Sine and Cosine

Introduction

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:

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 $\theta + 2 k \pi$ with $k \in \mathbb{Z}$.

Definition of sine and cosine

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 $\overset{―}{O Q}$ and the hypotenuse $\overset{―}{O P}$ of the right triangle inscribed in the unit circle, and since $\overset{―}{O P} = 1$, one obtains:

Similarly, the cosine of $\theta$ is defined as the $x$-coordinate of $P$. It coincides with the ratio between the leg $\overset{―}{O R}$ and the hypotenuse $\overset{―}{O P}$, so that:

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: \(sin^{2} ⁡ \theta + cos^{2} ⁡ \theta = 1\) Geometrically, this identity represents the Pythagorean theorem applied to the triangle $O P R$ inscribed in the unit circle, where $P R$ and $\overset{―}{O R}$ correspond to the legs, and $\overset{―}{O P}$ is the hypotenuse of unit length.

Trigonometric identities

• \[\text{1}. sin ⁡ ( 2 x ) = 2 sin ⁡ ( x ) cos ⁡ ( x )\]
• \[\text{2}. cos ⁡ ( 2 x ) = cos^{2} ⁡ ( x ) - sin^{2} ⁡ ( x )\]
• \[\text{3}. cos ⁡ ( 2 x ) = 1 - 2 sin^{2} ⁡ ( x )\]
• \[\text{4}. cos ⁡ ( 2 x ) = 2 cos^{2} ⁡ ( x ) - 1\]
• \[\text{5}. sin ⁡ ( x + y ) = sin ⁡ ( x ) cos ⁡ ( y ) + cos ⁡ ( x ) sin ⁡ ( y )\]
• \[\text{6}. cos ⁡ ( x + y ) = cos ⁡ ( x ) cos ⁡ ( y ) - sin ⁡ ( x ) sin ⁡ ( y )\]

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.

Periodicity

Sine and cosine take values between $- 1$ and $1$ because the lengths of segments $\overset{―}{O R}$ and $\overset{―}{P R}$ 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 $2 \pi$: \(sin ⁡ \theta = sin ⁡ ( \theta + 2 \pi k ) k \in \mathbb{Z}\) \(cos ⁡ \theta = cos ⁡ ( \theta + 2 \pi k ) k \in \mathbb{Z}\) 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 of that angle:

The ratio of the cosine to the sine of an angle $\theta$ is equal to the cotangent of that angle:

Common values

The following tables collect the values of sine at the most frequently encountered angles, expressed in radians.

The following tables collect the values of cosine at the most frequently encountered angles, expressed in radians.

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, $cos ⁡ ( - x ) = cos ⁡ ( x )$

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

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 ( $sinh ⁡ ( x ) , cosh ⁡ ( x )$ ) 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 $z = a + b i$ can be written as:

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:

\(\sqrt{a^{2} - x^{2}}\) \(\sqrt{x^{2} + a^{2}}\) \(\sqrt{x^{2} - a^{2}}\)

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 $[ - \pi , \pi ]$:

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. Here, the integral:

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.

applicationshyperbolic analogcomplex formfourier basisorthogonalitygraphsfunctionstrig relationscommon valuestangent and cotangentsymmetrysum formulasdouble anglepythagorean identityperiodicitynotationanglesunit circledefinitionstructuresidentitiesfoundations