Quarry School

Sine and cosine: every input works, and the output stays between −1 and 1

Explain it like I am five

Picture a dog on a tight leash exactly 1 meter long, tied to a post. It can run any number of laps, either way, yet stays within 1 meter of the post. That circle of radius 1 is the unit circle. East-west position is the x-coordinate, north-south the y-coordinate; east and north count as positive, west and south as negative. Sine is opposite over hypotenuse, here north-south distance over leash, and the leash is 1, so sin t is the y-coordinate itself; likewise cos t is the x-coordinate.

Example: is t = 500 allowed? The input t is the distance walked along the circle. One lap is 2π, about 6.28, so 500 is about 79.6 laps. The walk still ends somewhere, so sin 500 exists: about −0.468 in radian mode. Could sin t be 1.3? No: the dog would be 1.3 meters north, past the leash's end.

So for sine and cosine the domain (all allowed inputs) is every real number, and the range (all possible outputs) is [−1, 1]: every number from −1 to 1, both ends included.

In plain words

Picture a rider on a Ferris wheel with radius 1. You describe the rider with two numbers. The horizontal position is cosine, and the height above the center is sine. An input is the angle you turn through. The domain means all the inputs the function accepts. The range means all the outputs it can produce. You may turn through any angle, forward or backward, so sine and cosine accept every real number. You can turn for many laps, but the rider stays on the same wheel. The height and horizontal position stay between −1 and 1, including both ends.

x = cos θy = sin θP(cos t, sin t)
P always sits on the rim, 1 unit from the center, so its x-leg (cos t) and its y-leg (sin t) can never be longer than 1.
Reminder
  • Interval endpoints. Square brackets include an endpoint: [−1, 1] includes both −1 and 1.
  • Squaring a negative. (−0.6)2 = (−0.6)(−0.6) = 0.36, because matching signs give a positive product.
x = cos θy = sin θP(cos t, sin t)
P always sits on the rim, 1 unit from the center, so its x-leg (cos t) and its y-leg (sin t) can never be longer than 1.
input functionoutput domainsin t(−∞, ∞)cos t(−∞, ∞)
Both functions accept every real input.
input functionoutput rangesin t[−1, 1]cos t[−1, 1]
Both outputs stay between −1 and 1, including the endpoints.
input Functionoutput Domain (inputs allowed)sin tall real numbers, (−∞, ∞)cos tall real numbers, (−∞, ∞)
Read the output entry in the column under its input or category.
input Functionoutput Range (outputs possible)sin t[−1, 1]cos t[−1, 1]
Read the output entry in the column under its input or category.
Why it works. A walk of any length ends somewhere on the unit circle, and reading a coordinate never divides by anything. That gives every real input. Each point satisfies x2 + y2 = 1. Since x2 cannot be negative, y2 = 1 − x2 is at most 1, so −1 ≤ y ≤ 1. Interchanging x and y gives the same bound for x. These bounds give the entire range, rather than only limits: horizontal and vertical lines at any position between −1 and 1 meet the circle.
RuleFor sin t and cos t: domain (−∞, ∞), all real numbers; range [−1, 1]. Thus −1 ≤ sin t ≤ 1 and −1 ≤ cos t ≤ 1. Absolute value |a| means a number's distance from 0, so the notes' |sin t| ≤ 1 and |cos t| ≤ 1 say the same thing.
The same idea, five ways
Say it

Sine and cosine accept every real input and return a number from negative one through one.

Write it

The domain is all real numbers, and the range is the closed interval from −1 to 1.

In math
  • t is any real number
  • domain: (−∞, ∞)
  • range: [−1, 1]
  • −1 ≤ sin t ≤ 1
  • −1 ≤ cos t ≤ 1
  • On a graph, every input is allowed and every output height lies from −1 to 1.
  • {t | t is a real number}
  • {v | −1 ≤ v ≤ 1}
Like

A Ferris wheel can turn through any angle while its rider stays within one radius of the center.

