Periodic behavior made visible — sine, cosine, tangent, and reciprocal graphs; amplitude, period, phase shift, midline, asymptotes, transformations, and models of waves, tides, seasons, and oscillation
Chapter 4 established trigonometric functions as relationships between angles and coordinates on the unit circle. Sine and cosine emerged as the vertical and horizontal coordinates of a point tracing the unit circle. Tangent arose as their ratio. The six functions were defined, evaluated, and connected to special angles in both degrees and radians.
This chapter makes those functions visible. When you plot the output of sine, cosine, or tangent against the input angle — tracing the full rotation of the unit circle from left to right — a wave appears. That wave is not random. It has a precise shape, a predictable repeating cycle, and measurable features: the highest and lowest values, the distance between repetitions, the location of the center, and where the graph has been shifted horizontally. Every one of those features corresponds to a number you can read from the formula.
Understanding trigonometric graphs is not only a mathematical skill. It is the foundation for describing any quantity that repeats or oscillates over time. Sound is a wave. Light is a wave. The tides rise and fall in a periodic cycle. Electrical current alternates. Seasons repeat. A pendulum swings back and forth. A spinning wheel generates circular motion that projects onto a line as a sinusoidal wave. In every one of these situations, the mathematics of this chapter provides the model.
By the end of this chapter, you will be able to identify the amplitude, period, phase shift, and midline of a trigonometric function directly from its formula, graph it accurately, write a formula from a graph, and build models for periodic phenomena in science, engineering, and everyday life.
A buoy bobs on the ocean. At its highest point, it sits 2 meters above the average water line. At its lowest, it sits 2 meters below. It completes one full cycle every 8 seconds. If it starts at the average water line and moves upward, how high is the buoy after 2 seconds? After 6 seconds? After 20 seconds? By the end of this chapter, you will be able to write a single equation that answers these questions for any time value — and you will understand why that equation takes the shape it does.
A function is called periodic if its values repeat in a regular cycle as the input increases. Instead of always producing new outputs, the function circles back and begins the same pattern again. The length of one complete cycle — the smallest positive value P for which f(x + P) = f(x) for all x in the domain — is called the period of the function.
Think of a wheel spinning at a constant rate. The height of a point on the rim goes up, reaches the top, comes back down, reaches the bottom, and then returns to where it started. That is one period. After one full rotation, the entire pattern begins again. This is exactly what happens on the unit circle: as the angle increases from 0 to 2π, the point traces one complete circle. The y-coordinate (sine) and the x-coordinate (cosine) each complete one cycle. As the angle continues past 2π, the pattern repeats.
Periodic behavior is not only mathematical. Daylight hours repeat each year. Ocean tides rise and fall every 12.4 hours. Sound vibrates air molecules back and forth hundreds or thousands of times per second. Alternating electrical current reverses direction 60 times per second in most of North America. The seasons follow a year-long cycle. Heart rate produces repeating beats. In all these situations, a periodic function provides the mathematical model.
A function f is periodic with period P if P is the smallest positive number such that: f(x + P) = f(x) for every x in the domain of f.
The function returns to the same output value after every interval of length P.
On the unit circle, a point at angle θ has coordinates (cos θ, sin θ). As θ increases from 0 to 2π, the point traces one full revolution and returns to its starting position. The coordinates at angle θ + 2π are identical to those at angle θ. This is why both sin(θ + 2π) = sin θ and cos(θ + 2π) = cos θ — the period of sine and cosine is exactly 2π radians.
| Periodic Phenomenon | Period (approximate) | Trigonometric Model |
|---|---|---|
| Rotation of a wheel (at 1 rev/sec) | 1 second | Sine or cosine |
| Ocean tide (semi-diurnal) | 12.4 hours | Sine or cosine |
| A sound wave at concert A (440 Hz) | 1/440 second | Sine |
| Annual temperature cycle | 12 months | Sine or cosine |
| AC electrical current (North America) | 1/60 second | Sine |
The period of sin(x) and cos(x) is 2π. In calculus, the derivative of sin(x) is cos(x) only when x is measured in radians. When x is in degrees, a conversion factor appears. Radians make the mathematics of periodic functions clean, especially when limits and derivatives are involved. This is one major reason why radians are preferred in precalculus and beyond.
