Trigonometric functions

Trigonometric functions are important tools in mathematics that help us understand angles and their relationships in triangles. In mathematics, these functions connect an angle in a right-angled triangle to the ratios of the lengths of its sides. They are very useful in many areas of science and engineering, including navigation, solid mechanics, celestial mechanics, and geodesy.

The most commonly used trigonometric functions are sine, cosine, and tangent. Each of these has a reciprocal function called cosecant, secant, and cotangent, which are used less often. These functions help us solve problems involving angles and waves, making them essential for studying repeating patterns in nature and technology through methods like Fourier analysis.

Originally, these functions were defined only for angles less than 90 degrees, called acute angles. But today, we can use them for any angle by imagining a point moving around a circle with a radius of one unit, called the unit circle. This idea lets us extend these functions to all real numbers and even to complex numbers using special mathematical series and equations.

Notation

Trigonometric functions have special short names used in math problems. Today, we commonly use "sin" for sine, "cos" for cosine, and "tan" for tangent, among others. These names originally described certain lines in circles but later became used to talk about angles in many kinds of math.

When we see a number like a 2 written above a trigonometric symbol, it usually means we are squaring the result — for example, sin²x means (sin x) × (sin x). However, when we see a –1, this special mark means we are finding the inverse function, not flipping the number upside down. So sin⁻¹x means the angle whose sine is x, also called arcsin x.

Right-angled triangle definitions

In a right-angled triangle, if you know one of the smaller angles, called θ, you can find special ratios between the lengths of the sides. These ratios are what we call trigonometric functions. The longest side, called the hypotenuse, is opposite the right angle. The side opposite the angle θ is called the opposite side, and the side next to θ (but not the hypotenuse) is called the adjacent side.

These ratios stay the same for any right-angled triangle with the same angle θ. This makes trigonometric functions very useful in many areas of science and math. There are special ways, like mnemonics, to help remember these ratios easily.

sine sin ⁡ θ = o p p o s i t e h y p o t e n u s e {\displaystyle \sin \theta ={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}}	cosecant csc ⁡ θ = h y p o t e n u s e o p p o s i t e {\displaystyle \csc \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}}
cosine cos ⁡ θ = a d j a c e n t h y p o t e n u s e {\displaystyle \cos \theta ={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}}	secant sec ⁡ θ = h y p o t e n u s e a d j a c e n t {\displaystyle \sec \theta ={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}}
tangent tan ⁡ θ = o p p o s i t e a d j a c e n t {\displaystyle \tan \theta ={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}}	cotangent cot ⁡ θ = a d j a c e n t o p p o s i t e {\displaystyle \cot \theta ={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}}

Summary of relationships between trigonometric functions

Function	Description	Relationship
		using radians	using degrees
sine	⁠opposite/hypotenuse⁠	sin ⁡ θ = cos ⁡ ( π 2 − θ ) = 1 csc ⁡ θ {\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}}	sin ⁡ x = cos ⁡ ( 90 ∘ − x ) = 1 csc ⁡ x {\displaystyle \sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}}
cosine	⁠adjacent/hypotenuse⁠	cos ⁡ θ = sin ⁡ ( π 2 − θ ) = 1 sec ⁡ θ {\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}	cos ⁡ x = sin ⁡ ( 90 ∘ − x ) = 1 sec ⁡ x {\displaystyle \cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,}
tangent	⁠opposite/adjacent⁠	tan ⁡ θ = sin ⁡ θ cos ⁡ θ = cot ⁡ ( π 2 − θ ) = 1 cot ⁡ θ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}}	tan ⁡ x = sin ⁡ x cos ⁡ x = cot ⁡ ( 90 ∘ − x ) = 1 cot ⁡ x {\displaystyle \tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}}
cotangent	⁠adjacent/opposite⁠	cot ⁡ θ = cos ⁡ θ sin ⁡ θ = tan ⁡ ( π 2 − θ ) = 1 tan ⁡ θ {\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}}	cot ⁡ x = cos ⁡ x sin ⁡ x = tan ⁡ ( 90 ∘ − x ) = 1 tan ⁡ x {\displaystyle \cot x={\frac {\cos x}{\sin x}}=\tan \left(90^{\circ }-x\right)={\frac {1}{\tan x}}}
secant	⁠hypotenuse/adjacent⁠	sec ⁡ θ = csc ⁡ ( π 2 − θ ) = 1 cos ⁡ θ {\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}}	sec ⁡ x = csc ⁡ ( 90 ∘ − x ) = 1 cos ⁡ x {\displaystyle \sec x=\csc \left(90^{\circ }-x\right)={\frac {1}{\cos x}}}
cosecant	⁠hypotenuse/opposite⁠	csc ⁡ θ = sec ⁡ ( π 2 − θ ) = 1 sin ⁡ θ {\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}}	csc ⁡ x = sec ⁡ ( 90 ∘ − x ) = 1 sin ⁡ x {\displaystyle \csc x=\sec \left(90^{\circ }-x\right)={\frac {1}{\sin x}}}

