The parent graphs of sine and cosine, amplitude and period, the five key points for graphing a sine bx and a cosine bx, frequency and sine models of oscillation, and the tangent graph with its vertical asymptotes.
Subject: Algebra 2 · 65 slides · symbolic lesson
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Title
Algebra 2 · Chapter 14 — Trigonometric Graphs, Identities, and Equations
Graph Sine, Cosine, and Tangent Functions
Objectives
Five outcomes. The unit circle unrolled into a wave.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 908-913 — the lesson these objectives are drawn from
Warm-up
Chapter 13 evaluated the sine of any angle you like.
Discussion prompt
Compute the sine at 0, pi over 2, pi, 3 pi over 2 and 2 pi. Plot those five points with the angle on the horizontal axis. What shape is forming?
Hint: The unit circle's y-coordinate, read as the angle turns.
Answer:
\[ 0, \; 1, \; 0, \; -1, \; 0 \]
A wave: up to 1, back through 0, down to negative 1, and back to 0. Then the whole thing repeats forever, because the angle 2 pi lands where 0 did.
That repetition is what makes sine a periodic function, and the graph is what this whole chapter is about.
Concept
Plotting a trigonometric function against the angle turns the unit circle into a repeating graph. Sine and cosine give unbroken waves of amplitude one and period two pi; tangent gives branches separated by vertical asymptotes with period pi.
period — The horizontal length of one cycle, the shortest repeating portion of a periodic graph. For y equal to a sine bx it is two pi over the size of b, and for y equal to a tangent bx it is pi over the size of b.
\[ \text{Amplitude} = |a|; \qquad \text{Period} = \frac{2\pi}{|b|} \]
The reciprocal of the period is the frequency, which counts cycles per unit of time and is how oscillations are usually described in practice.
Figure (svg): Two columns comparing the sine and cosine graphs with the tangent graph
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 908-911
Section
Section 1
Concept
Both y equal to sine x and y equal to cosine x have domain all real numbers, range from negative one to one, amplitude one and period two pi. They differ only in where their cycle begins.
\[ -1 \le \sin x \le 1; \qquad -1 \le \cos x \le 1 \]
Sine crosses zero at 0 and every multiple of pi; cosine crosses at the odd multiples of pi over two, a quarter turn later.
Figure (svg): The graphs of sine and cosine over two full periods
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 908-908 — Characteristics of y = sin x and y = cos x
Picture it
Both parent graphs over two full periods.
Figure (svg): The graphs of sine and cosine over two full periods
The same wave appears twice, shifted by a quarter period. Sine begins at the middle of its range and cosine at the top.
Worked example
The boxed characteristics on page 908.
\[ \text{State the domain, range, amplitude, period and zeros of } y = \sin x \text{ and } y = \cos x. \]
Domain
Why: Every angle has a sine and a cosine.
Range and amplitude
Why: Between -1 and 1; half that spread.
\[ \text{range } -1\text{ to } 1,\text{ amplitude } 1 \]
Period
Why: The graph repeats after one full turn.
\[ 2 \pi \]
Zeros
Why: Where the y or x coordinate is zero.
\[ \sin e\text{ at multiples of } \pi;\text{ cosine at odd multiples of } \frac{\pi}{2} \]
Figure (svg): The graphs of sine and cosine over two full periods
\[ \text{amplitude } 1, \; \text{period } 2\pi \]
Verify: check the amplitude formula
Why: Amplitude is half the difference between the maximum and the minimum, which is half of 1 minus negative 1, or 1. That definition will matter in the next idea, where the maximum and minimum move away from plus and minus one.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 908-908
Fill the middle
The definition on page 908.
Fill in the blanks
\text1 = \tfrac______(M-m) = \tfrac______(1-(-1)) = ___
Why: One. Amplitude is half the total vertical spread, not the maximum itself, though for these parent graphs the two happen to agree.