To graph y = sin(x), recall that sin(x) is the y-coordinate of the point on the unit circle at angle x. As x increases from 0 to 2π, that y-coordinate moves from 0 up to 1, back down through 0 to −1, and then back up to 0. When you plot these output values against the input angle, tracing the vertical coordinate as the angle sweeps around the circle, you obtain the familiar S-shaped wave.
The key values to know precisely come directly from the special angles on the unit circle. At x = 0, sin(0) = 0. At x = π/2, sin(π/2) = 1. At x = π, sin(π) = 0. At x = 3π/2, sin(3π/2) = −1. At x = 2π, sin(2π) = 0. These five anchor points — zero, maximum, zero, minimum, zero — are all you need to sketch one complete cycle. After x = 2π, the pattern repeats.
| x (radians) | sin(x) | Description |
|---|---|---|
| 0 | 0 | Starting zero (beginning of cycle) |
| π/6 | 1/2 | Rising |
| π/4 | √2/2 ≈ 0.707 | Rising |
| π/3 | √3/2 ≈ 0.866 | Rising |
| π/2 | 1 | Maximum |
| π | 0 | Middle zero (falling through) |
| 3π/2 | −1 | Minimum |
| 2π | 0 | Ending zero (one full cycle complete) |
Draw a horizontal axis labeled x, with tick marks at multiples of π/2: 0, π/2, π, 3π/2, 2π, and extending to −2π on the left. Draw a vertical axis. Plot these five anchor points on one cycle from 0 to 2π: (0, 0), (π/2, 1), (π, 0), (3π/2, −1), (2π, 0). Connect with a smooth, continuous S-shaped curve — rising from (0,0) to the maximum at (π/2, 1), descending through (π, 0) to the minimum at (3π/2, −1), and rising back to (2π, 0). Repeat this pattern to the left for negative values of x. Label the maximum 1 and the minimum −1 on the vertical axis.
Domain: (−∞, ∞) — defined for all real numbers
Range: [−1, 1]
Period: 2π
Amplitude: 1 (the maximum distance from the midline)
Midline: y = 0 (the horizontal axis)
Zeros: x = nπ for every integer n (the function crosses the x-axis)
Symmetry: odd function — the graph has rotational symmetry about the origin
Every sine or cosine cycle is anchored by five key points: start, quarter cycle, half cycle, three-quarter cycle, and end. For y = sin(x) on [0, 2π], these are (0,0), (π/2, 1), (π, 0), (3π/2, −1), (2π, 0). Learning to find the equivalents of these five points on any transformed sine or cosine function is the core graphing skill of this chapter.
The graph of y = cos(x) is produced the same way as the sine graph — by tracking a coordinate on the unit circle — but this time it is the x-coordinate rather than the y-coordinate. As the angle increases from 0, the x-coordinate on the unit circle begins at 1 (the rightmost point), moves through 0 at the top, reaches −1 at the leftmost point, moves through 0 at the bottom, and returns to 1 at the end of one revolution. The cosine graph therefore begins at its maximum value and descends, rather than beginning at zero and rising as sine does.
The cosine function shares all of the same basic properties as sine — the same domain, the same range [−1, 1], the same period 2π, the same amplitude 1, and the same midline y = 0. The only difference is the phase relationship: the cosine graph is a horizontal shift of the sine graph. Specifically, cos(x) = sin(x + π/2). This means every feature of the cosine graph appears exactly π/2 units to the left of the corresponding feature on the sine graph.
| x (radians) | cos(x) | Description |
|---|---|---|
| 0 | 1 | Maximum (beginning of cycle) |
| π/2 | 0 | Descending zero |
| π | −1 | Minimum |
| 3π/2 | 0 | Ascending zero |
| 2π | 1 | Maximum (one full cycle complete) |
Draw horizontal and vertical axes as for the sine graph. Plot the five anchor points for one cycle on [0, 2π]: (0, 1), (π/2, 0), (π, −1), (3π/2, 0), (2π, 1). Connect with a smooth wave — starting at the maximum (0, 1), descending smoothly through (π/2, 0), reaching the minimum (π, −1), rising through (3π/2, 0), and returning to maximum (2π, 1). The shape is like a gentle arch that dips down to a trough and back up again. Extend the pattern in both directions. Label maximum 1 and minimum −1.
