The Tangent Function

Sine and cosine measure positions. Tangent measures slope. If a ray leaves the origin at angle $\theta$ and you ask "how steep is it?", the answer is $\tan\theta$. That single fact, tangent equals slope, is what makes tangent the function calculus reaches for when computing instantaneous rates of change. The unit-circle definition makes the slope interpretation precise; the algebra makes it computable; and the graph makes the asymptotes visible.

Definition

For any $\theta$ where $\cos\theta \neq 0$,

$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$

The domain excludes the angles where cosine vanishes, these are $\theta = \pi/2 + n\pi$ for integer $n$. At those angles tangent has vertical asymptotes: as $\theta$ approaches $\pi/2$ from below, $\cos\theta \to 0^+$ while $\sin\theta \to 1$, so the ratio shoots to $+\infty$. From above, it dives to $-\infty$.

Tangent as slope

Draw the ray from the origin at angle $\theta$. It hits the unit circle at $(\cos\theta, \sin\theta)$. The slope of that ray is rise over run:

$\text{slope} = \dfrac{\sin\theta - 0}{\cos\theta - 0} = \tan\theta$

So tangent is the slope of the ray. Equivalently: the line $y = (\tan\theta)\,x$ passes through the origin at angle $\theta$.

• Surveying: Tangent gives the slope (rise over run); a surveyor measuring the height of a tower from a known distance uses $\tan \theta = h/d$ with $\theta$ from a theodolite.
• Computer Graphics: A camera's field-of-view is set by $\tan(\text{fov}/2)$; the projection matrix in OpenGL/Vulkan literally contains this number, and choosing the right FOV avoids the "fish-eye" look.
• Civil Engineering: Road-grade signs ("7% grade ahead") are tangent values: a 7% grade is $\tan \theta = 0.07$, or about a 4^\circ incline. Highway engineering codes specify maximum tangents for safety.

(Drag the orange point. The dotted vertical line at $x = 1$ is the tangent line in the literal sense: the point where the extended ray hits it has height $\tan\theta$. Hide projections if the picture gets busy.)

Standard values

From the unit-circle table for sine and cosine: $\tan 0 = 0$, $\tan(\pi/6) = 1/\sqrt{3} = \sqrt{3}/3$, $\tan(\pi/4) = 1$, $\tan(\pi/3) = \sqrt{3}$, $\tan(\pi/2)$ is undefined. Beyond $\pi/2$ the function picks up signs from the quadrant pattern: positive in QI and QIII, negative in QII and QIV.

Period $\pi$, not $2\pi$

Here is the surprise. Sine and cosine repeat every $2\pi$, but tangent repeats every $\pi$. Why? Rotating by $\pi$ sends $(\cos\theta, \sin\theta)$ to $(-\cos\theta, -\sin\theta)$. Both signs flip, so the ratio stays the same: $(-\sin\theta)/(-\cos\theta) = \sin\theta / \cos\theta$. Half a turn around the unit circle gives a different point but the same slope of the ray, because a ray and its reverse have the same slope, just pointing the other way.

The Pythagorean cousin

Dividing the identity $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$ produces another identity:

$\tan^2\theta + 1 = \sec^2\theta, \qquad \text{where } \sec\theta = \dfrac{1}{\cos\theta}$

This shows up constantly in calculus, the derivative of $\tan x$ is $\sec^2 x$, for example.

Try it

• Predict first: what is $\tan(\pi/4)$, and what slope does the $45°$ ray have? Snap to $\pi/4$ and verify the readout shows $\tan(\pi/4) = 1$.
• Pick an angle slightly less than $\pi/2$. Watch the tangent readout balloon. Now nudge past $\pi/2$, it flips sign and is large and negative.
• If a line through the origin has slope $\sqrt{3}$, at what angle does it make with the $x$-axis? (Hint: it is one of the standard angles.)

Pause: tangent has period $\pi$, half the period of sine and cosine. Where does the second factor of $2$ come from? It comes from the sign-cancellation when you rotate by $\pi$. Try to articulate this in one sentence before continuing.

A trap to watch for

A frequent slip is treating $\tan(\pi/2)$ as $\infty$ as if it were a number. It is undefined. Saying "$\tan(\pi/2) = \infty$" is informal shorthand for "$\tan\theta$ grows without bound as $\theta \to \pi/2^-$," but $\infty$ is not a value of the function. Similarly, watch the sign: $\tan$ goes to $+\infty$ from one side and $-\infty$ from the other; it does not pass smoothly through. In a calculus computation, never plug in $\theta = \pi/2 + n\pi$; instead, take a one-sided limit.

What you now know

You can compute $\tan\theta$ as a ratio of sine to cosine, locate its asymptotes, evaluate it at standard angles, interpret it as the slope of a ray from the origin, and apply the identity $\tan^2\theta + 1 = \sec^2\theta$. The next section builds the trigonometric power tools, the addition formulas, that let you compute $\sin(A + B)$ and $\cos(A + B)$ from sine and cosine of $A$ and $B$ individually.

Quick check

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References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 11, §4, tangent and its geometric interpretation as slope.
• Spivak, M. (2008). Calculus (4th ed.). Publish or Perish. Chapter 15: analytic properties of trigonometric functions, including the derivative of tangent.
• Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage. Section 1.2: tangent as the slope of an inclined line.