Worked example
The boxed characteristics on page 908.
\[ \text{Explain why sine and cosine have their zeros at different places.} \]
Recall the definitions
Why: On the unit circle sine is y and cosine is x.
Find where y is zero
Why: At the two ends of the horizontal axis.
\[ 0, \pi, 2 \pi,... \]
Find where x is zero
Why: At the two ends of the vertical axis.
\[ \frac{\pi}{2}, 3 \pi / 2,... \]
Conclude
Why: The two axes are a quarter turn apart.
Figure (svg): The solution to Worked example read the zeros off the circle shown as a ladder of expressions, one row per algebraic move
\[ \sin x = 0 \text{ at } x = k\pi; \quad \cos x = 0 \text{ at } x = \tfrac{\pi}{2}+k\pi \]
Verify: check where each graph peaks instead
Why: Sine peaks where cosine is zero and cosine peaks where sine is zero, which is the same quarter-turn offset seen from the other side. The two graphs are the same curve, one shifted a quarter period from the other.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 908-908
Trap
\[ y = \sin x \text{ on } 0 \le x \le 2\pi \]
Treat that one cycle as the whole graph
Why: The picture in the book shows a limited stretch.
\[ \text{domain: } 0 \le x \le 2\pi \quad \text{(wrong)} \]
The domain is ALL real numbers. One cycle is drawn because the rest is an exact copy, not because the graph ends.
\[ \sin(x+2\pi) = \sin x \text{ for every } x \]
Extend by copying the cycle
Why: Left and right, without limit.
\[ \text{domain: all real numbers} \]
Negative angles are perfectly allowed too, which is why the parent graphs are drawn on both sides of zero.
Matching
The two parent graphs.
Match the pairs
Why: The two sets of zeros interleave, alternating every quarter turn. That is the graphical statement of the fact that sine and cosine are the same curve a quarter period apart.
Sorting
Both sine and cosine.
Sort into buckets
Sort each statement.
Asymptotes belong to tangent, not to sine or cosine — because sine and cosine have r underneath, and r is never zero.
Prediction
Commit before reasoning.
Predict first
Why is the sine graph identical on 0 to 2 pi and on 2 pi to 4 pi?
Correct: Because adding 2 pi returns the terminal side to where it was, giving a coterminal angle with the same sine.
\[ \sin(x+2\pi) = \sin x \]
Why: Coterminal angles from Lesson 13.2 share every trigonometric value, and 2 pi is exactly one full turn. So the whole graph is one cycle stamped out over and over, in both directions. This is why periodic functions are the natural language for anything that cycles — days, tides, sound, alternating current — and why the rest of this chapter is worth the trouble.
Section
Section 2
Concept
For y equal to a sine bx or a cosine bx, the amplitude is the size of a and the period is two pi divided by the size of b. Increasing a makes the graph taller; increasing b makes the cycles shorter.
\[ \text{Amplitude} = |a|; \qquad \text{Period} = \frac{2\pi}{|b|} \]
Because the two constants act independently, each can be read off a graph without knowing the other.
Figure (svg): How the constants a and b stretch the graph
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 909-909 — Amplitude and Period
Picture it
Example 1: y equal to 4 sine x, and y equal to cosine 4x.
Figure (svg): How the constants a and b stretch the graph
The first is stretched vertically and repeats at the usual rate; the second keeps its height and repeats four times as often.
Worked example
Example 1, both parts.
\[ \text{Find the amplitude and period of } y = 4\sin x \text{ and } y = \cos 4x. \]
First: read a and b
Why: Four in front, and 1 inside.
\[ a = 4, b = 1 \]
First: apply the formulas
Why: Size of 4; two pi over 1.
\[ \text{amplitude } 4,\text{ period } 2 \pi \]
Second: read a and b
Why: One in front, and 4 inside.
\[ a = 1, b = 4 \]
Second: apply the formulas
Why: Size of 1; two pi over 4.
\[ \text{amplitude } 1,\text{ period } \frac{\pi}{2} \]
Figure (svg): How the constants a and b stretch the graph
\[ 4, \; 2\pi; \qquad 1, \; \tfrac{\pi}{2} \]
Verify: check the second period against the picture
Why: A period of pi over 2 means four complete cycles fit into the usual 2 pi, which is exactly what multiplying the input by 4 should do: the graph reaches the end of a cycle in a quarter of the input. Larger b, shorter period.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 909-909
Fill the middle
Example 1b.