Both functions: domain (−∞, ∞), range [−1, 1], period 2π, amplitude 1, midline y = 0
Sine begins: at zero (0, 0), rises to maximum, falls through zero to minimum, returns
Cosine begins: at maximum (0, 1), falls to zero, reaches minimum, rises to zero, returns
Relationship: cos(x) = sin(x + π/2) and sin(x) = cos(x − π/2)
Sine symmetry: odd — sin(−x) = −sin(x) — rotational symmetry about origin
Cosine symmetry: even — cos(−x) = cos(x) — reflective symmetry about the y-axis
When the basic sine or cosine function is multiplied by a constant A, the graph stretches or compresses vertically. This vertical scale factor is captured by a quantity called the amplitude. The amplitude of a periodic function is the maximum distance the graph travels from the midline — in other words, half the total distance between the maximum and minimum values.
For y = A sin(x), the maximum output is |A| and the minimum is −|A|. The graph oscillates between these two extreme values, and the amplitude equals |A|. Note that amplitude is always a positive quantity — it measures distance, not direction. If A is negative, the graph is reflected across the midline (it begins by going down instead of up), but the amplitude is still |A|.
Amplitude controls the height of the wave. A sound wave with larger amplitude is louder. A water wave with larger amplitude is taller. An oscillating spring with larger amplitude moves farther from its rest position. In every physical context, amplitude measures the intensity or magnitude of the oscillation.
Amplitude = |A|
Maximum value: |A|
Minimum value: −|A|
If A > 0: same orientation as y = sin(x) or y = cos(x)
If A < 0: reflected across the midline — graph is vertically flipped
Amplitude = |3| = 3
Maximum value: 3 Minimum value: −3
Range: [−3, 3]
Amplitude = |−2| = 2
y = cos(x) starts at maximum (0, 1)
y = −2 cos(x) starts at minimum (0, −2) [reflected and scaled]
Amplitude is defined as a distance, so it is always non-negative. For y = −3 sin(x), the amplitude is 3, not −3. The negative sign tells you about the reflection, not the amplitude. Never report a negative amplitude.
For each function: state the amplitude, maximum value, minimum value, and range. For functions with a negative coefficient, describe the reflection.
The period of a trigonometric function is the horizontal length of one complete cycle. For y = sin(x) and y = cos(x), the period is 2π. But when the input is multiplied by a constant B — writing y = sin(Bx) — the period changes. A larger value of B causes the function to complete its cycle faster (smaller period). A smaller value of B causes the cycle to stretch out (larger period).
To see why, consider that y = sin(x) completes one cycle when its argument x goes from 0 to 2π. For y = sin(Bx) to complete one cycle, the argument Bx must go from 0 to 2π, which means x goes from 0 to 2π/B. The period is therefore 2π divided by B.
Period P = 2π / |B|
If B > 1: the period is less than 2π — the graph is horizontally compressed (faster cycle)
If 0 < B < 1: the period is greater than 2π — the graph is horizontally stretched (slower cycle)
If B < 0: use |B| for the period; the negative introduces a reflection in the horizontal direction
Period P = 2π / 4 = π/2
One cycle runs from x = 0 to x = π/2
Quarter points: 0, π/8, π/4, 3π/8, π/2
Anchor points: (0, 1), (π/8, 0), (π/4, −1), (3π/8, 0), (π/2, 1)
P = 2π / B
8 = 2π / B
B = 2π / 8 = π/4
In physics and engineering, frequency is the number of cycles completed per unit of time — it is the reciprocal of the period: f = 1/P. For a sound wave with a period of 1/440 seconds, the frequency is 440 Hz (hertz, or cycles per second). When working with trigonometric models, period and frequency describe the same property from opposite perspectives.
For each function: state the period and list the x-values of the five anchor points for one cycle starting at x = 0.
The midline of a trigonometric graph is the horizontal line that runs exactly halfway between the maximum and minimum values. For y = sin(x) and y = cos(x), the midline is the x-axis itself — the line y = 0. The function oscillates equally above and below this line.
When a constant D is added to the function — writing y = sin(x) + D or y = cos(x) + D — the entire graph shifts vertically. Every output value increases by D. The maximum becomes 1 + D, the minimum becomes −1 + D, and the midline moves from y = 0 to y = D. The shape, period, and amplitude are completely unchanged. Only the vertical position of the graph has shifted.
