# Section
2.6 The Tangent Function in Trigonometry

It is unfortunate that the word “tangent” in trigonometry has a different
meaning than in geometry. In geometry, and speaking loosely, a tangent is a
line that touches a curve in just one point. However, the tangent function
in trigonometry is a function that gives the *slope* of a line. Figure
2.6.1 shows that the slope of a line that passes through the origin and
makes an angle of θ radians is $\dfrac{\sin(\theta)}{\cos(\theta)}$. This is
why, in trigonometry, tan(θ) is defined as
$$\tan(θ)=\dfrac{\sin(\theta)}{\cos(\theta)},$$where θ is the measure in
radians of the arc that the line subtends, measured from 0 radians. It gives
the slope of a line.

*Figure 2.6.1. Slope of the ray from $(0,0)$ that subtends an arc of
$\theta$ radii is $\dfrac{\sin(\theta)}{\cos(\theta)}$.*

The quotient $\left( \dfrac{\sin(\theta)}{\cos(\theta)} \right)$ gives the
size of $\sin(\theta)$ in units of $\cos(\theta)$. A value of the quotient
does not have the radius as a unit of measure, as does sine and cosine. But
to graph $y = \tan(x)$ in the same coordinate system as sine and
cosine, the quotient must have the circle’s radius as a unit of measure. We
can do that by using similar triangles.

The animation in Figure 2.6.2 uses the relationship
$$\dfrac{\sin(\theta)}{\cos(\theta)}=\dfrac{\tan(\theta)}{1}.$$By displaying
the value of $\tan(\theta)$ at $x =1$, the directed segment that represents
the value of $\tan(\theta)$ is measured in units of the circle’s radius. We
can thus graph $y = \tan(x)$ in the same coordinate system as cosine and
sine because “1” stands for the same thing for all of them — the length of
the circle’s radius.

*Figure 2.6.2. Value of tan(θ) in units of the circle’s radius.*

*Reflection 2.4*: Convince yourself that the value labeled
$\tan(\theta)$ gives the slope of the line, and that this value is measured
in units of the circle’s radius.

*Reflection 2.5*: Play the animation in Figure 2.15, stopping it when
the value of $\cos(\theta)$ is close to 0. What happens to the quotient
sin($\theta$)/cos($\theta$) as $\cos(\theta)$ is near 0 but less than 0?
What happens to the quotient sin($\theta$)/cos($\theta$) as $\cos(\theta)$
is near 0 but greater than 0?

*Reflection 2.6*: Why does $\tan(\theta)$ make sudden jumps from
positive infinity to negative infinity?

Reflection 2.7: Is $\tan(\theta)$ always increasing over every
interval contained in the domain of its independent variable? That is, does
it ever decrease as $\theta$ increases continuously?

*Reflection 2.8*: Why is $\theta$ given as the input to sine, cosine,
and tangent functions in Figure 2.6.2 instead of *x*? When is it
appropriate to write “sin(*x*)”, “cos(*x*)”, and “tan(*x*)”?