = \sin(750° - 2 \times 360°) = \sin(30°).

Step 2: Identify the quadrant. Determine which quadrant the angle falls in (or whether it is a quadrantal angle lying on an axis). This determines the sign of the function value.

Step 3: Find the reference angle. Compute the acute angle between the terminal side and the xx-axis. The reference angle identifies which set of exact values (from the special right triangles) to use.

Step 4: Evaluate and assign the sign. Look up the function value at the reference angle, then apply the sign from Step 2.

Example: evaluate cos⁡(7π4)\cos\left(\frac{7\pi}{4}\right).

The angle 7π4\frac{7\pi}{4} is in [0,2π)[0, 2\pi), so no coterminal reduction is needed. It lies in Quadrant IV (since 3π2<7π4<2π\frac{3\pi}{2} < \frac{7\pi}{4} < 2\pi). The reference angle is 2π−7π4=π42\pi - \frac{7\pi}{4} = \frac{\pi}{4}. Cosine is positive in Quadrant IV. Therefore cos⁡(7π4)=+cos⁡(π4)=22\cos\left(\frac{7\pi}{4}\right) = +\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}.

Example: evaluate tan⁡(240°)\tan(240°).

The angle 240°240° is in Quadrant III (180°<240°<270°180° < 240° < 270°). The reference angle is 240°−180°=60°240° - 180° = 60°. Tangent is positive in Quadrant III (both sine and cosine are negative, and a negative divided by a negative is positive). Therefore tan⁡(240°)=+tan⁡(60°)=3\tan(240°) = +\tan(60°) = \sqrt{3}.

This procedure works uniformly for every angle and every function. For non-standard angles — those whose reference angle is not 30°30°, 45°45°, or 60°60° — a calculator is needed for the reference angle evaluation, but the sign-determination step remains the same.

Finding All Function Values from One Known Value

A common task in trigonometry is: given the value of one trigonometric function and the quadrant of the angle, find the values of all six functions. The Pythagorean identity and the reciprocal/quotient relationships make this possible.

Suppose sin⁡θ=35\sin\theta = \frac{3}{5} and θ\theta is in Quadrant II. The Pythagorean identity gives:

cos⁡2θ=1−sin⁡2θ=1−925=1625\cos^2\theta = 1 - \sin^2\theta = 1 - \frac{9}{25} = \frac{16}{25}


So cos⁡θ=±45\cos\theta = \pm\frac{4}{5}. Since θ\theta is in Quadrant II, where cosine is negative: cos⁡θ=−45\cos\theta = -\frac{4}{5}.

From here, the remaining four functions follow:

tan⁡θ=sin⁡θcos⁡θ=3/5−4/5=−34\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{3/5}{-4/5} = -\frac{3}{4}


cot⁡θ=1tan⁡θ=−43\cot\theta = \frac{1}{\tan\theta} = -\frac{4}{3}


sec⁡θ=1cos⁡θ=−54\sec\theta = \frac{1}{\cos\theta} = -\frac{5}{4}


csc⁡θ=1sin⁡θ=53\csc\theta = \frac{1}{\sin\theta} = \frac{5}{3}


The quadrant information is essential. Without it, cos⁡θ\cos\theta could be +45+\frac{4}{5} or −45-\frac{4}{5}, and half the remaining values would change sign. The Pythagorean identity determines the magnitude; the quadrant determines the sign.

This process also works starting from tangent. If tan⁡θ=−724\tan\theta = -\frac{7}{24} and cos⁡θ>0\cos\theta > 0 (placing θ\theta in Quadrant IV), then a right triangle with legs 7 and 24 has hypotenuse 72+242=25\sqrt{7^2 + 24^2} = 25. In Quadrant IV, sine is negative and cosine is positive, so sin⁡θ=−725\sin\theta = -\frac{7}{25} and cos⁡θ=2425\cos\theta = \frac{24}{25}. The reciprocals and remaining ratios follow immediately.

The technique extends to starting from secant, cosecant, or cotangent — the Pythagorean identities 1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta and 1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta play the analogous role.