Quarry School

6. Cosine starts at the top and shares the same wave shape

Explain it like I am five

Picture the same Ferris wheel of radius 1, with the rider starting at the far right, level with the hub. Sine tracked the height. Cosine tracks the sideways distance from the hub: right is +, left is −. After turning x, the rider's point is (cos x, sin x): sideways distance first, height second. So cosine is the point's x-coordinate and sine its y-coordinate: two numbers, not the axes themselves.

Follow one lap and graph (turn, sideways distance). At x = 0 the rider is 1 unit right, so the graph starts at the top, (0, 1). At π2, straight above the hub: sideways 0. At π, 1 unit left: −1. At 3π2, straight below: 0. At 2π, back: 1. Join (0, 1), (π2, 0), (π, −1), (3π2, 0) and (2π, 1) smoothly: sine's wave shape, starting at the top.

Why cos(−x) = cos x: turning the same amount clockwise mirrors the rider across the level line through the hub. The height flips sign, but the sideways distance stays, so cosine is even: opposite inputs give equal outputs.

In plain words

Picture the same wheel you used for sine, with its seat beginning at the far right. Sine records the seat's height above the center. Cosine records how far the seat is to the right of the center. At the start, that sideways distance is 1 on a wheel of radius 1, so the cosine graph begins at the top. A quarter turn later the seat is directly above the center, and its sideways distance is zero. Half a turn later it is 1 unit to the left, recorded as −1. The resulting graph has the same smooth wave shape as sine, with a different starting point.

−2π−ππ2π−11midline y = 0amplitude 1one perioddomainrange
Cosine begins at output A and repeats with the marked period.
Reminder
  • Unit circle definitions. Sine reads vertical coordinate and cosine horizontal coordinate. At circle point (1, 0), cos 0 = 1 and sin 0 = 0.
  • Range notation. [−1, 1] includes both endpoints and means −1 ≤ y ≤ 1. The interval (−∞, ∞) describes every real input.
  • Quarter turns. A full turn is 2π, so its quarter is 2π4 = π2.
  • Signed multiplication and absolute value. −43 × (−1) = 43, and |−43| = 43. Reflection changes signs, not amplitude.
  • Fraction division and substitution. 2π ÷ 35 = 10π3. Check 35 × 10π3 = 2π.
  • Integer repetitions. An Integer is a whole number such as −2, 0, or 3. Three full turns add 6π, and cos(x + 6π) = cos x.
  • Odd versus even. Sine changes sign: sin(−π2) = −1. Cosine retains output: cos(−π) = cos π = −1.
Why it works. A unit circle point can travel through every real input angle, and its horizontal coordinate changes without jumps. That gives cosine its all real number domain and Continuous graph. Its horizontal coordinate stays between −1 and 1, so its range matches sine's. A full turn returns the point to the same position, giving period 2π. Equal clockwise and counterclockwise turns have the same horizontal coordinate, so cos(−x) = cos x. A quarter turn makes the old horizontal coordinate the new height, which gives sin(x + π2) = cos x.
RuleRule: y = cos x has domain (−∞, ∞), range [−1, 1], period 2π, and cos(−x) = cos x. It begins at (0, 1). Also cos x = sin(x + π2), a shift of sine π2 left. For A cos(ωx), amplitude = |A| and T = 2π|ω| when A ≠ 0 and ω ≠ 0.
The same idea, five ways
Say it

Say: high, middle, low, middle, high.

Write it

Cosine reads the circle's horizontal coordinate and makes an even wave of period 2π.

In math
  • y = cos x
  • domain: (−∞, ∞)
  • range: [−1, 1]
  • −1 ≤ cos x ≤ 1
  • cos(x + 2nπ) = cos x for every integer n
  • cos(−x) = cos x
  • cos x = sin(x + π2)
Like

Record the wheel seat's sideways position instead of height.