In real-world models, the vertical shift D often carries important meaning. If a sinusoidal function models the height of ocean water, D represents the average depth — the water level around which the tide oscillates. If it models temperature, D represents the average temperature for the cycle.
Midline: y = D
Maximum value: D + |A|
Minimum value: D − |A|
Range: [D − |A|, D + |A|]
The midline shifts up when D > 0 and down when D < 0. The amplitude and period are not affected by D.
Midline: y = 2
Maximum: 2 + 3 = 5
Minimum: 2 − 3 = −1
Range: [−1, 5]
Midline D = (8 + 2) / 2 = 10/2 = 5
Amplitude A = (8 − 2) / 2 = 6/2 = 3
Period = 2π means B = 1
Equation: y = 3 sin(x) + 5
For each function: state the midline, amplitude, maximum value, minimum value, and range.
A phase shift is a horizontal translation of a trigonometric graph — the entire wave moves left or right along the x-axis without changing its shape, height, or repeating cycle length. When the argument of sine or cosine includes a subtraction inside the parentheses — such as sin(Bx − C) — the graph is shifted horizontally.
The phase shift is the amount by which the graph has moved. Its value is C/B. To understand why, consider that sin(x) begins its standard cycle at x = 0, where the argument equals 0. For y = sin(Bx − C), the cycle begins when Bx − C = 0, which gives x = C/B. This is the new starting point of the cycle. When C/B is positive, the shift is to the right. When C/B is negative, the shift is to the left.
This is one of the most common sources of sign errors in trigonometry. The standard form is sin(Bx − C), not sin(Bx + C). When the function is written as sin(Bx + C), the phase shift is −C/B, which is a shift to the left. It is easy to confuse the direction. The safest approach is to always factor the B from inside the parentheses before reading the shift.
Phase shift = C/B
If C/B > 0: the graph shifts to the right by C/B units
If C/B < 0: the graph shifts to the left by |C/B| units
Equivalently, the cycle begins at x = C/B instead of x = 0.
Caution: if the function is written as y = A sin(Bx + C) + D, factor out B first: y = A sin[B(x + C/B)] + D, revealing a shift of −C/B (to the left).
Phase shift = C/B = π/2
The graph shifts π/2 units to the right.
The cycle begins at x = π/2 instead of x = 0.
y = cos(3x + π) = cos[3(x + π/3)]
Standard form: cos(Bx − C) where C = −π/3
Phase shift = C/B = (−π/3)/1... but we already see the shift directly:
the graph shifts π/3 units to the LEFT.
In y = sin(Bx − C): subtract means shift right. In y = sin(Bx + C): add means shift left. This is counterintuitive because adding shifts left. Always factor B from the argument before deciding direction: y = sin(Bx + C) = sin[B(x + C/B)] shifts C/B units to the left because the "+C/B" inside is moving the starting point to the left.
For each function, state the phase shift and its direction. Then state the x-value where the first cycle begins.
The four parameters — A (amplitude), B (period control), C (phase shift), and D (vertical shift) — combine in the general form y = A sin(Bx − C) + D. Reading this function correctly requires you to identify each parameter and understand its role before drawing a single point. This section develops a systematic procedure for graphing any transformed sine or cosine function and for writing an equation to match any given graph.
The procedure is: read A, B, C, D from the formula; compute the amplitude |A|, the period 2π/|B|, the phase shift C/B, and the midline y = D; locate the starting point of one cycle at x = C/B; divide the period into four equal parts to find the five anchor points; apply the amplitude and midline to find the y-values at those points; and sketch the smooth wave through all five points. Then extend the pattern as needed.
Step 1: Identify A, B, C, D directly from the formula (or factor B first if needed)
Step 2: Compute amplitude = |A|, period P = 2π/|B|, phase shift = C/B, midline y = D
Step 3: Locate the starting x of one cycle: x₀ = C/B. End of cycle: x₀ + P
Step 4: Divide the cycle into four equal intervals. The five x-values are: x₀, x₀ + P/4, x₀ + P/2, x₀ + 3P/4, x₀ + P
Step 5: Assign y-values at those five points. For y = A sin(Bx − C) + D: D, D+|A|, D, D−|A|, D (if A > 0); reversed if A < 0
Step 6: Draw the midline y = D as a dashed reference line. Plot all five points and connect with a smooth wave.