Radians versus degrees

In geometry, angles can be measured in degrees, where a right angle is 90° and a full turn is 360°. This is often used in elementary mathematics.

However, in more advanced math like calculus, angles are usually measured in radians. A radian measures the angle by the length of the arc it cuts out on the unit circle. One radian is about 57.3°, and a full turn is about 6.28 radians, or 2π rad. Radians make calculations easier and are the preferred unit in higher-level math.

Unit-circle definitions

All trigonometric functions can be described using a unit circle, which is a circle with a radius of one centered at a point we call the origin. By looking at points on this circle, we can define important trigonometric functions like sine and cosine.

The unit circle helps us understand angles that go beyond just right-angled triangles. It shows that trigonometric functions repeat their values in regular patterns, called periods, which makes them useful for studying things that repeat over and over again.

Algebraic values

The algebraic expressions for important angles in trigonometry start with the zero angle and go up to the right angle. For example, the sine of 0 degrees is 0, the sine of 30 degrees is 1/2, and the sine of 90 degrees is 1. These values can be remembered easily using square roots with a denominator of 2.

For angles that are multiples of three degrees, the exact values of sine and cosine can be expressed using square roots and can be constructed with a ruler and compass. For other angles, the values involve more complex numbers or special types of numbers called algebraic numbers.

Angle, θ, in		sin ⁡ ( θ ) {\displaystyle \sin(\theta )}	cos ⁡ ( θ ) {\displaystyle \cos(\theta )}	tan ⁡ ( θ ) {\displaystyle \tan(\theta )}
radians	degrees
0 {\displaystyle 0}	0 ∘ {\displaystyle 0^{\circ }}	0 {\displaystyle 0}	1 {\displaystyle 1}	0 {\displaystyle 0}
π 12 {\displaystyle {\frac {\pi }{12}}}	15 ∘ {\displaystyle 15^{\circ }}	6 − 2 4 {\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}	6 + 2 4 {\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}	2 − 3 {\displaystyle 2-{\sqrt {3}}}
π 6 {\displaystyle {\frac {\pi }{6}}}	30 ∘ {\displaystyle 30^{\circ }}	1 2 {\displaystyle {\frac {1}{2}}}	3 2 {\displaystyle {\frac {\sqrt {3}}{2}}}	3 3 {\displaystyle {\frac {\sqrt {3}}{3}}}
π 4 {\displaystyle {\frac {\pi }{4}}}	45 ∘ {\displaystyle 45^{\circ }}	2 2 {\displaystyle {\frac {\sqrt {2}}{2}}}	2 2 {\displaystyle {\frac {\sqrt {2}}{2}}}	1 {\displaystyle 1}
π 3 {\displaystyle {\frac {\pi }{3}}}	60 ∘ {\displaystyle 60^{\circ }}	3 2 {\displaystyle {\frac {\sqrt {3}}{2}}}	1 2 {\displaystyle {\frac {1}{2}}}	3 {\displaystyle {\sqrt {3}}}
5 π 12 {\displaystyle {\frac {5\pi }{12}}}	75 ∘ {\displaystyle 75^{\circ }}	6 + 2 4 {\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}	6 − 2 4 {\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}	2 + 3 {\displaystyle 2+{\sqrt {3}}}
π 2 {\displaystyle {\frac {\pi }{2}}}	90 ∘ {\displaystyle 90^{\circ }}	1 {\displaystyle 1}	0 {\displaystyle 0}	undefined

Definitions in analysis

Trigonometric functions are important in mathematics because they relate angles to the sides of triangles. They are used in many areas of science, like navigation and physics, to study patterns that repeat.

