3.8 The Tangent Function

The tangent function gives the slope of the terminal ray on the unit circle, so $\tan\theta = \frac{\sin\theta}{\cos\theta}$ when $\cos\theta \ne 0$. Its graph has a period of $\pi$, vertical asymptotes wherever $\cos\theta = 0$, and a range of all real numbers.

The Tangent Function Summary

The tangent function gives the slope of the terminal ray on the unit circle. Since slope is change in y over change in x, tangent can also be written as tan θ = sin θ / cos θ whenever cos θ is not 0.

AP Precalculus Topic 3.8 focuses on building and transforming the tangent graph. Tangent has period π, vertical asymptotes at θ = π/2 + kπ, zeros at θ = kπ, and transformations of the form y = a tan(b(θ + c)) + d.

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Why This Matters for the AP Precalculus Exam

Unit 3 (Trigonometric and Polar Functions) carries a large share of the exam, and the tangent function is the place where you connect unit circle reasoning to a function that is not sinusoidal. On both calculator and non-calculator questions, you may be asked to construct or read tangent graphs, locate asymptotes and zeros, and describe how the graph increases and changes concavity between asymptotes. Knowing tan θ = sin θ / cos θ also sets up later work with inverse trig functions and identities, where being fluent with tangent saves time.

Key Takeaways

• tan θ is the slope of the terminal ray, which equals sin θ / cos θ wherever cos θ is not 0.
• The period of tangent is π, half the period of sine and cosine.
• Vertical asymptotes occur at θ = π/2 + kπ for integers k, because cos θ = 0 there.
• Tangent increases on each interval between consecutive asymptotes and changes from concave down to concave up.
• The range is all real numbers, and zeros occur at θ = kπ.
• The general form y = a tan(b(θ + c)) + d controls vertical stretch, period, phase shift, and vertical shift, just like sinusoidal transformations.

Building the Tangent Function from the Unit Circle

The tangent function is defined using an angle θ in standard position on the unit circle, a circle of radius 1 centered at the origin. Where the terminal ray crosses the circle, you get a point P with coordinates (cos θ, sin θ). The tangent of the angle is the slope of that terminal ray.

Since slope is the change in y over the change in x, and the point on the unit circle is (cos θ, sin θ), you get:

$\tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cos \theta \neq 0$

So tangent is both the slope of the terminal ray and the ratio of sine to cosine. The restriction cos θ ≠ 0 matters because dividing by zero is undefined.

Why the Asymptotes Appear

As you sweep θ counterclockwise from 0, the slope of the terminal ray grows. At θ = π/2, the terminal ray is vertical, and a vertical line has undefined slope. In ratio terms, cos(π/2) = 0, so sin θ / cos θ tries to divide by zero. That is where you get a vertical asymptote.

This repeats every half rotation. Tangent has a vertical asymptote at:

$\theta = \frac{\pi}{2} + k\pi$

for any integer k. The graph shoots toward positive infinity approaching an asymptote from one side and comes up from negative infinity on the other side.

Period of π

Look at two angles like θ = π/6 and θ = 7π/6. Their terminal rays lie along the same line, so they have the same slope and the same tangent value. Because slope values repeat every half revolution, tangent has a period of π, not 2π like sine and cosine.

Key Features of the Graph

• Zeros: tan θ = 0 wherever sin θ = 0, which is at θ = kπ.
• Range: all real numbers, since the function climbs without bound as it approaches each asymptote.
• Increasing behavior: between any two consecutive asymptotes, tangent is always increasing.
• Concavity: between consecutive asymptotes, the graph changes from concave down to concave up. The point where this switch happens (at the zeros) is a point of inflection.

A common point of confusion: even though the graph drops from positive infinity to negative infinity at each asymptote, the function is still increasing on each separate interval. The jump is a break in the graph, not a decrease.

Transformations of the Tangent Function

Tangent uses the same transformation structure you learned for sinusoidal functions, even though its graph is not a sine wave. The general form is:

$y = a \tan(b(\theta + c)) + d$

Here is what each parameter does:

• a (vertical dilation): stretches or compresses the graph vertically by a factor of |a|. If a < 0, the graph reflects over the x-axis. Tangent has no amplitude in the sinusoidal sense because it has no maximum or minimum, so think of a as a vertical stretch, not an amplitude.
• b (period change): the period becomes π/|b|. Larger |b| squeezes the graph and makes the period shorter; smaller |b| stretches it. If b < 0, the graph reflects over the y-axis.
• c (phase shift): shifts the graph horizontally by -c units. So tan(θ + c) moves the graph left when c is positive.
• d (vertical shift): shifts the whole graph, including the line through its points of inflection, up by d units (down if d is negative).

To find the new asymptote locations after a transformation, set the inside expression equal to where the parent asymptotes are:

$b(\theta + c) = \frac{\pi}{2} + k\pi$

Solving for θ gives the shifted asymptotes.

How to Use This on the AP Precalculus Exam

MCQ

• Match a tangent equation to its graph by checking the period (π/|b|), asymptote spacing, and whether a < 0 causes a reflection.
• Identify asymptote locations by finding where the inside of the function makes cosine zero, or by solving b(θ + c) = π/2 + kπ.
• Read zeros and inflection points off a graph at θ = kπ for the parent function.

Free Response

• When you describe the graph, use precise language: state the period, the asymptote locations, where the function increases, and where concavity changes.
• If a question gives a transformed tangent function, identify a, b, c, and d clearly and explain each effect. Clean notation makes your reasoning easy to follow.
• Connect tan θ = sin θ / cos θ when a question asks you to justify why an asymptote exists at a specific input.

Common Trap

Remember that the tangent period is π/|b|, not 2π/|b|. Using the sinusoidal period formula here is one of the most frequent mistakes.

Common Misconceptions

• Tangent has an amplitude. It does not. Tangent has no maximum or minimum value, so |a| is a vertical stretch factor, not an amplitude.
• The period is 2π. Tangent repeats every π, half the period of sine and cosine.
• Tangent decreases at the asymptotes. The drop from positive infinity to negative infinity is a break in the graph. On each interval between asymptotes, tangent is increasing.
• Asymptotes happen where sine is zero. Asymptotes happen where cosine is zero (θ = π/2 + kπ). Zeros of tangent happen where sine is zero (θ = kπ).
• Phase shift direction. In tan(θ + c), a positive c shifts the graph left by c units, not right.
• Tangent only works with right triangles. The right triangle ratio (opposite over adjacent) is one view, but on the unit circle tangent is the slope of the terminal ray and is defined for all angles where cos θ ≠ 0.

Related AP Precalculus Guides

• 3.3 Sine and Cosine Function Values
• 3.6 Sinusoidal Function Transformations
• 3.4 Sine and Cosine Function Graphs
• 3.7 Sinusoidal Function Context and Data Modeling
• 3.9 Inverse Trigonometric Functions
• 3.12 Equivalent Representations of Trigonometric Functions