Amplitude = |2| = 2
Period P = 2π / 3
Phase shift = π / 3 (to the right)
Midline y = 1
Maximum value = 1 + 2 = 3
Minimum value = 1 − 2 = −1
Range = [−1, 3]
Cycle starts at x₀ = π/3
Cycle ends at x₀ + P = π/3 + 2π/3 = π
Quarter-cycle width = (2π/3) / 4 = π/6
Five anchor x-values: π/3, π/2, 2π/3, 5π/6, π
Corresponding y-values: 1, 3, 1, −1, 1
(midline, max, midline, min, midline)
D = (5 + (−1)) / 2 = 2
A = (5 − (−1)) / 2 = 3
Period 4π = 2π/B → B = 1/2
Cosine's maximum at x = π means phase shift = π → C/B = π → C = π · (1/2) = π/2
Equation: y = 3 cos((x/2) − (π/2)) + 2
Draw horizontal and vertical axes. Draw the midline y = 1 as a dashed horizontal line. Mark the five anchor points: (π/3, 1), (π/2, 3), (2π/3, 1), (5π/6, −1), (π, 1). Connect with a smooth sinusoidal wave that rises from the midline to the maximum, descends through the midline to the minimum, and returns. Label maximum y = 3 and minimum y = −1. Mark the period 2π/3 as the horizontal distance from start to end of the cycle. Extend the pattern one full cycle to the left as well.
For each function: state amplitude, period, phase shift, midline, maximum value, minimum value, range, and five anchor points for one cycle. Then sketch the graph.
The tangent function is fundamentally different from sine and cosine. While sin and cos are bounded — their outputs never exceed 1 or fall below −1 — the tangent function is unbounded. As its input approaches certain values, its output grows without limit in either the positive or negative direction. These values where tangent is undefined become vertical asymptotes on the graph.
Recall that tan(x) = sin(x)/cos(x). Tangent is undefined wherever cos(x) = 0, which occurs at x = ±π/2, ±3π/2, ±5π/2, and so on — the odd multiples of π/2. Between two consecutive asymptotes lies one complete cycle of the tangent function. Unlike sine and cosine, whose period is 2π, the period of tan(x) is π. The graph passes through zero once per cycle at the midpoint between consecutive asymptotes.
Within each cycle, the tangent curve rises from near −∞ at the left asymptote, crosses through zero at the center, and continues rising toward +∞ near the right asymptote. The shape is a steep S-curve or stretched diagonal that becomes nearly vertical close to the asymptotes.
| x (radians) | tan(x) | Description |
|---|---|---|
| −π/2 | undefined | Vertical asymptote (approaches −∞ from right) |
| −π/4 | −1 | Below zero |
| 0 | 0 | Zero (center of cycle) |
| π/4 | 1 | Above zero |
| π/2 | undefined | Vertical asymptote (approaches +∞ from left) |
Domain: all real numbers except x = π/2 + nπ for any integer n
Range: (−∞, ∞) — unbounded in both directions
Period: π
Zeros: x = nπ for every integer n
Vertical asymptotes: x = π/2 + nπ for every integer n (odd multiples of π/2)
No amplitude: the function is unbounded, so amplitude is not defined for tangent
Midline: y = 0
Draw horizontal and vertical axes. Draw dashed vertical lines at x = −3π/2, −π/2, π/2, and 3π/2 to represent the asymptotes. On the interval (−π/2, π/2), plot the anchor points (−π/4, −1), (0, 0), and (π/4, 1). Connect with a smooth S-shaped curve that approaches −∞ near x = −π/2 and approaches +∞ near x = π/2. Repeat the same shape in the intervals (−3π/2, −π/2) and (π/2, 3π/2). Each branch is a separate curve that does not cross the asymptotes. Label the zeros at x = 0, ±π, ±2π.
Sine and cosine complete one full cycle as the unit circle makes one full revolution (2π radians). Tangent, as the ratio sin/cos, repeats its pattern in half that distance. After π radians, both sin and cos have opposite signs, but their ratio sin/cos is the same as it was π radians earlier. The sign pattern of the ratio repeats every π, so the period of tangent is π.