Fill in the blanks
\text2 = \frac______ = \frac______ = \frac______}
Why: Pi over 2. Four cycles now fit where one used to, because the input is being multiplied by 4 before the cosine sees it.
Worked example
Guided Practice 1 to 4.
\[ \text{Find the amplitude and period of } y = 2\cos x, \; y = 5\sin x, \; f(x) = \sin\pi x, \; g(x) = \cos 4\pi x. \]
First and second
Why: Coefficients 2 and 5, both with b = 1.
\[ \text{amplitudes } 2\text{ and } 5,\text{ period } 2 \pi \]
Third: b is pi
Why: Two pi over pi.
\[ \text{amplitude } 1,\text{ period } 2 \]
Fourth: b is 4 pi
Why: Two pi over 4 pi.
\[ \text{amplitude } 1,\text{ period } \frac{1}{2} \]
Note the pattern
Why: A pi inside cancels the pi in the period.
Figure (svg): The solution to Worked example four more shown as a ladder of expressions, one row per algebraic move
\[ 2\pi; \; 2\pi; \; 2; \; \tfrac{1}{2} \]
Verify: notice why a pi inside is convenient
Why: When b contains a factor of pi the period comes out a plain number, which is exactly what applications want: a wave with period 2 seconds is easier to describe than one with period 2 pi. That is why models of real oscillations almost always have a pi inside.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 909-909
Error analysis
A student states the amplitude and period of y = cos 4x.
Annotate
On: \( \text{amplitude } 4, \quad \text{period } 2\pi \)
The constant in FRONT stretches vertically; the constant INSIDE compresses horizontally. Reading which position a number occupies settles which job it does.
Matching
Read a from the front and b from inside.
Match the pairs
Why: Only the first has an amplitude other than 1, and only it leaves the period alone. The other three all change the period and none touches the height, which is the independence of a and b in action.
Ranking
Shortest first.
Put in order
Why: The periods are 1/2, about 1.57, 2, about 2.09 and about 6.28. The larger the value of b, the shorter the period, regardless of what the coefficient in front is doing.
Prediction
Commit before reasoning.
Predict first
Compare y = sin x with y = 5 sin x. Do they have the same period?
Correct: Yes — a stretches the graph vertically only, so both repeat every 2 pi.
\[ 5\sin(x+2\pi) = 5\sin x \]
Why: Multiplying the OUTPUT by 5 makes every height five times bigger but does not change when those heights occur, so the zeros and peaks stay exactly where they were. Only a change to the INPUT, which is what b does, can move them. That separation is why amplitude and period can be read off a graph independently, and it holds for every function, not just these.
Section
Section 3
Concept
To graph one cycle, mark the period on the horizontal axis and divide it into quarters. Those five x-values carry the intercepts, the maximum and the minimum, and joining them draws the curve.
\[ 0, \; \tfrac{1}{4}T, \; \tfrac{1}{2}T, \; \tfrac{3}{4}T, \; T \quad \text{where } T = \tfrac{2\pi}{b} \]
Sine starts at zero and rises; cosine starts at the maximum. Otherwise the two follow the same pattern of quarters.
Figure (svg): The five key x-values for one cycle of a sine and a cosine graph
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 909-910 — Graphing key points
Picture it
The key x-values for a sine graph and for a cosine graph.
Figure (svg): The five key x-values for one cycle of a sine and a cosine graph
Sine goes middle, top, middle, bottom, middle. Cosine goes top, middle, bottom, middle, top. Same heights, different starting place.