See it
x = cos θy = sin θcos 0 = 1
The starting horizontal coordinate is 1.
The same idea, other ways
As the circle's sideways position

At inputs 0, π2, π, 3π2, and 2π the seat is right, above, left, below, and right again. The horizontal coordinate gives each output in the table.

input xoutput cos x01[[π|2]]0π−1[[3π|2]]02π1↓ evaluate: input given, read the output below it
Read each column as an input and its cosine output.
As sine starting earlier

Sine's first top is at π2. Move that top left by π2 and it lands at zero. Every other bend moves with it, producing the cosine wave.

−π−π/2π/2π3π/22π−11
The solid sine graph shifts left by π over 2 from the dashed parent sine.
As matching outputs at opposite inputs

cos(π3) = 12 and cos(−π3) = 12. Opposite inputs give the same height, which is what Even function means.

−π−π/2π/2π−11amplitude 1one periodsame heightsame height
Cosine begins at output A and repeats with the marked period.
xcos x
01
π20
π−1
3π20
2π1
.1Cosine graph and its five key points

Each quarter turn gives an anchor for the cosine graph. The seat begins farthest right, so its output is 1. At the top of the circle its sideways position is zero. At the left it is −1. At the bottom it is zero, and at the right it is 1 again. These five Key points make one Cycle. The curve between them is smooth and repeats forever.

  • Rule: Domain is all real numbers, (−∞, ∞), and the graph is Continuous.
  • Rule: Range is [−1, 1], amplitude is 1, and Midline is y = 0.
  • Rule: Period (T) is 2π; a Cycle is one complete repetition.
  • Rule: Cosine decreases on [0, π] and increases on [π, 2π]. Decreasing means the height falls as input increases; increasing means it rises.
  • Rule: Cosine's least positive period is 2π: if p repeats every output, cos p = cos 0 = 1, which first occurs at one full turn. Thus no shorter positive shift repeats the full graph.
π/2π3π/22π−11amplitude 1one perioddomainrange
Cosine begins at output A and repeats with the marked period.
Reminder
  • Graph coordinates. At angle π the circle point is (−1, 0), but the cosine graph point is (π, −1): input first, cosine output second.
  • Domain and range. Domain means allowed inputs; range means achieved outputs. Cosine accepts x = 100 but cannot output 2.
The same idea, five ways
Say it

Say: high, middle, low, middle, high.

Write it

Cosine begins and ends a cycle at its maximum, with its minimum halfway through.

In math
  • cos 0 = 1
  • cos(π2) = 0
  • cos π = −1
  • cos(3π2) = 0
  • cos(2π) = 1
  • cos(x + 2nπ) = cos x for every integer n
Like

Read sideways position after each quarter turn of the wheel.

See it
input xoutput cos x01[[π|2]]0π−1[[3π|2]]02π1↓ evaluate: input given, read the output below it
Read each column as an input and its cosine output.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: The first cosine point is (0, 0).
That is sine's start. The starting horizontal circle coordinate is 1.
✓ Instead: Cosine begins at (0, 1).
Tips and tricks
  • Tip: know the cosine start cold: high, middle, low, middle, high. Rebuild the input locations with quarter turns.
.2Even function and symmetry about the y-axis

Fold the graph along its vertical y-axis. Cosine's two halves line up because inputs the same distance from zero give the same height. That is Symmetry about the y-axis. An Even function is defined by equal outputs at opposite inputs, f(−x) = f(x). On the wheel, equal clockwise and counterclockwise turns give the same sideways position, although the heights on the circle differ.

  • Rule: cos(−x) = cos x for every real input.
  • Rule: Every cosine graph point (x, y) has a matching point (−x, y).
  • Rule: Even describes opposite inputs with equal outputs. It does not require positive outputs.
−2π−ππ2π−11amplitude 1one period
Cosine begins at output A and repeats with the marked period.
Reminder
  • Location of a minus sign. −cos x negates output; cos(−x) changes input. At x = 0 these give −1 and 1.
  • Special triangle values. π3 is 60°. Its adjacent leg over hypotenuse is 12.
The same idea, five ways
Say it

Say: changing the input sign leaves cosine unchanged.

