# Sine and Cosine
Source: algebrica.org — CC BY-NC 4.0
https://algebrica.org/sine-and-cosine/
## Introduction
Sine and cosine are the two primary trigonometric functions. Given an oriented angle \\( \theta \\), represented on the [unit circle](../unit-circle.md) 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 of sine and cosine
Consider an oriented angle \\( \theta \\) and the point \\( P \\) on the [unit circle](../unit-circle.md) 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
\\]
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.
- - -
## Fundamental trigonometric identity
The values of sine and cosine satisfy a property known as the [fundamental trigonometric identity](../pythagorean-identity.md):
\\[ \sin^2\theta + \cos^2\theta = 1 \\]
Geometrically, this identity represents the [Pythagorean theorem](../pythagorean-theorem.md) applied to the triangle \\( OPR \\) inscribed in the unit circle, where \\( PR \\) and \\( \overline{OR} \\) correspond to the legs, and \\( \overline{OP} \\) is the hypotenuse of unit length.
- - -
## Trigonometric identities
+ \\[
\text{1. } \quad \sin(2x) = 2\\,\sin(x)\cos(x)
\\]
+ \\[
\text{2. } \quad \cos(2x) = \cos^{2}(x) - \sin^{2}(x)
\\]
+ \\[
\text{3. } \quad \cos(2x) = 1 - 2\sin^{2}(x)
\\]
+ \\[
\text{4. } \quad \cos(2x) = 2\cos^{2}(x) - 1
\\]
+ \\[
\text{5. } \quad \sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)
\\]
+ \\[
\text{6. } \quad \cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y)
\\]
> These identities capture the most 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](../trigonometric-identities.md).
- - -
## 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](../integers.md) 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](../functions.md) with a period of \\( 2 \pi \\):
\\[ \sin\theta = \sin(\theta + 2 \pi k) \quad k \in \mathbb{Z} \\]
\\[ \cos\theta = \cos(\theta + 2 \pi k) \quad k \in \mathbb{Z} \\]
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](../tangent-and-cotangent.md) 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](../tangent-and-cotangent.md) 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](../sine-function.md) \\(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](../determining-the-domain-of-a-function.md), 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](../even-and-odd-functions.md), \\( \sin(-x) = -\sin(x)\\)
- - -
The [cosine function](../cosine-function.md) \\( 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](../even-and-odd-functions.md), \\( \cos(-x) = \cos(x) \\)
- - -
## 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](../unit-circle.md) 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](../hyperbola.md)
\\[
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:
\\[
\cosh(x) = \frac{e^{x} + e^{-x}}{2}
\\]
\\[
\sinh(x) = \frac{e^{x} - e^{-x}}{2}
\\]
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](../hyperbolic-sine-and-cosine.md) ( \\( \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](../complex-numbers-trigonometric-form.md). 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](../complex-numbers-introduction.md) 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](../indefinite-integrals.md), particularly in the technique known as [trigonometric substitution](../trigonometric-substitution-for-integrals.md), where expressions of the form:
\\[\sqrt{a^2 - x^2}\\]
\\[\sqrt{x^2 + a^2}\\]
\\[\sqrt{x^2 - a^2}\\]
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](../vectors.md) in Euclidean geometry. There, two vectors are orthogonal if their dot product is zero. Here, the integral:
\\[
\langle f, g \rangle =
\int_{-\pi}^{\pi} f(x)g(x)\\,dx
\\]
plays an analogous role (it acts as an inner product). When this integral vanishes, the functions behave as mutually perpendicular directions in a functional 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](../fourier-series.md).
