Foundations of the Tangent Function

Teaching the tangent function to high school students often presents a unique challenge because it behaves so differently from sine and cosine. Where sine and cosine oscillate softly between −1 and 1, the tangent function grows without bound, repeats more frequently, and has dramatic vertical asymptotes. A clear, step‑by‑step guide that builds from concrete triangle geometry through the unit circle and into graphing helps students internalise these differences and gain genuine confidence.

Begin by grounding the tangent function in a right‑angled triangle. Remind students of the mnemonic SOH‑CAH‑TOA, and zero in on tan(θ) = opposite / adjacent. Use several right triangles with different side lengths but the same acute angle to show that the ratio remains constant—this is the essence of why trigonometry works. Draw a triangle with legs 3 and 4, hypotenuse 5, and mark the angle opposite the 3‑unit side. Ask: “If the angle changes, does that ratio still hold?” Let them see that the tangent is a function of the angle, not of the triangle’s size.

Next, extend the idea beyond acute angles. Many students first encounter tangent as a “slope” of a line; you can preview this by showing that on a coordinate plane, the slope of a line from the origin to a point (x, y) is y / x. This foreshadows the unit circle definition and the idea that tangent can be negative when the point lies in Quadrants II or IV.

For additional support, the Khan Academy unit circle lessons offer interactive visuals that reinforce the triangle‑to‑circle transition.

The Unit‑Circle Definition of Tangent

Transition to the unit circle by stating that for a point (x, y) on the circle (radius = 1), tan(θ) = y / x, provided x ≠ 0. This may initially confuse students because the “opposite” and “adjacent” sides become lengths that are smaller than or equal to 1. Use a diagram of the unit circle with a right triangle drawn inside: the horizontal leg is cos θ, the vertical leg is sin θ, and the slope of the hypotenuse from the origin to the point is sin θ / cos θ, which is exactly tan θ.

Emphasise that because the denominator cos θ can be zero, tan θ is undefined when cos θ = 0—that is, at θ = 90° (π/2) and every 180° thereafter. This is a key conceptual leap: the function “blows up” at those angles. Give students a quick memory device: “Cosine equals zero, tangent has no friend.”

Draw a unit circle with the four quadrants labelled. For each quadrant, list the sign of tan θ (positive in QI and QIII; negative in QII and QIV). Ask students to verbalise why: both sine and cosine are positive in QI so their quotient is positive; in QII sine is positive and cosine is negative → negative quotient; and so on. This exercise cements the relationship between tan and the signs of sine and cosine.

Use the Desmos graphing calculator to animate a point moving around the unit circle while a second screen simultaneously traces the graph of tan θ. The visual feedback of the point “jumping” to infinity at the asymptotes is powerful.

Step‑by‑Step Guide to Graph the Tangent Function

Step 1: Plot key angles on the unit circle

Have students draw a unit circle and mark the special angles: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, and 360°. For each angle, write the coordinates (cos θ, sin θ). This serves as a reference for all later calculations.

Step 2: Compute tangent values for these angles

Create a table with columns for θ (in degrees and radians), sin θ, cos θ, and tan θ = sin θ / cos θ. For angles where cos θ = 0, write “undefined” rather than a number. A completed table might look like:

  • 0° (0): tan = 0
  • 30° (π/6): tan ≈ 0.577
  • 45° (π/4): tan = 1
  • 60° (π/3): tan ≈ 1.732
  • 90° (π/2): undefined
  • 180° (π): tan = 0
  • 270° (3π/2): undefined
  • 360° (2π): tan = 0

Encourage students to notice that the values for 30° and 210° (or 45° and 225°) are equal, foreshadowing the π‑periodicity.

Step 3: Draw vertical asymptotes

On graph paper or using a digital tool, direct students to draw dashed vertical lines at every odd multiple of π/2: at θ = −π/2, π/2, 3π/2, 5π/2, and so on. Remind them that the graph will approach these lines but never cross them. This is a good moment to recall why tan θ = sin θ / cos θ: the denominator becomes zero at those lines.

Step 4: Plot zeros and the general shape

Mark the points where tan θ = 0: at multiples of π (0°, 180°, 360°, …). Between each asymptote and the adjacent zero, the graph must rise or fall toward infinity. For example, from θ = −π/2 to 0, tan θ goes from negative infinity (approaching from the right) up to 0. Then from 0 to π/2, it climbs from 0 to positive infinity.

Have students draw a smooth curve through the plotted points, making sure it approaches the asymptotes steeply and crosses the zeros with a linear‑looking transition. Repeat for one or two more periods so they see the repeating pattern. The graph of y = tan θ has a distinctive S‑like shape that repeats every π units, not 2π like sine and cosine.

Step 5: Check with technology

After students have drawn their graphs manually, compare with a graphing calculator or the Desmos graph of tan(x). Let them adjust their curves to match the correct steepness and position of the asymptotes. This self‑check step builds accuracy and confidence.