Write it

Cosine is even, so its graph mirrors across the y-axis.

In math
  • cos(−x) = cos x
  • f(−x) = f(x)
  • (x, y) pairs with (−x, y)
Like

Fold the graph vertically through zero and the two halves meet.

See it
−π−π/2π/2π−11amplitude 1one period−[[π|3]][[π|3]]
Cosine begins at output A and repeats with the marked period.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: Cosine is even, so cos π must be positive.
Even means equal outputs at opposite inputs. Here cos π = cos(−π) = −1.
✓ Instead: Equal negative outputs also confirm even symmetry.
Tips and tricks
  • Tip: even means equal at opposite inputs. Odd means opposite outputs, as sin(−x) = −sin x.
.3Negative cosine

Put a mirror along the graph's horizontal middle line, y = 0. A point 1 unit above that line moves to 1 unit below. That is what the outside minus in −cos x does. The input stays in place; the output changes sign. The wave begins at the bottom instead of the top. Its amplitude, period, domain, and range stay the same. Its left and right halves still match.

  • Rule: −cos x is cos x reflected across the x-axis.
  • Rule: Its amplitude is 1, period is 2π, and range is [−1, 1].
  • Rule: It remains even: −cos(−x) = −cos x.
π/2π3π/22π−11amplitude 1one period
Cosine begins at output A and repeats with the marked period.
Reminder
  • Negating outputs. −(−1) = 1, while −0 = 0. A bottom becomes a top but an intercept stays in place.
  • Amplitude from extrema. Half the full height is 1−(−1)2 = 22 = 1.
The same idea, five ways
Say it

Say: low, middle, high, middle, low.

Write it

An outside minus reflects cosine's heights across y = 0.

In math
  • y = −cos x
  • (x, y) becomes (x, −y)
  • −cos 0 = −1
Like

Reflect each height in a horizontal mirror without moving its input.

See it
π/2π3π/22π−11amplitude 1one period
Cosine begins at output A and repeats with the marked period.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: −cos x has amplitude −1 or period −2π.
The outside minus reverses outputs, not distance measurements.
✓ Instead: A = −1, amplitude = 1, and period = 2π.
Tips and tricks
  • Tip: an outside minus swaps top and bottom. The zeros and period stay fixed.
.4Sinusoidal function and the quarter cycle shift

Imagine two recordings of the same wheel motion, with one recording starting a quarter turn earlier. Their shapes match after you slide one recording sideways. A Sinusoidal function has the sine or cosine wave shape, possibly resized or moved. A Horizontal shift moves every graph point left or right without changing its height. Cosine is sine shifted π2 left. Its expression contains x + π2 because the old sine angle is reached at an earlier input x. For a wave, Horizontal shift is also called Phase shift. Here you learn the quarter-cycle shift that relates sine and cosine; later transformations use the same idea for other distances.

  • Rule: cos x = sin(x + π2) for every real x.
  • Rule: This shifts the parent sine graph π2 left.
  • Rule: A Cycle is a full repetition; π2 is one quarter of the parent period 2π.
  • Rule: Nonconstant waves formed by resizing sine or cosine with A and ω are sinusoidal.
−π−π/2π/2π3π/22π−11
The solid sine graph shifts left by π over 2 from the dashed parent sine.
Reminder
  • Solving a linear equation. x + π2 = π2 becomes x = 0 after subtracting the same angle from both sides; 0 checks in the original equation.
  • Adding angle fractions. 5π6 + π2 = 5π6 + 3π6 = 4π3.
  • Reference angle and signs. A reference angle is the acute tilt from the nearest horizontal axis. At 4π3, subtract π to get π3. In Quadrant III sine is negative, giving −32. At 5π6, subtract from π to get π6.
The same idea, five ways
Say it

Say: cosine is sine moved a quarter cycle to the left.

Write it

Adding π2 inside sine makes each old feature occur π2 earlier on the input axis.

In math
  • cos x = sin(x + π2)
  • old input = new input + π2
  • new input = old input − π2
Like

Slide a recording of the same motion earlier while keeping all heights.

