CF 102697152 - Unit Circle

Rating: -
Tags: -
Solve time: 49s
Verified: yes

Solution

Problem Understanding

The task is a direct application of the unit circle. The input is one integer angle measured in degrees, between 0 and 359 inclusive. Imagine the ray obtained by rotating the positive x-axis counterclockwise by that angle. Its intersection with the unit circle has coordinates

(cosθ,sinθ).

The required output is only the x-coordinate, so we need to print cosθ. The input is in degrees, while Python's trigonometric functions expect radians, so the angle must first be converted using

θ rad =θ deg 180 π .

The small input range means there is no algorithmic difficulty. There is only one integer to process, and it has at most 360 possible values. Even a constant-time mathematical-library call is comfortably within the limits. The official problem gives a one-second time limit and 256 MB of memory, so there is no need for any precomputation or approximation scheme.

The main edge cases come from angles where cosine is exactly known mathematically but floating-point arithmetic may represent the value slightly differently. For example,

corresponds to the point (1,0), so the correct x-coordinate is 1. A careless implementation that converts degrees to radians incorrectly could produce a value different from 1.

Another boundary case is

whose point is (−1,0), so the answer is −1. A solution that forgets that cosine changes sign in the second and third quadrants could incorrectly print a positive value.

The quarter-turn case is also useful:

Mathematically, the answer is 0. A floating-point implementation using cos(pi / 2) may produce a tiny value such as 6.123233995736766e−17 instead of exactly zero. That is still the correct numerical answer within normal floating-point comparison tolerance, and the judge accepts floating-point output accordingly.

Finally, the largest allowed angle,

is only one degree short of a full revolution. Its cosine is positive and very close to 1. Treating 359 as though it were 360 would silently produce the wrong result.

Approaches

A literal brute-force approach could store the cosine of every integer angle from 0 through 359 and then look up the requested value. Building such a table requires 360 cosine evaluations, and looking up the answer afterward takes constant time. If we instead imagine a naive sequential search through the table, the worst case requires 360 comparisons, or 359 failed comparisons followed by the final match. This is already fast enough for this problem because the domain contains only 360 possible angles.

However, that approach completely misses the mathematical structure of the task. The x-coordinate on a unit circle is defined directly by cosine. There is no need to enumerate possible angles because the input itself is already the angle we need. The brute-force method works because the domain is tiny, but it adds unnecessary storage and computation. The observation that the desired coordinate is exactly cos(θ) reduces the entire problem to one degree-to-radian conversion and one library call.

Approach	Time Complexity	Space Complexity	Verdict
Brute Force	O(360), effectively O(1)	O(360)	Accepted, but unnecessary
Direct Cosine	O(1)	O(1)	Accepted

Algorithm Walkthrough

Read the integer angle in degrees. There is exactly one value because the problem contains a single test case.
Convert the angle from degrees to radians using angle * pi / 180. This is necessary because Python's math.cos function uses radians.
Compute the cosine of the converted angle. On the unit circle, cosine is precisely the x-coordinate of the corresponding point.
Print the resulting floating-point value. No manual rounding is needed because the output is allowed to be a floating-point number.

Why it works

For every angle θ, the point where the corresponding ray meets the unit circle is (cosθ,sinθ). The problem asks specifically for the x-coordinate, so that coordinate is exactly cosθ. The only representation change required by the programming language is converting the input from degrees to radians before calling the cosine function. Thus the algorithm computes precisely the mathematical quantity requested.

Python Solution

Pythonimport sysimport math
input = sys.stdin.readline
angle = int(input())radians = angle * math.pi / 180.0answer = math.cos(radians)
print(answer)

The first line imports sys for the requested fast-input convention and math for the cosine function and the constant π.

The input is an integer, so int(input()) is sufficient. There is no loop because the problem contains only one angle.

The expression angle * math.pi / 180.0 performs the degree-to-radian conversion. Using floating-point arithmetic here is necessary because radians are generally not integral.

math.cos then computes the x-coordinate directly. Python's floating-point representation can produce tiny errors around values such as zero, but these are normal floating-point approximations and are well within the tolerance expected for a floating-point output problem.

There is no integer-overflow concern because the angle is at most 359, and Python integers also grow automatically. There is no boundary-index calculation or iteration, so there are no off-by-one issues in the implementation.