Worked example
Example 1, both parts.
\[ \text{Graph } y = 4\sin x \text{ and } y = \cos 4x \text{ over one period.} \]
First: quarter the period 2 pi
Why: Zero, pi/2, pi, 3 pi/2, 2 pi.
First: attach the heights
Why: Sine starts at 0 and peaks at a.
\[ (0, 0), (\frac{\pi}{2}, 4), (\pi, 0), (3 \pi / 2, -4), (2 \pi, 0) \]
Second: quarter the period pi/2
Why: Zero, pi/8, pi/4, 3 pi/8, pi/2.
Second: attach the heights
Why: Cosine starts at a.
\[ (0, 1), (\frac{\pi}{8}, 0), (\frac{\pi}{4}, -1), (3 \pi / 8, 0), (\frac{\pi}{2}, 1) \]
Figure (svg): The five key x-values for one cycle of a sine and a cosine graph
\[ \!\left(\tfrac{\pi}{2},4\right); \qquad \!\left(\tfrac{\pi}{4},-1\right) \]
Verify: check the second graph's intercepts
Why: They fall at pi over 8 and 3 pi over 8, which are one quarter and three quarters of the period pi over 2 — not at pi over 2 and 3 pi over 2 as they would be for the parent graph. Every key x-value scales with the period.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 909-909
Fill the middle
Example 1a.
Fill in the blanks
\text2 \tfrac______(2\pi) = \frac______}, \text___ 4
Why: Pi over 2. A sine graph reaches its maximum a quarter of the way through each cycle, whatever the period happens to be.
Worked example
Example 2.
\[ \text{Graph } y = \tfrac{1}{2}\cos 2\pi x. \]
Find the amplitude and period
Why: Half; and two pi over two pi.
\[ \text{amplitude } \frac{1}{2},\text{ period } 1 \]
Quarter the period
Why: Zero, 1/4, 1/2, 3/4, 1.
Attach the heights
Why: Cosine starts at the maximum.
\[ \frac{1}{2}, 0, -\frac{1}{2}, 0, \frac{1}{2} \]
Draw and extend
Why: Copy the cycle sideways.
Figure (svg): The solution to Worked example a fractional amplitude shown as a ladder of expressions, one row per algebraic move
\[ \!\left(0,\tfrac{1}{2}\right), \!\left(\tfrac{1}{2},-\tfrac{1}{2}\right), \!\left(1,\tfrac{1}{2}\right) \]
Verify: check that the period really is 1
Why: Two pi over 2 pi is exactly 1, so the graph completes a full cycle between x equal to 0 and x equal to 1. A period of 1 makes the graph unusually easy to read, which is why 2 pi is such a common choice for b in applications.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 910-910
Trap
\[ y = \cos 4x \]
Put the minimum at pi
Why: The parent cosine has its minimum there.
\[ (\pi, -1) \quad \text{(wrong)} \]
The period here is pi over 2, so the minimum sits at half of that: pi over 4. At x equal to pi the graph is back at its maximum.
\[ T = \tfrac{\pi}{2} \;\Longrightarrow\; \text{minimum at } \tfrac{1}{2}T = \tfrac{\pi}{4} \]
Compute the period first, then take quarters of IT
Why: Never quarters of 2 pi.
\[ \!\left(\tfrac{\pi}{4},-1\right) \]
Every key x-value is a fraction of the actual period, so changing b moves all five of them together.
Fill the middle
Example 2.
Fill in the blanks
T = 1: \; \text3 \tfrac______ \text___ \tfrac___}___
Why: Three quarters. For a cosine graph the two zeros in each cycle fall at one quarter and three quarters of the period.
Comparison
Fill the blanks. One period, five heights.
Comparison matrix
| Position in the cycle | y = a sin bx | y = a cos bx |
|---|---|---|
| Start | 0 | a |
| One quarter | a | 0 |
| Half | 0 | -a |
| Three quarters | -a | 0 |
Both columns cycle through the same five heights; cosine is simply one quarter ahead. Shifting a sine graph a quarter period to the left produces the cosine graph exactly.