See it
−π−π/2π/2π3π/22π−11
Sine with x plus π over 2 lies on top of cosine.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: sin(x + π2) moves sine to the right.
At x = 0 the old top angle π2 has already been reached. The feature occurs earlier.
✓ Instead: It shifts left by π2. A right shift would use sin(x − π2).
Tips and tricks
  • Tip: follow one known point instead of memorizing the sign. Solve new input plus shift = old input, then substitute back.
Strategy: step by step
  1. 1. Start the parent cosine graph at (0, 1), then read the unit circle's horizontal coordinate at each quarter turn.
  2. 2. For A cos(ωx), read the outside A and inside ω as for sine. Use |A| for amplitude and 2π ÷ |ω| for period.
  3. 3. Keep midline y = 0 and range [−|A|, |A|]. A negative A changes the start from the top to the bottom.
  4. 4. Compare opposite inputs using cos(−u) = cos u. Equal outputs give symmetry about the y-axis. An Even function has equal outputs at opposite inputs. Symmetry about the y-axis means (x, y) matches (−x, y), so the left and right halves mirror each other.
  5. 5. To compare with sine, use cos x = sin(x + π2). Each old sine feature occurs π2 earlier, so the graph moves left.
Strategy
Strategy: describe a cosine wave
1
Is A or ω zero?
YesDescribe the constant function instead of assigning a smallest positive period.
NoUse amplitude |A| and period 2π ÷ |ω|.
↓
2
Is A negative?
YesThe graph begins at its bottom, y = A.
NoThe graph begins at its top, y = A.
↓
3
Are you comparing cos x with cos(−x)?
YesThe outputs agree because cosine is even.
NoIf an outside minus gives −cos x, reverse the output sign.
  1. Use the same A and ω formulas as for sine.
  2. At x = 0 cosine equals 1, so the final output equals A.
  3. Use opposite inputs to check even symmetry.
Worked exampleThe same height and width formulas for cosine

For y = −43 cos(35x), find amplitude, period, range, starting point, and whether opposite inputs give equal outputs. This asks for size, repetition length, achieved heights, and mirror symmetry.

−3π−2π−ππ2π3π−11amplitude 1.33333one periodrange
Cosine begins at output A and repeats with the marked period.
  1. A = −43 and ω = 35.The outside coefficient multiplies output, and the inside coefficient multiplies input.
  2. Amplitude = |−43| = 43.Amplitude is a distance, even when the coefficient is negative.
  3. T = 2π ÷ 35 = 2π × 53 = 10π3.The inside angle needs a full turn; divide by its coefficient using the reciprocal.
  4. Put T back: 35 × 10π3 = 2π.This verifies the proposed input length for a full turn.
  5. Range = [−43, 43].Multiplying cosine's two endpoints −1 and 1 by the coefficient gives the two extreme heights.
  6. At x = 0, y = −43 cos 0 = −43, so the start is (0, −43).The parent cosine output is 1, and the outside minus sends it to the bottom.
  7. At −x, y = −43 cos(−35x) = −43 cos(35x).Cosine is even, so changing the entire angle's sign leaves its output unchanged.
Answer
  • Amplitude = 43.
  • Period = 10π3.
  • Range = [−43, 43].
  • Starting point = (0, −43).
  • The function is even, with symmetry about the y-axis.
Check At half a period, x = 5π3, the inside angle is π, so y = −43 × (−1) = 43. This reaches the other endpoint of the range and confirms the bottom start.
Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.
Rung 1Rung 1: a quarter cycle of a faster wave

Explain how y = 3 cos(2x) comes from y = 3 sin(2x) by moving the sine graph sideways. Find the leftward distance and verify where the first sine top moves. This asks for a graph movement, rather than a change in height.

