Sine and Cosine
Summarize
Definition
Sine and cosine are the two primary trigonometric functions. Given an oriented angle $\theta$, represented on the unit circle by a point $P$, the sine and cosine of $\theta$ are defined respectively as the $y$-coordinate and the $x$-coordinate of $P$. The unit circle is the circle of radius $1$ centered at the origin, described by the equation:
$$ x^2+y^2=1 $$
An oriented angle is positive when described by a counterclockwise rotation and negative when described by a clockwise rotation. All angles differing by an integer multiple of $2\pi$ identify the same point on the unit circle, and are therefore represented as $\theta+2k\pi$ with $k \in \mathbb{Z}$.
Definition 1. Consider an oriented angle $\theta$ and the point $P$ on the unit circle associated with $\theta$. The sine of $\theta$ is defined as the $y$-coordinate of $P$. It coincides with the ratio between the leg $\overline{OQ}$ and the hypotenuse $\overline{OP}$ of the right triangle inscribed in the unit circle, and since $\overline{OP} = 1$, one obtains:
$$ \sin(\theta) = \frac{\overline{OQ}}{\overline{OP}} = \frac{\overline{OQ}}{1} = y_P $$
Definition 2. Similarly, the cosine of $\theta$ is defined as the $x$-coordinate of $P$. It coincides with the ratio between the leg $\overline{OR}$ and the hypotenuse $\overline{OP}$, so that:
$$ \cos(\theta) = \frac{\overline{OR}}{\overline{OP}} = \frac{\overline{OR}}{1} = x_P $$
The sine and cosine of an angle are therefore nothing more than the projections of the point $P$ onto the coordinate axes: sine onto the $y$-axis and cosine onto the $x$-axis.
Definition via the area of a circular sector
The geometric description above introduces sine and cosine as coordinates of a point on the unit circle, but it relies on the intuitive notion of arc length, which has not been defined independently of the trigonometric functions themselves. A rigorous treatment based on integral calculus avoids this logical dependency by measuring an oriented angle through the area of the corresponding circular sector, a quantity that can be computed directly through a definite integral.
The construction begins with an analytic definition of $\pi$ as twice the area enclosed by the upper half of the unit circle:
$$ \pi := 2\int_{-1}^{1}\sqrt{1-x^2} \ dx $$
In this formulation $\pi$ is identified as a real number determined by a definite integral, without any reference to the length of a curve.
Definition 3. For $-1 \leq x \leq 1$, let $A(x)$ denote the area of the circular sector bounded by the horizontal axis, by the half-line joining the origin to the point $(x, \sqrt{1-x^2})$, and by the arc of the unit circle connecting this point to $(1, 0)$. The area $A(x)$ decomposes as the sum of the area of a right triangle and the area of a region lying under the upper half of the circle:
$$ A(x) = \frac{x\sqrt{1-x^2}}{2} + \int_x^1 \sqrt{1-t^2} \ dt $$
The function $A$ is continuous on $[-1, 1]$ and decreases monotonically from $A(-1) = \pi/2$ to $A(1) = 0$. For $-1 < x < 1$, the Fundamental Theorem of Calculus yields the derivative:
$$ A'(x) = -\frac{1}{2\sqrt{1-x^2}} $$
The negative sign confirms that $A$ is strictly decreasing on the interior of its domain.
The cosine and sine functions on the interval $[0, \pi]$ are now defined as the coordinates of the unique point on the unit circle that bounds a sector of area $\theta/2$. For $0 \leq \theta \leq \pi$, the cosine of $\theta$ is the unique value in $[-1, 1]$ satisfying:
$$ A(\cos\theta) = \frac{\theta}{2} $$
The sine of $\theta$ is then defined by:
$$ \sin\theta = \sqrt{1 - \cos^2\theta} $$
Existence and uniqueness of $\cos\theta$ are guaranteed by the continuity of $A$ on $[-1, 1]$, combined with the intermediate value theorem applied to the values it attains between $0$ and $\pi/2$. The fundamental trigonometric identity $\sin^2\theta + \cos^2\theta = 1$ holds by construction.