Prediction
Commit before reasoning.
Predict first
You have drawn y = (1/2) cos 2 pi x on 0 to 1. What happens on 1 to 2?
Correct: An exact copy of the first cycle, and the same again on every interval of length 1.
\[ f(x+1) = f(x) \text{ for all } x \]
Why: The period is 1, so adding 1 to x returns the function to the same value. Drawing one cycle and copying it left and right is the standard way to produce the whole graph, and the textbook says so explicitly. This is also why a single cycle carries all the information: everything the function will ever do, it does within one period.
Section
Section 4
Concept
The frequency counts cycles per unit of time and is the reciprocal of the period. Given a maximum value and a frequency, the model y equal to a sine bt follows at once, with a the maximum and b two pi times the frequency.
\[ \text{frequency} = \frac{1}{\text{period}} = \frac{b}{2\pi} \]
Sound, light, alternating current and pendulums are all described by frequency rather than period, so this conversion is the usual first step in a model.
Figure (svg): A pure tone modelled as a sine wave of 2000 cycles per second
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 910-910 — Model with a sine function
Picture it
Example 3: 2000 hertz at a maximum pressure of 2 millipascals.
Figure (svg): A pure tone modelled as a sine wave of 2000 cycles per second
Two thousand cycles a second means a period of one two-thousandth of a second, and b comes out as 4000 pi.
Worked example
Example 3.
\[ \text{A pure tone has frequency } 2000 \text{ Hz and maximum pressure } 2. \text{ Write and graph } P = a\sin bt. \]
Read a from the maximum
Why: Two millipascals.
\[ a = 2 \]
Convert frequency to b
Why: Two thousand equals b over 2 pi.
\[ b = 4000 \pi \]
Write the model
Why: Two sine of 4000 pi t.
\[ P = 2 \sin(4000 \pi t) \]
Find the key points
Why: Period is 1/2000; quarter it.
\[ \max\text{ at } \frac{1}{8000}, \min\text{ at } \frac{3}{8000} \]
Figure (svg): A pure tone modelled as a sine wave of 2000 cycles per second
\[ P = 2\sin(4000\pi t) \]
Verify: check the period against the frequency
Why: Two pi over 4000 pi is one two-thousandth, and its reciprocal is 2000 — the stated frequency. Converting back is the reliable check, because the factor of 2 pi is easy to drop or double.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 910-910
Fill the middle
Example 3, Step 1.
Fill in the blanks
2000 = \frac4000___ \;\Longrightarrow\; b = ___\pi
Why: Four thousand pi. Multiplying both sides by 2 pi turns cycles per second into radians per second.
Worked example
Guided Practice 9.
\[ \text{How does the model change at a frequency of } 1000 \text{ Hz?} \]
Recompute b
Why: One thousand equals b over 2 pi.
\[ b = 2000 \pi \]
Write the new model
Why: The amplitude is unchanged.
\[ P = 2 \sin(2000 \pi t) \]
Find the new period
Why: Two pi over 2000 pi.
\[ \frac{1}{1000}\text{ second} \]
Compare
Why: Half the frequency.
Figure (svg): The solution to Worked example halve the frequency shown as a ladder of expressions, one row per algebraic move
\[ P = 2\sin(2000\pi t) \]
Verify: check what the listener would hear
Why: Halving the frequency drops the tone by exactly one octave, and the pressure amplitude of 2 is unchanged, so the loudness is the same. The graph is the old one stretched horizontally by a factor of two with its height untouched — a and b doing separate jobs again.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 911-911
Error analysis
A student models a 2000 hertz tone.
Annotate
On: \( P = 2\sin(2000t) \)
The 2 pi converts between cycles and radians. Leaving it out multiplies the period by 2 pi and divides the frequency by the same, which is a factor of more than six.
Fill the middle
Example 3, Step 2.
