Table of Contents
Understanding Tangent
In this chapter we focus specifically on the tangent trigonometric ratio. You should already be familiar with the basic right–triangle setup and the ideas of sine and cosine from earlier chapters in this section.
Tangent in a Right Triangle
For an acute angle $\theta$ in a right triangle, the tangent of $\theta$ is defined using the lengths of the sides:
- $\text{opposite}$: the side opposite the angle $\theta$
- $\text{adjacent}$: the side next to $\theta$ (but not the hypotenuse)
The tangent of $\theta$ is the ratio
$$
\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}.
$$
This definition applies to angles between $0^\circ$ and $90^\circ$ in a right triangle.
Example: If for an angle $\theta$ the opposite side has length $3$ and the adjacent side has length $4$, then
$$
\tan \theta = \frac{3}{4}.
$$
Notice that unlike sine and cosine, tangent can be any real number (positive, negative, greater than $1$, or between $-1$ and $0$, etc.), depending on the side lengths.
Relationship Between Tangent, Sine, and Cosine
Tangent is closely related to sine and cosine. Using the definitions
$$
\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}, \quad
\cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}},
$$
we can express tangent in terms of sine and cosine.
Divide sine by cosine:
$$
\frac{\sin \theta}{\cos \theta}
= \frac{\dfrac{\text{opposite}}{\text{hypotenuse}}}{\dfrac{\text{adjacent}}{\text{hypotenuse}}}
= \dfrac{\text{opposite}}{\text{adjacent}}
= \tan \theta.
$$
So the key identity for tangent is
$$
\tan \theta = \frac{\sin \theta}{\cos \theta},
$$
whenever $\cos \theta \neq 0$.
This identity is very important because:
- It shows that tangent “blows up” (is undefined) when $\cos \theta = 0$.
- It lets you compute tangent if you already know sine and cosine.
When Tangent Is Defined and Undefined
Because $\tan \theta = \dfrac{\sin \theta}{\cos \theta}$, tangent is only defined when the denominator $\cos \theta$ is not zero.
On the unit circle (covered in another chapter), cosine is zero at angles like $90^\circ$ and $270^\circ$ (or $\dfrac{\pi}{2}$ and $\dfrac{3\pi}{2}$ in radians), so:
- $\tan 90^\circ$ is undefined.
- $\tan 270^\circ$ is undefined.
- More generally, $\tan \theta$ is undefined for angles where $\cos \theta = 0$.
In right-triangle terms, this corresponds to trying to form a ratio with an adjacent side of length $0$, which is impossible.
Signs of Tangent in Different Quadrants
As angles extend beyond $0^\circ$ to $360^\circ$ (or $0$ to $2\pi$ radians), sine and cosine can be positive or negative. Because
$$
\tan \theta = \frac{\sin \theta}{\cos \theta},
$$
the sign of tangent depends on the signs of sine and cosine:
- When sine and cosine have the same sign (both positive or both negative), tangent is positive.
- When sine and cosine have opposite signs, tangent is negative.
This gives the usual pattern in the four quadrants:
- Quadrant I: $\sin \theta > 0$, $\cos \theta > 0$
$\Rightarrow \tan \theta > 0$ (positive) - Quadrant II: $\sin \theta > 0$, $\cos \theta < 0$
$\Rightarrow \tan \theta < 0$ (negative) - Quadrant III: $\sin \theta < 0$, $\cos \theta < 0$
$\Rightarrow \tan \theta > 0$ (positive) - Quadrant IV: $\sin \theta < 0$, $\cos \theta > 0$
$\Rightarrow \tan \theta < 0$ (negative)
You do not need the full unit circle to remember this; just recall that tangent is a ratio of sine and cosine.
Special Angle Values of Tangent
For some common angles, tangent takes especially simple values. Assuming angles in degrees:
- $\tan 0^\circ = 0$
- $\tan 30^\circ = \dfrac{1}{\sqrt{3}}$ (often written $\dfrac{\sqrt{3}}{3}$)
- $\tan 45^\circ = 1$
- $\tan 60^\circ = \sqrt{3}$
- $\tan 90^\circ$ is undefined
These values come from special right triangles (the $45^\circ$–$45^\circ$–$90^\circ$ triangle and the $30^\circ$–$60^\circ$–$90^\circ$ triangle). You do not need the full derivations here, but these are worth memorizing because they appear often.
Geometric Interpretation: Tangent as a Slope
One of the most useful interpretations of tangent is in terms of slope.
Consider an angle $\theta$ in standard position in the coordinate plane: its vertex at the origin, one side along the positive $x$-axis, and the other side rotating counterclockwise.
If a point $(x, y)$ lies on the terminal side of the angle with $x \neq 0$, then
$$
\tan \theta = \frac{y}{x}.
$$
But $\dfrac{y}{x}$ is also the slope of the line through the origin and the point $(x, y)$. So:
- $\tan \theta$ is the slope of the line that makes an angle $\theta$ with the positive $x$-axis.
From this point of view:
- Large positive values of $\tan \theta$ correspond to steep lines slanting upward.
- Large negative values correspond to steep lines slanting downward.
- $\tan \theta = 0$ corresponds to a horizontal line.
This interpretation becomes very important in algebra and analytic geometry, where you work with slopes and linear functions.
Tangent and Right-Triangle Problems
Tangent is especially convenient when you know one side (opposite or adjacent) and need to find the other, and the hypotenuse is not directly involved.
Typical uses:
- Given an angle and the length of the adjacent side, find the opposite side:
$$
\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
\quad\Rightarrow\quad
\text{opposite} = (\tan \theta)\cdot(\text{adjacent}).
$$ - Given an angle and the length of the opposite side, find the adjacent side:
$$
\text{adjacent} = \frac{\text{opposite}}{\tan \theta}.
$$
In many practical situations involving heights and distances (such as measuring the height of a building using a measured distance and an angle of elevation), tangent is the simplest ratio to use when the hypotenuse is not directly measured.
The Inverse Tangent Function (Brief Introduction)
When you know a tangent value and want the angle, you use the inverse tangent function, often written as $\tan^{-1}$ or $\arctan$.
If
$$
\tan \theta = k,
$$
then
$$
\theta = \tan^{-1}(k)
$$
gives an angle whose tangent is $k$.
Key points here:
- Calculators usually have a
tankey and antan⁻¹(oratan) key. - $\tan^{-1}(k)$ returns one principal angle; other angles with the same tangent differ by full turns (adding or subtracting integer multiples of $180^\circ$ or $\pi$ radians).
The detailed study of inverse trigonometric functions, their ranges, and how to solve equations with them is covered in other chapters; here you only need to recognize that $\tan^{-1}$ undoes tangent in a certain sense.
Summary of Main Facts About Tangent
- Right triangle definition:
$$
\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}.
$$ - Relation to sine and cosine:
$$
\tan \theta = \dfrac{\sin \theta}{\cos \theta}, \quad \cos \theta \neq 0.
$$ - Tangent is undefined whenever $\cos \theta = 0$.
- Tangent can take any real value (not restricted between $-1$ and $1$).
- Tangent equals the slope of a line making angle $\theta$ with the positive $x$-axis.
- Special values (degrees): $0^\circ \mapsto 0$, $45^\circ \mapsto 1$, $60^\circ \mapsto \sqrt{3}$, and $90^\circ$ is undefined.
- Inverse tangent, $\tan^{-1}$, is used to find angles from a known tangent ratio.