The extension of $\sin$ and $\cos$ from $[0, \pi]$ to the entire real line proceeds in two stages. For $\pi \leq \theta \leq 2\pi$, the values are obtained by reflection:
$$ \begin{align} \sin\theta &= -\sin(2\pi - \theta) \\[6pt] \cos\theta &= \cos(2\pi - \theta) \end{align} $$
For an arbitrary real number $\theta$, write $\theta = 2k\pi + \theta'$ with $k \in \mathbb{Z}$ and $\theta' \in [0, 2\pi]$, and set:
$$ \begin{align} \sin\theta &= \sin\theta' \\[6pt] \cos\theta &= \cos\theta' \end{align} $$
This procedure produces functions defined on all of $\mathbb{R}$ and periodic of period $2\pi$, in full agreement with the geometric description given earlier.
Within this analytic framework the derivatives $\sin'(\theta) = \cos\theta$ and $\cos'(\theta) = -\sin\theta$ are not postulated, but obtained as theorems. They follow by recognizing $\cos$ as the inverse of the function $B(x) = 2A(x)$ and applying the differentiation rule for inverse functions.
Fundamental trigonometric identity
The values of sine and cosine satisfy a property known as the fundamental trigonometric identity:
$$ \sin^2\theta + \cos^2\theta = 1 $$
Geometrically, this identity represents the Pythagorean theorem applied to the triangle $OPR$ inscribed in the unit circle, where $\overline{PR}$ and $\overline{OR}$ correspond to the legs, and $\overline{OP}$ is the hypotenuse of unit length.
Trigonometric identities
The most frequently encountered identities involving sine and cosine fall into two families: the double-angle formulas, which express $\sin(2x)$ and $\cos(2x)$ in terms of $\sin x$ and $\cos x$, and the addition formulas, which decompose $\sin(x+y)$ and $\cos(x+y)$ into products of the two functions evaluated separately at $x$ and at $y$.
$$ \begin{align} &\sin(2x) = 2\sin(x)\cos(x) \\[6pt] &\cos(2x) = \cos^{2}(x) - \sin^{2}(x) \\[6pt] &\cos(2x) = 1 - 2\sin^{2}(x) \\[6pt] &\cos(2x) = 2\cos^{2}(x) - 1 \\[6pt] &\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y) \\[6pt] &\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y) \end{align} $$
These identities capture the essential relationships between sine and cosine. They follow directly from the geometry of the unit circle and form the foundation of many trigonometric transformations. For a broader overview, refer to the full collection of trigonometric identities.
For example, starting from the addition formulas for sine and cosine, we want to prove that $\cos(3x) = 4\cos^3(x) - 3\cos(x)$ and obtain an analogous expression for $\sin(3x)$ in terms of $\sin(x)$ alone.
Writing $3x = 2x + x$ and applying the addition formula for the cosine, we obtain:
$$ \cos(3x) = \cos(2x)\cos(x) - \sin(2x)\sin(x) $$
To reach a formula in $\cos(x)$ alone, we replace the double-angle terms with $\cos(2x) = 2\cos^2(x) - 1$ and $\sin(2x) = 2\sin(x)\cos(x)$, and then expand the products:
$$ \cos(3x) = 2\cos^3(x) - \cos(x) - 2\sin^2(x)\cos(x) $$
The residual $\sin^2(x)$ is removed through the Pythagorean identity $\sin^2(x) = 1 - \cos^2(x)$, and collecting like terms yields:
$$ \cos(3x) = 4\cos^3(x) - 3\cos(x) $$
The same strategy applies to $\sin(3x)$. The addition formula for the sine gives:
$$ \sin(3x) = \sin(2x)\cos(x) + \cos(2x)\sin(x) $$
This time the appropriate variant of the double-angle identity is $\cos(2x) = 1 - 2\sin^2(x)$, which already involves $\sin^2(x)$. Substituting it together with $\sin(2x) = 2\sin(x)\cos(x)$ and expanding produces:
$$ \sin(3x) = 2\sin(x)\cos^2(x) + \sin(x) - 2\sin^3(x) $$
Using $\cos^2(x) = 1 - \sin^2(x)$ to eliminate the last residual cosine and simplifying, we arrive at the analogous identity:
$$ \sin(3x) = 3\sin(x) - 4\sin^3(x) $$
Periodicity
Sine and cosine take values between $-1$ and $1$ because the lengths of segments $\overline{OR}$ and $\overline{PR}$ cannot exceed the radius, which is equal to 1.