Fill in the blanks
T = \tfrac8000___: \; \text___ \tfrac______T = \frac______}
Why: One eight-thousandth of a second. Quartering a period that is itself a fraction just multiplies the denominator by four.
Matching
b is two pi times the frequency.
Match the pairs
Why: The bottom two show the conversion from the other direction: a period of 1 needs b equal to 2 pi. Notice that halving the frequency halves b in both pairs, so period and frequency really are reciprocals.
Prediction
Commit before reasoning.
Predict first
A tone's frequency doubles while its maximum pressure stays at 2. What changes about the graph and about the sound?
Correct: The graph's cycles get half as long and the pitch rises by an octave; the loudness is unchanged.
\[ f \to 2f \;\Longrightarrow\; T \to \tfrac{1}{2}T, \; a \text{ unchanged} \]
Why: Frequency controls pitch and amplitude controls loudness, and they are exactly the b and a of the model. Doubling the frequency halves the period, squeezing the graph horizontally without changing its height. A doubling of frequency is precisely one musical octave, which is why the same note an octave up has twice the frequency — and why a and b staying independent matters far outside mathematics.
Section
Section 5
Concept
The tangent graph has period pi rather than two pi, no amplitude at all, and vertical asymptotes at the odd multiples of pi over two, where cosine is zero. For y equal to a tangent bx the period is pi over the size of b and the asymptotes are at odd multiples of pi over twice that size.
\[ \text{Period} = \frac{\pi}{|b|}; \qquad \text{asymptotes at odd multiples of } \frac{\pi}{2|b|} \]
Because the range is all real numbers, the tangent graph has no maximum or minimum and therefore no amplitude to speak of.
Figure (svg): The tangent graph with its vertical asymptotes
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 911-911 — Characteristics of y = a tan bx
Picture it
The parent tangent graph over two full periods.
Figure (svg): The tangent graph with its vertical asymptotes
Each branch climbs from far below to far above between two asymptotes, crossing zero exactly halfway.
Worked example
Example 4.
\[ \text{Graph one period of } y = 2\tan 3x. \]
Find the period
Why: Pi over 3.
\[ T = \frac{\pi}{3} \]
Find the asymptotes
Why: Odd multiples of pi over 6.
\[ x = -\frac{\pi}{6}\text{ and } \frac{\pi}{6} \]
Mark the intercept
Why: Halfway between them.
\[ (0, 0) \]
Mark the halfway points
Why: At plus and minus pi over 12, height plus and minus a.
\[ (\frac{\pi}{12}, 2)\text{ and } (-\frac{\pi}{12}, -2) \]
Figure (svg): The tangent graph with its vertical asymptotes
\[ T = \tfrac{\pi}{3}, \; \text{asymptotes at } x = \pm\tfrac{\pi}{6} \]
Verify: check the halfway height
Why: At x equal to pi over 12 the argument 3x is pi over 4, whose tangent is 1, so y is 2 times 1, which is 2 — matching a. That is why the halfway points always sit at plus and minus a, whatever b is: the argument there is always pi over 4.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 912-912
Fill the middle
Example 4.
Fill in the blanks
y = 2\tan 3x: \; T = \frac3___ = \frac______}
Why: Pi over 3. Tangent's parent period is pi, so its formula uses pi over b rather than 2 pi over b.
Worked example
Guided Practice 10, 11 and 13.
\[ \text{Find the period and asymptotes of } y = 3\tan x, \; y = \tan 2x, \; g(x) = 5\tan\pi x. \]
First: b is 1
Why: Pi over 1; asymptotes at odd multiples of pi/2.
Second: b is 2
Why: Pi over 2; asymptotes at odd multiples of pi/4.
\[ \text{period } \frac{\pi}{2} \]
Third: b is pi
Why: Pi over pi; asymptotes at odd multiples of 1/2.
\[ \text{period } 1 \]
Note the coefficient in front
Why: Three, one and five.