There are several ways to define these functions without using geometry. One common method uses special kinds of mathematical series called power series. These series help us understand how sine and cosine behave, even for very small or very large numbers.

Another way looks at how these functions change and uses rules called differential equations. These rules help us see patterns in how sine and cosine grow and shrink, which is useful in many areas of science and engineering.

Periodicity and asymptotes

The sine and cosine functions repeat their values in regular patterns, called periods. They have a period of 2π, meaning that after every 2π, the functions start their pattern again. For example, sin(z + 2π) is the same as sin(z), and the same goes for cosine.

These functions also have smaller patterns called semiperiods. For sine and cosine, this semiperiod is π, and it changes the sign of the value. For example, sin(z + π) is the opposite of sin(z). Tangent and cotangent also repeat every π. Additionally, shifting sine by π/2 gives the cosine value, and shifting cosine by π/2 gives the opposite of the sine value.

Function	Definition	Domain	Set of principal values
y = arcsin ⁡ x {\displaystyle y=\arcsin x}	sin ⁡ y = x {\displaystyle \sin y=x}	− 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1}	− π 2 ≤ y ≤ π 2 {\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}
y = arccos ⁡ x {\displaystyle y=\arccos x}	cos ⁡ y = x {\displaystyle \cos y=x}	− 1 ≤ x ≤ 1 {\displaystyle -1\leq x\leq 1}	0 ≤ y ≤ π {\textstyle 0\leq y\leq \pi }
y = arctan ⁡ x {\displaystyle y=\arctan x}	tan ⁡ y = x {\displaystyle \tan y=x}	− ∞	− π 2
y = arccot ⁡ x {\displaystyle y=\operatorname {arccot} x}	cot ⁡ y = x {\displaystyle \cot y=x}	− ∞	0
y = arcsec ⁡ x {\displaystyle y=\operatorname {arcsec} x}	sec ⁡ y = x {\displaystyle \sec y=x}	x 1 {\displaystyle x1}	0 ≤ y ≤ π , y ≠ π 2 {\textstyle 0\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}}
y = arccsc ⁡ x {\displaystyle y=\operatorname {arccsc} x}	csc ⁡ y = x {\displaystyle \csc y=x}	x 1 {\displaystyle x1}	− π 2 ≤ y ≤ π 2 , y ≠ 0 {\textstyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 0}

Applications

Main article: Uses of trigonometry

Trigonometry helps us understand how angles and sides of triangles are connected. It has many useful rules, like the law of sines and the law of cosines, which help solve problems involving triangles. These rules are important in fields like navigation and engineering.

Trigonometric functions are also key in studying patterns that repeat, such as waves. They help describe movements like a swing or light waves, and they are used in a special math tool called a Fourier series to break down complex patterns into simpler parts.

History

Main article: History of trigonometry

The trigonometric functions we use today were created a long time ago, starting in the medieval period. Early mathematicians like Hipparchus and Ptolemy worked on a function called the chord. Later, Indian astronomers developed functions related to sine. By the 9th century, Islamic mathematicians knew all six main trigonometric functions and used them to solve triangles.

Over time, many important mathematicians studied these functions. In the 1600s, Gottfried Leibniz showed that sine is a special kind of function called a transcendental function. Later, Euler found clever ways to connect these functions to other mathematical ideas.

Etymology

Main article: History of trigonometry § Etymology

The word sine comes from the Latin word sinus, meaning "bend" or "fold". This was used to translate an Arabic word that originally came from Sanskrit, where it meant "bowstring".

The word tangent comes from the Latin tangens, meaning "touching", because a tangent line touches a circle. The word secant comes from the Latin secans, meaning "cutting", because a secant line cuts the circle.

The prefix "co-" in words like cosine comes from an old book by Edmund Gunter in 1620. He used it to mean "sine of the complementary angle".