Tangent

In the unit circle, the tangent of an angle alpha is the ratio of the sine to the cosine of that angle. $$ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} $$ It is commonly denoted by the symbols tan or tg.

The word "tangent" comes from the Latin "tangere," meaning "to touch."

Geometrically, the tangent is a line segment perpendicular to the x-axis, originating at point C and intersecting the extended side OP.

the tangent

Proof. For a given angle α, two similar triangles, OAP and OCK, are formed. Thus, we can establish the proportion between the segments $$ \overline{ KC } \ : \ \overline{ PA } = \overline{ OC } \ : \ \overline{ OA } $$ which leads to $$ \frac{ \overline{ KC } }{\ \overline{ PA } } = \frac{ \overline{ OC } } { \overline{ OA } } $$ In a unit circle, the radius is 1, so we can substitute OC=1: $$ \frac{ \overline{ KC } }{\ \overline{ PA } } = \frac{ 1 } { \overline{ OA } } $$ Now, we isolate the tangent segment KC: $$ \overline{ KC } = \frac{ \overline{ PA } } { \overline{ OA } } $$ Here, PA represents the sine of angle α, and OA is the cosine of angle α. $$ \overline{ KC } = \frac{ \sin \alpha } { \cos \alpha } $$ Since tan α is defined as sin α/cos α, this proves that the segment KC is equal to the tangent of angle α. $$ \overline{ KC } = \frac{ \sin \alpha } { \cos \alpha } = tan \ \alpha$$

The tangent is a periodic function with a period of [0 ; π).

the tangent function

Note. In the first quadrant, the tangent is positive and tends toward infinity as the angle approaches 90°. At 90°, the tangent is undefined because the cosine equals zero. In the second quadrant, the tangent is negative because the cosine is negative and the sine is positive. In the third quadrant, the tangent is positive because both the sine and cosine are negative. At 270°, the tangent is again undefined due to the cosine being zero. In the fourth quadrant, the tangent is negative because the sine is negative and the cosine is positive.
the sign of the tangent

The tangent does not exist at π/2 (90°) and 3/2π (270°).

At these points, vertical asymptotes are formed.

the vertical asymptotes in the tangent graph

Since the tangent is a periodic function, it has infinitely many asymptotes at the points

$$ \frac{ \pi }{ 2 } + k \cdot \pi \ \ \ \ k \in Z $$

Therefore, the domain of the tangent function consists of all real numbers, except for the points where the tangent is undefined.

$$ R - \{ \frac{ \pi }{ 2 } + k \cdot \pi \} $$

The range of the tangent, however, spans from -∞ to +∞, covering all real numbers.

$$ \tan \ \alpha \ \ : \ \ R - \{ \frac{ \pi }{ 2 } + k \cdot \pi \} \longrightarrow (-\infty ; +\infty ) $$

Unlike sine and cosine, the tangent can take on any real value.

the graph of the tangent

Note. The complete graph of the tangent function for all angles is called the tangentoide.

The tangent is a periodic function that repeats every π (or 180°).

$$ \tan \alpha = \tan( \alpha + n \cdot \pi ) $$

It is not a continuous function, as there are infinitely many discontinuities corresponding to the vertical asymptotes.

The tangent is undefined at π/2 (90°) and at every periodic angle of π/2 ± n·π.

$$ \alpha = \frac{\pi}{2} \pm n \cdot \pi $$

Note. The sin/cos ratio is undefined when the denominator is zero, which occurs when the cosine is zero. The cosine equals zero at π/2 (90°) and at every value obtained by adding an integer multiple of π to π/2, i.e., π/2+k·π (or 90°+k·180°). $$ \cos (\frac{\pi}{2}+k \cdot \pi) \ \ \ \ k \in Z $$

The tangent is an odd function, meaning that tan(-α) = - tan(α).

$$ \tan ( - \alpha ) = - \tan ( \alpha ) $$

The tangent of an oriented angle (α) and its opposite (-α) have the same value but opposite signs.

the tangent is an odd function

Key Angles

Here are some important angles for the tangent function:

Angle (Degrees) Angle (Radians) Tangent
$$ 0° $$ $$ 0 $$ $$ 0 $$
$$ 30° $$ $$ \frac{\pi}{6} $$ $$ \frac{\sqrt{3}}{3} $$
$$ 45° $$ $$ \frac{\pi}{4} $$ $$ 1 $$
$$ 60° $$ $$ \frac{\pi}{3} $$ $$ \sqrt{3} $$
$$ 90° $$ $$ \frac{\pi}{2} $$ $$ \nexists $$
$$ 180° $$ $$ \pi $$ $$ 0 $$
$$ 270° $$ $$ \frac{3 \pi}{2} $$ $$ \nexists $$

The tangent is a period ic function with a period of π.

The Inverse Tangent Function

The tangent is not inherently invertible because it is not a bijective function.

the graph of the tangent

However, if we limit the domain to the interval [-π/2, π/2], the tangent becomes bijective.

the tangent becomes bijective in the reduced interval

Thus, within the interval [-π/2, π/2], the tangent is an invertible function.

the tangent in the reduced interval

To graph the inverse tangent function, we rotate the tangent graph 90° counterclockwise.

rotation of the graph by 90°

Next, we reflect the graph horizontally.

the graph of the arctangent

The inverse of the tangent is known as the arctangent (arctan).

Note. The tangent can be made invertible in other intervals, as long as the function remains bijective. For example, the tangent is also invertible in the interval [π/2, 3π/2].

Origin of the Word "Tangent"

The term "tangent" was introduced by the Danish mathematician and physicist Thomas Fincke in the 16th century, originally in reference to sundials.

In a sundial, a rod (gnomon) casts a shadow on the ground at a certain angle α.

The ratio of the length of the rod to the length of the shadow is the tangent of the angle α.

the angle between the rod and the ground

And so on.

 
 

Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

FacebookTwitterLinkedinLinkedin
knowledge base

Tangent (trigonometry)