Figure (svg): The solution to Worked example three more tangent graphs shown as a ladder of expressions, one row per algebraic move
\[ \pi; \; \tfrac{\pi}{2}; \; 1 \]
Verify: check what a in front actually does
Why: It sets the height at the halfway points, so a larger a makes the branch steeper — but since the range is already all real numbers, it cannot make the graph taller. That is precisely why tangent has no amplitude even though a is doing visible work.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 912-912
Trap
\[ y = 2\tan 3x \]
Use two pi over b
Why: The formula from the last three ideas is applied.
\[ T = \frac{2\pi}{3} \quad \text{(wrong)} \]
Tangent repeats every pi, not every two pi, so its period is pi over b. The wrong answer is twice too long.
\[ T = \frac{\pi}{b} = \frac{\pi}{3} \]
Use pi over b for tangent
Why: A different parent period means a different formula.
\[ \text{asymptotes at } x = \pm\tfrac{\pi}{6} \]
Tangent is y over x, and both coordinates change sign together after half a turn, so the ratio returns after only pi.
Sorting
Compare with sine and cosine.
Sort into buckets
Sort each statement about y = tan x.
The two false statements are the two ways tangent differs most from sine and cosine, and both trace back to what sits in the denominator of each definition.
Matching
Pi over the coefficient inside.
Match the pairs
Why: The coefficients in front, 3, 1, 2 and 5, play no part at all in the period; only the number inside does. The last has a pi inside that cancels the pi in the formula, leaving the tidy period of 1.
Prediction
Commit before reasoning.
Predict first
Sine and cosine repeat every 2 pi. Why does tangent repeat after only pi?
Correct: Because after half a turn both x and y change sign, and their ratio is unchanged.
\[ \tan(x+\pi) = \frac{-y}{-x} = \frac{y}{x} = \tan x \]
Why: Rotating by pi sends the point x, y to negative x, negative y. Sine and cosine, which read those coordinates directly, both flip sign and need another half turn to return. But tangent is their QUOTIENT, and two sign flips cancel. So tangent comes back after pi while sine and cosine take 2 pi — and the same argument gives cotangent a period of pi as well.
Comparison
Fill the blanks. Same circle, three different pictures.
Comparison matrix
| Question | y = a sin bx | y = a cos bx | y = a tan bx |
|---|---|---|---|
| Period | 2 pi / |b| | 2 pi / |b| | pi / |b| |
| Amplitude | |a| | |a| | none |
| Starts a cycle at | zero | the maximum | an asymptote |
| Asymptotes | none | none | odd multiples of pi/(2|b|) |
Sine and cosine are the same wave a quarter period apart, and tangent is their quotient — which is why it repeats twice as often and breaks wherever the cosine on the bottom hits zero.
Pattern
Constants first, then quarters.
Tangent's period is pi over b, not 2 pi over b. Using the wrong formula doubles every answer.
OpenStax Algebra and Trigonometry 2e, §8.1 Graphs of the Sine and Cosine Functions §8.1
Check
The constant inside changes the period.
Check your understanding
What are the amplitude and period of y = cos 4x?
Answer: A
Why: The coefficient in front is 1 and b is 4, so the period is 2 pi over 4.
Check
Frequency is b over two pi.
Check your understanding
A pure tone has frequency 2000 hertz and maximum pressure 2. What is the model?
Answer: A
Why: a is the maximum 2, and 2000 = b over 2 pi gives b = 4000 pi.
Check
Tangent has its own period formula.
Check your understanding
What is the period of y = 2 tan 3x?
Answer: A
Why: Tangent's period is pi over b, not 2 pi over b.
Real world
A Ferris wheel of radius 25 metres has its centre 30 metres above the ground and completes one revolution every 40 seconds. A rider boards at the lowest point.
Discussion prompt
Write a model for the rider's height above the ground, and find their height after 30 seconds.
Hint: Boarding at the bottom means starting at the minimum, which is a cosine flipped.