If an integer multiple of a full revolution is added to an angle $\theta$, the sine and cosine values remain unchanged because the point $P$ returns to the same position on the unit circle. From this property, it follows that sine and cosine are periodic functions with a period of $2 \pi$:
$$ \begin{align} \sin\theta &= \sin(\theta + 2\pi k) \quad k \in \mathbb{Z} \\[6pt] \cos\theta &= \cos(\theta + 2\pi k) \quad k \in \mathbb{Z} \end{align} $$
This means that the functions repeat their values every $2 \pi$, reflecting the cyclic nature of circular motion.
Tangent and cotangent
The ratio of the sine to the cosine of an angle $\theta$ is equal to the tangent of that angle:
$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$
The ratio of the cosine to the sine of an angle $\theta$ is equal to the cotangent of that angle:
$$ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} $$
Common values
The following tables collect the values of sine at the most frequently encountered angles, expressed in radians.
$$ \begin{align} x &= -\pi/2 &\quad& \sin(-\pi/2) = -1 \\[6pt] x &= -\pi/3 &\quad& \sin(-\pi/3) = -\sqrt{3}/2 \\[6pt] x &= -\pi/4 &\quad& \sin(-\pi/4) = -\sqrt{2}/2 \\[6pt] x &= -\pi/6 &\quad& \sin(-\pi/6) = -1/2 \\[6pt] x &= 0 &\quad& \sin(0) = 0 \\[6pt] x &= \pi/6 &\quad& \sin(\pi/6) = 1/2 \\[6pt] x &= \pi/4 &\quad& \sin(\pi/4) = \sqrt{2}/2 \\[6pt] x &= \pi/3 &\quad& \sin(\pi/3) = \sqrt{3}/2 \\[6pt] x &= \pi/2 &\quad& \sin(\pi/2) = 1 \end{align} $$
The following tables collect the values of cosine at the most frequently encountered angles, expressed in radians.
$$ \begin{align} x &= -\pi/2 &\quad& \cos(-\pi/2) = 0 \\[6pt] x &= -\pi/3 &\quad& \cos(-\pi/3) = 1/2 \\[6pt] x &= -\pi/4 &\quad& \cos(-\pi/4) = \sqrt{2}/2 \\[6pt] x &= -\pi/6 &\quad& \cos(-\pi/6) = \sqrt{3}/2 \\[6pt] x &= 0 &\quad& \cos(0) = 1 \\[6pt] x &= \pi/6 &\quad& \cos(\pi/6) = \sqrt{3}/2 \\[6pt] x &= \pi/4 &\quad& \cos(\pi/4) = \sqrt{2}/2 \\[6pt] x &= \pi/3 &\quad& \cos(\pi/3) = 1/2 \\[6pt] x &= \pi/2 &\quad& \cos(\pi/2) = 0 \end{align} $$
Sine and cosine function
The sine function $f(x) = \sin(x)$ assigns to each angle $x$, expressed in radians, its corresponding sine value. Its graph is a periodic wave with a period of $2 \pi$ and an amplitude of 1, oscillating between -1 and 1. The function $f(x) = \sin x$ has all real numbers in its domain, but its range is $-1 \leq \sin(x) \leq 1$.
- Domain: $x \in \mathbb{R}$
- Range: $y \in \mathbb{R} : -1 \leq y \leq 1$
- Periodicity: periodic in $x$ with period $2 \pi$
- Parity: odd, $\sin(-x) = -\sin(x)$
The cosine function $f(x) = \cos(x)$ assigns to each angle $x$, expressed in radians, its corresponding cosine value. Its graph is a periodic wave with a period of $2 \pi$ and an amplitude of 1, oscillating between -1 and 1. The function $f(x) = \cos x$ has all real numbers in its domain, but its range is $-1 \leq \cos(x) \leq 1$.