See it
x = cos θy = sin θP(cos t, sin t)
Both coordinates of the rider stay within one radius of the center.
The same idea, other ways
As a wheel

You can rotate the wheel any number of turns. Its rider's height and horizontal reach stay within one radius of the center.

x = cos θy = sin θP(cos t, sin t)
The input can keep growing while the point stays on the same wheel.
As a machine

The domain is the machine's accepted input list. The range is its possible output list. The sine machine accepts 1000 radians, but it cannot return the height 1000.

1000 radiansread height onradius 1a number in [−1, 1]inputoutput
An unrestricted input does not mean an unrestricted output.
With numbers

The point (0.8, −0.6) is on the circle because 0.64 + 0.36 = 1. Its cosine is 0.8 and its sine is −0.6. Both fit in [−1, 1].

x = cos θy = sin θ(0.8, −0.6)
The point (0.8, −0.6) has horizontal position 0.8 and height −0.6.
Why the bound holds

If a height exceeded 1 in size, its square would exceed 1. Adding the nonnegative horizontal square could not make x2 + y2 equal 1.

x2 + y2 = 1
x2 ≥ 0
y2 ≤ 1
A coordinate bigger than the radius cannot lie on the circle.
FunctionDomain (inputs allowed)Range (outputs possible)
sin tall real numbers, (−∞, ∞)[−1, 1]
cos tall real numbers, (−∞, ∞)[−1, 1]
.1Sine reads height

The sine output is the rider's height measured from the center. Above the center gives a positive height; below gives a negative height.

  • sin t = y.
  • Domain: (−∞, ∞).
  • Range: [−1, 1].
x = cos θy = sin θP(cos t, sin t)
At the top, sine reaches its largest output, 1.
Worked exampleA height inside the wheel

The question asks whether the output −35 can be a sine value.

x = cos θy = sin θ(0.8, −0.6)
This circle point has the requested height −0.6.
  1. −35 = −0.6, which is between −1 and 1.Converting the fraction makes its position clear.
  2. The point (0.8, −0.6) lies on the circle, so the height −0.6 occurs.0.82 + (−0.6)2 = 0.64 + 0.36 = 1.
Answer
Yes. −35 is in the sine range.
Check The output's size is 35, which is less than 1.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The input must stay between −1 and 1 because the output must.
An angle input counts travel, while the output records a bounded coordinate.
✓ Instead: Every real input is accepted. Only the output is bounded.
Tips and tricks
  • Use this object's defining formula to check the example before relying on a remembered pattern.
.2Cosine reads horizontal position

The cosine output measures how far right or left the rider sits from the center. Right is positive, and left is negative.

  • cos t = x.
  • Domain: (−∞, ∞).
  • Range: [−1, 1].
x = cos θy = sin θP(cos t, sin t)
At the far left, cosine reaches its smallest output, −1.
Worked exampleA position beyond the rim

The question asks whether the output 75 can be a cosine value.

75 = 1.4
1.4 > 1
cos t cannot equal 1.4
The proposed output lies beyond the wheel's right edge.
  1. 75 = 1.4, which is greater than 1.Divide 7 by 5 to compare the proposed position with the radius.
  2. No circle point has horizontal coordinate 1.4.The radius is 1, so the circle's farthest right position is 1.
Answer
No. 75 is outside the cosine range.
Check A coordinate of 1.4 would already contribute 1.96 to x2 + y2, which cannot equal 1.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The input must stay between −1 and 1 because the output must.
An angle input counts travel, while the output records a bounded coordinate.
✓ Instead: Every real input is accepted. Only the output is bounded.
Tips and tricks
  • Use this object's defining formula to check the example before relying on a remembered pattern.
Strategy: step by step
  1. 1. Read the question: is it about an input t (domain) or about an output value (range)?
  2. 2. Domain of sin or cos: the answer is always yes, for any number in degrees or radians, positive or negative.
  3. 3. Range of sin or cos: check whether the value is between −1 and 1, endpoints included. If it is, yes; if it is bigger than 1 or smaller than −1, no.
  4. 4. Unmask disguised numbers before comparing: π ≈ 3.14 and 32 = 1.5 are outside, while 32 ≈ 0.87 is inside.
  5. Read a reference-table entry in the column under its input or category in the matching picture.
Strategy
Deciding domain or range membership
1
Does the question ask about the domain?
YesTreat the number as an input angle. Every real input is accepted.
NoTreat the number as an output. Check whether −1 ≤ output ≤ 1.
↓
2
Is the output within [−1, 1]?
YesIt is in the range.
NoIt is outside the range.
  1. 1. Decide whether the proposed number is an input angle or an output.
  2. 2. For a sine or cosine input, answer yes for every real number.
  3. 3. For an output, compare it with −1 and 1, including both endpoints.
Worked exampleDomain and range questions for sine and cosine