Answer:
\[ b = \frac{2\pi}{T} = \frac{2\pi}{40} = \frac{\pi}{20} \]
\[ h(t) = 30-25\cos\!\left(\frac{\pi t}{20}\right) \]
\[ h(30) = 30-25\cos\!\left(\frac{3\pi}{2}\right) = 30-25(0) = 30 \text{ m} \]
The rider is at 30 metres, exactly level with the centre and rising.
Two choices carry the model. The MINUS in front of the cosine flips the wave so it starts at the bottom rather than the top, which is what boarding at the lowest point requires. And the 30 out in front raises the whole wave so its centre is at axle height rather than at zero — that constant is not amplitude or period but a vertical shift, and Lesson 14.4 will treat it properly. Everything else is the b equal to 2 pi over the period that this lesson established.
Commit first
Answer, then rate your confidence honestly.
Predict first
Does the tangent graph have an amplitude?
Correct: No — amplitude is half the spread between a maximum and a minimum, and tangent has neither.
\[ \text{range of } \tan x: \; (-\infty,\infty) \]
Why: The tangent graph runs off to infinity in both directions on every branch, so its range is all real numbers and there is no highest or lowest value to measure. The constant a is still doing visible work — it sets the height at the halfway points and so controls how steeply the branch rises — but that is steepness, not amplitude. Only a bounded oscillation can have an amplitude, which is why sine and cosine have one and tangent does not.
Explain it
They can plot points and have never seen a periodic graph.
Discussion prompt
In four sentences or fewer, explain why the sine graph is a repeating wave.
Hint: Talk about going round a circle.
Answer:
Walk around a circle of radius 1 and keep track of how high you are. You start level with the centre, rise to the top, come back level, drop to the bottom, and return.
Plot that height against how far you have walked and you get one wave. Keep walking and you go round again, so the wave repeats forever.
Exit ticket
Name the weakest spot before you close the deck.
Predict first
Which of these would you least want handed to you cold?
Correct: Whichever you picked is tonight's ten minutes, and each has a one-line fix.
Why: For the constants, the one in front stretches vertically and the one inside compresses horizontally. For key points, compute the period first and take quarters of IT, never of 2 pi. For frequency, b is 2 pi times the frequency, and checking the period back is the safest test. For tangent, its parent period is pi, so the formula is pi over b.
Connect it up
Paper. Fifteen minutes.
Draw it
Build a trigonometric graphs page. Top left: draw both parent graphs over two periods on the same axes, marking their zeros in different colours and writing beside them why those zeros interleave. Top right: draw two graphs that share an amplitude but differ in period, and two that share a period but differ in amplitude, labelling a and b on each. Middle: pick a sine and a cosine function of your own with b not equal to 1, quarter each period, plot all five key points, and join them. Bottom left: write a sine model from a frequency and a maximum of your own choosing, then convert back to check the frequency you started from. Bottom right: draw one period of a tangent function, marking both asymptotes, the intercept and the two halfway points, and note beside it why the period is pi rather than 2 pi.
If any key point on your page sits at a quarter of 2 pi rather than a quarter of the actual period, all five are misplaced. Recompute the period first.
Recap
Five things, and the unit circle turned into a graph.
| If you see | Then |
|---|---|
| A constant in front | It sets the amplitude |
| A constant inside | It divides into 2 pi to give the period |
| A graph to sketch | Quarter the period and use the five key heights |
| A frequency | b is 2 pi times it |
| A tangent function | The period is pi over b |
| Asymptotes wanted | Odd multiples of pi over twice the size of b |
Lesson 14.2 adds the two things this lesson left out: a horizontal shift that moves the cycle sideways, and a vertical shift that raises the whole wave.
McDougal Littell Algebra 2 (Texas Edition), Ch. 14 Trigonometric Graphs, Identities, and Equations — Lesson 14.1 Graph Sine, Cosine, and Tangent Functions §14.1, pp. 908-913 — everything on these slides traces back here
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