The vertical asymptotes of the tangent function deserve careful attention because they arise directly from the definition: tan(x) = sin(x)/cos(x). Division by zero is undefined. Wherever cos(x) = 0, the tangent function cannot be evaluated, and the graph approaches ±∞ near that input value. The asymptotes of y = tan(x) occur at every value where cos(x) = 0: that is, at x = π/2 + nπ for every integer n.
When the tangent is transformed to y = A tan(Bx − C) + D, the same logic applies. The vertical asymptotes occur wherever the argument Bx − C equals ±π/2 + nπ. To find the asymptote equations, set the argument equal to π/2 and −π/2 (and their periodic repetitions) and solve for x.
Near an asymptote, the tangent function grows rapidly toward +∞ on one side and falls rapidly toward −∞ on the other side. It does not cross or touch the asymptote. The asymptote is a boundary that the curve approaches but never reaches. Each branch of the tangent curve is trapped between two consecutive asymptotes and fills the entire vertical range (−∞, +∞) within that interval.
Vertical asymptotes occur where the argument equals π/2 + nπ:
Bx − C = π/2 + nπ
x = (C + π/2 + nπ) / B for every integer n
Period of y = tan(Bx): P = π / |B|
Adjacent asymptotes are separated by exactly one period P = π/|B|.
Period = π / 2
Asymptotes where 2x = π/2 + nπ
x = π/4 + nπ/2
First positive asymptote (n = 0): x = π/4
Second positive asymptote (n = 1): x = π/4 + π/2 = 3π/4
The graphs of the three reciprocal functions — secant, cosecant, and cotangent — are derived directly from the graphs of cosine, sine, and tangent. Because they are reciprocals, they are undefined wherever the base function equals zero, and their asymptotes occur at exactly those zero locations.
Cosecant is the reciprocal of sine: csc(x) = 1/sin(x). It is undefined wherever sin(x) = 0, which is at x = nπ for every integer n. The graph consists of U-shaped arches opening upward and downward alternately. Where sine reaches its maximum value of 1, cosecant equals 1. Where sine reaches its minimum of −1, cosecant equals −1. Between those extremes, cosecant curves away from the x-axis toward the asymptotes at the zeros of sine. The range of cosecant is (−∞, −1] ∪ [1, ∞) — it is never between −1 and 1.
Secant is the reciprocal of cosine: sec(x) = 1/cos(x). Its asymptotes occur at the zeros of cosine: x = π/2 + nπ. The secant graph is similar in shape to cosecant — alternating arches opening upward and downward — but offset by π/2 because cosine and sine are related by that horizontal shift. The range of secant is also (−∞, −1] ∪ [1, ∞).
Cotangent is the reciprocal of tangent: cot(x) = cos(x)/sin(x). Its asymptotes occur where sin(x) = 0, at x = nπ. Like tangent, cotangent has period π. But its shape is reversed: where tangent rises steeply from left to right, cotangent falls steeply from left to right within each cycle. Cotangent passes through zero at x = π/2 + nπ.
| Function | Base Function | Asymptotes | Period | Range |
|---|---|---|---|---|
csc(x) | sin(x) | x = nπ (where sine = 0) | 2π | (−∞,−1] ∪ [1,∞) |
sec(x) | cos(x) | x = π/2 + nπ (where cosine = 0) | 2π | (−∞,−1] ∪ [1,∞) |
cot(x) | tan(x) (or sin/cos) | x = nπ (where sine = 0) | π | (−∞,∞) |
First, lightly sketch y = sin(x). Draw dashed vertical asymptotes at x = 0, ±π, ±2π (the zeros of sine). On the interval (0, π), where sine is positive, draw a U-shaped arch opening upward with its vertex at (π/2, 1) — the minimum of this arch, touching the maximum of sine. On the interval (π, 2π), where sine is negative, draw a U-shaped arch opening downward with its vertex at (3π/2, −1). Repeat this pattern in both directions. The cosecant graph never touches the x-axis; the arches approach the asymptotes without crossing them.
First, lightly sketch y = cos(x). Draw dashed vertical asymptotes at x = ±π/2, ±3π/2 (the zeros of cosine). On (−π/2, π/2), where cosine is positive, draw a U-shaped arch opening upward with vertex at (0, 1) touching the cosine maximum. On (π/2, 3π/2), where cosine is negative, draw a U-shaped arch opening downward with vertex at (π, −1). Repeat. Secant never equals zero and never exists between −1 and 1.