−π/2π/2π−22amplitude 3one periodmoved top
A faster wave moves left by π over 4 to reach the cosine starting point.
  1. Use the quarter-turn identity with angle 2x: 3 cos(2x) = 3 sin(2x + π2).The identity holds for every angle, so the entire angle 2x can replace x in it. Multiplying both equal outputs by 3 preserves equality.
  2. Write 2x + π2 = 2(x + π4).Distributing 2 gives 2x + 2π4, which is the original expression. A change of π4 in x changes the inside angle by π2.
  3. For the old sine top, solve 2x = π2 to get x = π4. Put it back: 2 × π4 = π2.The parent top occurs when its inside angle reaches a quarter turn. This finds and verifies the original input location.
  4. For the shifted top, solve 2x + π2 = π2: subtract π2, then divide by 2 to get x = 0. Put it back: 2 × 0 + π2 = π2.The new graph reaches the same top angle at an earlier input. Solving the inside-angle equation locates that moved feature.
  5. The top moves from (π4, 3) to (0, 3), so the shift is π4 left. T = 2π2 = π, and T4 = π4.The heights stay fixed during a horizontal move. This distance is a quarter of the faster wave's period.
Answer
  • 3 cos(2x) = 3 sin(2(x + π4)).
  • The shift is π4 left.
  • The first sine top moves from (π4, 3) to (0, 3).
Check Direct cosine substitution gives 3 cos 0 = 3 at the new top. At x = π4, cosine gives 3 cos(π2) = 0; the shifted sine gives 3 sin(π) = 0 as well. The period check 2 × π = 2π confirms that the leftward distance is one quarter cycle.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: Counterexample: cos(−x) = −cos x because negative angles have negative values.
An angle's sign gives turning direction, not cosine sign. At x = 0 the claimed equality would say 1 = −1.
✓ Instead: Cosine is even: cos(−x) = cos x. The function −cos x separately negates outputs.
✗ Not this: Counterexample: cos(0x) has period 2π0.
It is constantly cos 0 = 1, and division by zero is undefined. A constant has no smallest positive period.
✓ Instead: Its domain is all real numbers and range is {1}. Use the period formula for a nonconstant wave.
Tips and tricks
  • Tip: with ω > 0 and positive A, sine begins in the middle going up and cosine begins at the top. Negative outside coefficients reverse the corresponding direction or height. A negative ω reverses the sine direction again, while cosine is even.
  • Tip: even means equal at opposite inputs. Check cos(π3) and cos(−π3).
  • Tip: keep the same amplitude and period formulas together on the cheat sheet: |A| and 2π ÷ |ω|.
  • Tip: rebuild the cosine shift by moving sine's top from π2 to 0.
Trap. Starting cosine at the origin, or confusing cos(−x) with −cos x. Cosine starts at (0, 1). Changing its input sign leaves output unchanged, while an outside minus reflects its heights.
Keep in mind
  • Cosine starts at the top, (0, 1), while sine starts in the middle, (0, 0).
  • A minus inside disappears but a minus outside flips: cos(−x) = cos x, while −cos x is the whole wave upside down, starting at (0, −1).
  • Cosine is sine shifted π2 to the left: cos x = sin(x + π2), so every sine feature arrives π2 earlier.
  • Cosine is the x-number because, in the right triangle between the rider and the level line, the side next to the angle lies along the x-axis, and CAH is adjacent over hypotenuse, with hypotenuse 1.
Memory hookThe point is (cos x, sin x): alphabetical like (x, y), so cosine is the x-number and sine the y-number. Cosine starts at the crown, (0, 1).
Flash cards: say the answer out loud, then flip
Where does y = cos x start?
At the top: (0, 1)
What is an even function?
A function with f(−x) = f(x): opposite inputs give equal outputs.
Is cosine even or odd, and what does that do to its graph?
Even: cos(−x) = cos x, so the graph mirrors across the y-axis.
Find the amplitude, period and starting point of y = −7 cos(5x).
  • Amplitude: 7
  • Period: 2π5
  • Start: (0, −7)
Find the range of y = 2.5 cos x.
[−2.5, 2.5]
Does y = cos x start at the origin?
  • No. It starts at (0, 1)
  • sine is the one that starts at the origin.