The question asks whether each proposed input or output is allowed. Answer yes or no. (a) Is t = 1265 in the domain of cos t? (b) Is −0.6 in the range of sin t? (c) Is 32 in the range of cos t? (d) Is 1 in the range of sin t?

x = cos θy = sin θ(0.8, −0.6)
The point (0.8, −0.6) shows that the proposed sine output −0.6 is possible.
−11
−0.6 and 1 belong to [−1, 1], while 1.5 lies outside it.
  1. (a) Yes. cos 1265 means: walk 1265 units around the unit circle and read the x-coordinate of the stopping point.The walk always ends somewhere, so the domain of cosine is all real numbers.
  2. (b) Yes. −0.6 is between −1 and 1.The range of sine is [−1, 1].
  3. (c) No. 32 = 1.5, which is bigger than 1.No point of the unit circle is 1.5 units to the right of the center; the rim is only 1 unit away.
  4. (d) Yes. At the top of the circle P = (0, 1), so sin 90° = 1.1 is an endpoint, and the bracket in [−1, 1] includes it.
Answer
  • (a) yes
  • (b) yes
  • (c) no
  • (d) yes
Check For (b), find an actual point with height −0.6: (0.8, −0.6) is on the unit circle because 0.82 + (−0.6)2 = 0.64 + 0.36 = 1.
Counterexamples: what it is not, and the tempting wrong moves
✗ Not this: The input 5 is impossible because 5 > 1.
The bound applies to the output, rather than the input angle.
✓ Instead: sin 5 and cos 5 are defined because 5 is a real input.
✗ Not this: cos t = 32 is possible because 32 is a real number.
The cosine output is a horizontal coordinate on a circle of radius 1, so 1.5 is too far from the center.
✓ Instead: 32 is outside the range [−1, 1].
Tips and tricks
  • Write 'input' above a domain question and 'output' above a range question before comparing numbers.
  • For the range, picture the wheel inside a square whose edges are x = ±1 and y = ±1.
Trap. Mixing up the two questions. The domain is about the input t, and every t is allowed for sine and cosine. The range is about the output, which must be between −1 and 1. The input t = 5 is fine for sine; the output 5 is impossible.
Keep in mind
  • Domain is about the input and range about the output: t = 7 is a fine input for sine, but 7 is an impossible output.
  • Unmask disguised numbers before comparing with 1: 2 ≈ 1.41 and 75 = 1.4 are too big for sine or cosine, while −56 ≈ −0.83 fits.
  • Here x and y are the point's coordinates, signed distances measured along the axes, not the axes themselves: at the point (0.28, −0.96), cos t = 0.28 and sin t = −0.96, negative because the point is below the center.
  • The ends count: sin t = 1 at the top point (0, 1), because the square bracket in [−1, 1] includes 1.
Memory hookAlphabetical order: the point is (cos t, sin t), since c comes before s as x comes before y. The 1 meter leash keeps both between −1 and 1.
Flash cards: say the answer out loud, then flip
What is the domain of a function?
All the inputs it accepts.
What are the domain and range of sin t and cos t?
  • Domain: all real numbers.
  • Range: [−1, 1].
Is −910 a possible value of sin t?
Yes: −0.9 is between −1 and 1.
Is 87 a possible value of cos t?
No: 87 ≈ 1.14, bigger than 1.
Is t = 25 an allowed input for sin t, even though 25 is bigger than 1?
Yes. The limit of 1 applies to the output, not the input.
On the unit circle, which coordinate is sin t and which is cos t?
  • sin t is y, the height.
  • cos t is x, the left-right position.