Worked Examples

The statement provides the sample with angle 30.

angle	radians	cosine	output
30	30π/180=π/6	0.8660254037844387	0.8660254037844387

The conversion gives π/6, whose cosine is 3 /2, approximately 0.8660254037844387. This confirms that the program is using degrees as the input unit while supplying radians to math.cos.

A second useful example is 180 degrees.

angle	radians	cosine	output
180	180π/180=π	-1.0	-1.0

Here the ray points directly along the negative x-axis. The unit-circle point is (−1,0), so the x-coordinate is -1. This example exercises the sign change that occurs after crossing 90 degrees.

A third boundary-oriented example is 359 degrees.

angle	radians	cosine	output
359	359π/180	approximately 0.9998476951563913	0.9998476951563913

The point is almost back at (1,0), but not exactly there. This demonstrates why simply treating the largest input as a full 360-degree rotation would be incorrect.

Complexity Analysis

Measure	Complexity	Explanation
Time	O(1)	One degree-to-radian conversion and one cosine evaluation
Space	O(1)	Only a constant number of floating-point variables are stored

The input domain is only 360 possible angles, and the algorithm does not depend on the size of that domain through iteration. It performs a fixed amount of work and uses constant memory, so it easily fits the one-second and 256 MB limits given by the problem.

Test Cases

Because this problem has exactly one input value rather than an array or collection, the usual "all values equal" category is not applicable. The closest equivalent is checking repeated executions with the same angle, which must always produce the same result.

For the assertions, comparing floating-point output with a tolerance is preferable to requiring an exact decimal representation.

Pythonimport sysimport ioimport math

def solve(data: str) -> str:    angle = int(data.strip())    radians = angle * math.pi / 180.0    return f"{math.cos(radians)}\n"

def run(inp: str) -> str:    return solve(inp)

def assert_float_output(inp: str, expected: float, message: str) -> None:    actual = float(run(inp).strip())    assert math.isclose(actual, expected, rel_tol=1e-12, abs_tol=1e-12), message

# Provided sampleassert_float_output(    "30\n",    0.8660254037844387,    "sample 1")
# Custom: minimum-size angleassert_float_output(    "0\n",    1.0,    "angle 0 should produce x = 1")
# Custom: opposite point on the circleassert_float_output(    "180\n",    -1.0,    "angle 180 should produce x = -1")
# Custom: quarter turn, where floating-point cosine may be extremely close to zeroassert_float_output(    "90\n",    0.0,    "angle 90 should produce x = 0")
# Custom: maximum allowed angleassert_float_output(    "359\n",    math.cos(math.radians(359)),    "angle 359 must not be confused with 360")
# Custom: repeated same value behaves deterministicallyassert_float_output(    "45\n",    math.sqrt(2) / 2,    "repeated executions have the same mathematical result")

Test input	Expected output	What it validates
`30`	`0.8660254037844387`	Provided sample and degree-to-radian conversion
`0`	`1.0`	Minimum input and positive x-axis
`180`	`-1.0`	Negative cosine and opposite point
`90`	`0.0` within tolerance	Floating-point behavior around zero
`359`	`0.9998476951563913` approximately	Maximum input and near-full rotation
`45`	`0.7071067811865476` approximately	Repeated deterministic evaluation and a non-cardinal angle

Edge Cases

For angle 0, the exact input is

The algorithm converts it to 0 radians and computes cos0=1, producing 1.0. A manual quadrant-based implementation can easily introduce an unnecessary special case here, while the direct formula handles it naturally.

For angle 180, the input is

The converted angle is π, and math.cos(pi) returns -1.0. The invariant from the algorithm walkthrough is preserved because cosine already encodes the correct sign of the x-coordinate in every quadrant.

For angle 90, the input is

The mathematical answer is zero. A floating-point library may return a value extremely close to zero rather than exactly zero because π/2 cannot be represented exactly in binary floating point. The program does not round or truncate that value, which avoids introducing an arbitrary precision policy into the solution. The judge's floating-point comparison treats the tiny numerical error as equivalent to zero.

For the upper boundary, the input is

The algorithm converts it to 359π/180 radians and computes approximately 0.9998476951563913. This differs from the answer for 360 degrees, which would be exactly 1. The input restriction ends at 359, so the program correctly processes the supplied angle rather than reducing it to a nearby cardinal direction.