Draw dashed vertical asymptotes at x = 0, ±π, ±2π. On (0, π), the cotangent curve falls from near +∞ just right of x = 0, passes through (π/4, 1), crosses zero at (π/2, 0), passes through (3π/4, −1), and falls toward −∞ approaching x = π. The shape is a descending S-curve that mirrors the rising S-curve of tangent. Repeat the pattern between each pair of consecutive asymptotes.
Students often confuse which function contributes the asymptotes. The rule is simple: the asymptotes of a reciprocal function occur at the zeros of the base function. Cosecant's asymptotes come from sine's zeros; secant's asymptotes come from cosine's zeros; cotangent's asymptotes come from sine's zeros (not cosine's). When in doubt, write out the base function and find where it equals zero.
A sinusoidal model is a function of the form y = A sin(Bx − C) + D or y = A cos(Bx − C) + D that is fitted to describe real-world data that repeats in a cycle. The four parameters carry specific meanings in every context: |A| is the amplitude — the half-range of oscillation; B determines the period — the length of one complete cycle; C/B is the phase shift — the horizontal position of the first peak or zero; and D is the vertical shift — the average or equilibrium value about which the oscillation occurs.
To build a sinusoidal model from data or a described situation, the procedure is: find the maximum and minimum values; compute D = (max + min)/2 and |A| = (max − min)/2; determine the period from the data and use B = 2π/P; determine the phase shift from the location of the first maximum, minimum, or zero. Then write the equation and verify it matches the data.
Step 1: Find maximum value M and minimum value m from the data or description
Step 2: D = (M + m) / 2 (midline = average of max and min)
Step 3: |A| = (M − m) / 2 (amplitude = half the total range)
Step 4: P = length of one complete cycle → B = 2π / P
Step 5: Determine phase shift from the location of the first max, min, or zero
Step 6: Write y = A sin(Bx − C) + D or y = A cos(Bx − C) + D and verify
D = (4.5 + 0.5) / 2 = 2.5 meters (average tide level)
A = (4.5 − 0.5) / 2 = 2 meters (amplitude)
P = 12 hours → B = 2π/12 = π/6
First maximum at t = 3 → phase shift 3
Cosine form: C/B = 3 → C = 3 · (π/6) = π/2
Model: y = 2 cos((π/6)t − π/2) + 2.5
D = (90 + 30) / 2 = 60°F
A = (90 − 30) / 2 = 30°F
P = 12 months → B = 2π/12 = π/6
Minimum at m = 1: use −cos or shifted cosine.
−cos(Bm − C) reaches minimum when argument = 0 → Bm − C = 0 → m = 1
C = B · 1 = π/6
Model: T = −30 cos((π/6)m − π/6) + 60
Both sine and cosine can model the same phenomenon — they differ only by a phase shift. In practice, choose cosine when the first notable feature in the data is a maximum or minimum (cosine starts at an extreme). Choose sine when the first notable feature is a zero crossing moving upward (sine starts at zero and rises). Either choice gives an equivalent model; the phase shift adjusts accordingly.
Trigonometric graphing concentrates several common misconceptions in a small space. A student who misreads one parameter can produce a graph that is wrong in two or three ways simultaneously. The following errors appear repeatedly. Understanding each one before it trips you up will save significant time on exams and homework.
Correct: The coefficient in front of the trig function is the amplitude. The coefficient inside the argument is B, which controls the period via P = 2π/B. For y = sin(3x), amplitude = 1 (no coefficient outside) and period = 2π/3. For y = 3 sin(x), amplitude = 3 and period = 2π.
Correct: When the argument is Bx + C (addition inside), the shift is to the LEFT by C/B. Factor first: sin(x + π/3) = sin[1·(x − (−π/3))], revealing a shift of −π/3, which is π/3 to the left. Subtraction inside → right. Addition inside → left.
Correct: Amplitude is always non-negative. For y = −4 sin(x), amplitude = |−4| = 4. The negative sign indicates a vertical reflection — the graph is flipped — but the amplitude is 4, not −4.
Correct: The vertical shift D = 3 raises the midline from y = 0 to y = 3. The graph oscillates between 3 − 2 = 1 and 3 + 2 = 5, not between −2 and 2. Always draw the midline first and measure amplitude from it.