- Domain: $x \in \mathbb{R}$
- Range: $y \in \mathbb{R} : -1 \leq y \leq 1$
- Periodicity: periodic in $x$ with period $2\pi$
- Parity: even, $\cos(-x) = \cos(x)$
A detailed treatment of the sine function and the cosine function, including notable values, limits, derivatives, and integrals, is provided in their respective entries.
Sine and cosine in the hyperbolic setting
In the circular case, the sine and cosine of an angle $\theta$ are obtained from the unit circle of radius $1$, where the point on the circumference provides the coordinates $(\cos\theta, \sin\theta)$. A closely related construction exists in the hyperbolic context, where the reference curve is the equilateral hyperbola
$$ x^{2} - y^{2} = 1 $$
Here, instead of an angle determined by a circular sector, one considers a hyperbolic sector whose area identifies a parameter $x$. The point on the hyperbola associated with this area has coordinates:
$$ \begin{align} \cosh(x) &= \frac{e^{x} + e^{-x}}{2} \\[6pt] \sinh(x) &= \frac{e^{x} - e^{-x}}{2} \end{align} $$
These expressions mirror the circular definitions but arise from a different geometric framework. Just as $\cos\theta$ and $\sin\theta$ describe how a point moves around the unit circle, the hyperbolic sine and cosine ($\sinh(x), \cosh(x)$) describe how a point evolves along the hyperbola as the hyperbolic sector grows.
Trigonometric structure of complex numbers
Sine and cosine are also the building blocks of the trigonometric form of a complex number. Any complex number $z = a + bi$ can be written as:
$$z = r(\cos\theta + i\sin\theta)$$
where $r = \sqrt{a^2 + b^2}$ is the modulus and $\theta = \arctan(b/a)$ is the argument. In this representation, sine and cosine no longer describe a point on a circle, but the direction and magnitude of a complex number in the plane.
Applications in integration
The identities and properties of sine and cosine are not limited to trigonometry. They become essential tools in integrals, particularly in the technique known as trigonometric substitution, where expressions of the form:
$$ \begin{align} \sqrt{a^2 - x^2} \\[6pt] \sqrt{x^2 + a^2} \\[6pt] \sqrt{x^2 - a^2} \end{align} $$
are simplified by replacing the variable $x$ with a suitable trigonometric function. The approach works precisely because the Pythagorean identities of sine and cosine turn the expression under the square root into a perfect square, eliminating the radical entirely.
Orthogonality of sine and cosine
Beyond their geometric meaning on the unit circle, sine and cosine possess a deeper analytical property that emerges when they are considered over an entire period. When integrated across a full symmetric interval, trigonometric functions with different frequencies behave independently from one another. This phenomenon is known as orthogonality. More precisely, for any integers $n$ and $m$, the following relations hold on the interval $[-\pi, \pi]$:
$$ \begin{aligned} \int_{-\pi}^{\pi} \sin(nx)\cos(mx) \ dx &= 0 \\[6pt] \int_{-\pi}^{\pi} \cos(nx)\cos(mx) \ dx &= \begin{cases} \pi & n = m \neq 0 \\ 0 & n \ne m \end{cases} \\[6pt] \int_{-\pi}^{\pi} \sin(nx)\sin(mx) \ dx &= \begin{cases} \pi & n = m \\ 0 & n \ne m \end{cases} \end{aligned} $$
These identities express the fact that trigonometric waves with distinct frequencies do not overlap when averaged through integration over $[-\pi,\pi]$. In other words, the contribution of one frequency disappears when tested against a different one across a complete period. This situation is analogous to perpendicular vectors in Euclidean geometry. There, two vectors are orthogonal if their dot product is zero. For continuous functions on $[-\pi,\pi],$ the corresponding inner product is:
$$ \langle f, g \rangle = \int_{-\pi}^{\pi} f(x)g(x) \ dx $$
When this integral is zero, the functions are orthogonal in this function space.
This property reveals that sine and cosine form a structurally independent system of oscillations. Because of this orthogonality, it becomes possible to isolate individual harmonic components inside a periodic function, an idea developed systematically in the theory of Fourier Series.