Key Properties of the Tangent Function

Once students can graph tan θ, formalise the properties:

  • Periodicity: The period is π (180°). Unlike sine and cosine which repeat after 2π, tan repeats after π. Show algebraically: tan(θ + π) = sin(θ + π) / cos(θ + π) = (−sin θ) / (−cos θ) = tan θ.
  • Odd symmetry: tan(−θ) = −tan θ. The graph is symmetric about the origin. Students can check that their plotted points satisfy this.
  • No amplitude: The range is all real numbers (−∞, ∞). There is no maximum or minimum value.
  • Asymptotes: Vertical asymptotes at θ = π/2 + kπ, for any integer k.
  • Zeros: The function equals zero at θ = kπ, for any integer k.
  • Increasing behavior: Over each interval between consecutive asymptotes, tan θ is strictly increasing.

Provide a short exercise: given a graph of tan θ with only one period shown, ask students to write the equations of the asymptotes, identify the zeros, and state the period. If they can do it for one period, they can extend it left and right.

Real‑World Applications of the Tangent Function

Students often ask, “When will I ever use this?” The tangent function has immediate practical applications:

  • Angles of elevation and depression: If a person looks up at a flagpole, the angle of elevation θ satisfies tan θ = height / distance from the flagpole. Given two pieces of information, students can solve for the third. Provide word problems: “A tree casts a 12‑m shadow when the sun is at an angle of 40°. How tall is the tree?”
  • Slope of a line: The slope m of a line making an angle θ with the positive x‑axis is m = tan θ. This connects trigonometry directly to algebra. Example: “A ramp has an incline angle of 20°. What is its slope?”
  • Navigation and surveying: Surveyors use the tangent function to measure distances across rivers or lakes using angle measurements from a baseline. Share a brief narrative: “A surveyor stands 50 m from a building and measures an angle of elevation of 62° to the top. How tall is the building?”
  • Physics – projectile motion: The tangent of the launch angle appears in range equations. Although beyond high school scope, mentioning it can pique curiosity.

For more real‑world examples, the Math Is Fun trigonometry page provides clear word problems and diagrams.

Common Mistakes and How to Avoid Them

Anticipating errors is part of a good teaching guide. Common pitfalls include:

  • Forgetting undefined values: Students may try to compute tan 90° using their calculator without checking mode (degrees vs. radians). Emphasise that the calculator will give a very large number (or an error in some settings) because it’s not truly defined. Always ask “Is cos θ zero?” before evaluating.
  • Misplacing asymptotes: Some students think asymptotes occur at all multiples of π (180°). Drill: “Asymptotes are at 90°, 270°, 450°, … — odd multiples of 90°.” Use the mnemonic “asymptotes at the sine‑cosine crossing where cosine is zero.”
  • Confusing period with sine/cosine: Because they learned sine and cosine first, many students inadvertently draw tan with a 2π period. Emphasise that the tangent repeats twice as fast—every π. Show this by asking them to compare tan 0° and tan 180° (both zero).
  • Sketching the curve incorrectly between asymptotes: The default mistake is to make the curve straight or to curve it the wrong way (e.g., curving up then down). Demonstrate that within each interval tan is increasing, so it must go from −∞ (at the left asymptote) rising to +∞ (at the right asymptote) without any horizontal flattening.

A quick troubleshooting exercise: give students a partially drawn graph with three asymptotes but one of the S‑shapes drawn backwards. Have them spot the error and correct it.

Assessment Ideas and Practice Problems

To solidify learning, use a mix of conceptual and procedural tasks:

  • Conceptual: “Without using a calculator, determine whether tan 200° is positive or negative. Explain your reasoning.”
  • Procedural: “Find tan θ for θ = 135° using the unit circle. Show your work.”
  • Graphing: “Plot two full periods of y = tan θ. Label the asymptotes and zeros.”
  • Application: “From a point 100 m from the base of a cliff, the angle of elevation to the top is 35°. How high is the cliff?”
  • Challenge: “If tan θ = 2 and θ is in QIII, find sin θ and cos θ. (Hint: Draw a right triangle in the correct quadrant.)”

These problems can be done as a set of 10–15 exercises, gradually increasing in difficulty. A solution key with step‑by‑step reasoning is invaluable for self‑study.

For additional practice, the Purplemath lesson on graphing tangent offers worked examples and practice questions with explanations.

Conclusion

A well‑designed step‑by‑step guide transforms the tangent function from an intimidating collection of weird rules into a logical, visual, and even beautiful part of mathematics. By starting with right‑triangle ratios, moving to the unit circle, then meticulously graphing the function and analysing its properties, students build a robust mental model. Remind them that the key differences—period π, vertical asymptotes, no amplitude—are what make tangent both challenging and fascinating. Frequent practice with graphing, table‑building, and word problems, supplemented by interactive digital tools, cements these ideas. When students can confidently explain why tan θ is undefined at 90° and why its graph looks the way it does, they have truly mastered a fundamental trigonometric function.