Correct: The asymptotes of y = tan(x) occur at x = π/2 + nπ — the zeros of cosine, not the zeros of tangent. The zeros of tangent (where tan(x) = 0) occur at x = nπ, which is exactly where the curve crosses the x-axis. Asymptotes and zeros are at different locations.
Correct: The five anchor points divide one cycle into FOUR equal intervals, each of width P/4. The five points are at x₀, x₀ + P/4, x₀ + P/2, x₀ + 3P/4, x₀ + P — that is, start of cycle, quarter cycle, half cycle, three-quarter cycle, end of cycle. There are four intervals between the five points, not five.
Correct: The sine and cosine graphs are smooth, continuous curves. At the maximum and minimum, the curve is rounded — it has zero slope at that point. The top and bottom of each cycle are smooth arches, not pointed spikes. Only asymptotes involve behavior that approaches infinity; maxima and minima are smooth turning points.
Each problem describes a student's work or claim. Identify the error and state the correct answer or corrected approach.
Chapter 6 will study trigonometric identities and equations. An identity is an equation that is true for every value in the domain of the functions involved — it describes a structural relationship rather than asking you to find a specific value. Knowing the graphs of all six trigonometric functions prepares you for identity work in two important ways.
First, the graphs help you understand whether a proposed identity is plausible. If you claim that sin(x) = cos(x) for all x, you need only recall that the sine and cosine graphs intersect at only two points per cycle — they are clearly not identical. On the other hand, the observation that sin(x + π/2) = cos(x) is visually obvious from the graphs: the cosine curve is the sine curve shifted π/2 units to the left. The graph confirms the identity before any algebra begins.
Second, the graphs help you solve trigonometric equations. A trigonometric equation asks for the values of x that make the equation true — this is a different question from evaluating a function or verifying an identity. Graphically, solving sin(x) = 0.5 means finding the x-values where the horizontal line y = 0.5 intersects the sine curve. Because the sine curve is periodic, there are infinitely many solutions. Chapter 6 will develop the algebraic methods for finding them precisely, but the graphical understanding from Chapter 5 provides the foundation.
Consider the equation 2 sin(x) − 1 = 0. Solving gives sin(x) = 1/2. Graphically, this asks where the horizontal line y = 1/2 intersects the graph of y = sin(x). From the unit circle, sin(π/6) = 1/2 and sin(5π/6) = 1/2. Within each period of 2π, there are exactly two solutions. On the interval [0, 2π], the solutions are x = π/6 and x = 5π/6. The graph makes the number of solutions per cycle visible before any algebraic work begins.
For each function, state: amplitude, period, phase shift, midline, maximum value, minimum value, and range.
For each function, find the five anchor points for one complete cycle and sketch the graph. Label the midline.
For each situation: identify the four parameters A, B, C, D; write the equation; use the model to answer the question asked; and verify the model at the boundary conditions.
Each problem describes a student's claim or work. Identify the error and write the correct answer.
What is wrong? State the correct period and explain how to compute it.
Identify the direction error. What is the correct phase shift and direction?
What is the error? Find the correct maximum, minimum, and range.
Explain the error completely. What actually happens to csc(x) when sin(x) = 0?
Are these the correct asymptotes for cotangent? If not, state the correct asymptote locations and explain why.
What are the correct values of A and D? Explain the formulas used to find them.
Chapter 5 made trigonometric functions visible. You can now recognize and produce the wave shape of any transformed sine, cosine, tangent, or reciprocal function, and you can build models from real periodic data. Chapter 6 will move inside those graphs, asking a different question: what algebraic relationships hold between the six trigonometric functions? An identity is a relationship that is always true — valid for every input where the expressions are defined. The Pythagorean identity sin²(x) + cos²(x) = 1 is the most fundamental, arising directly from the equation of the unit circle. From it flow the reciprocal identities, quotient identities, even-odd identities, cofunction identities, and eventually the sum, difference, double-angle, and half-angle identities. You will use these identities to simplify expressions, verify relationships, and solve trigonometric equations — finding the exact set of x-values where a trigonometric equation holds. The graphical intuition from Chapter 5 — knowing approximately where curves intersect, understanding periodic repetition, and recognizing the shape of the wave — will support every step.