What is a radian and how is it used in Pure Mathematics 1 for Edexcel International A Levels?

A radian is a unit of angular measure used in mathematics. It is defined as the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius. In Pure Mathematics 1 for Edexcel International A Levels, radians are used to measure angles in trigonometry, calculus, and other mathematical contexts, providing a natural way to describe angles in terms of the circle's geometry.

How do you convert degrees to radians in the context of Edexcel International A Levels Mathematics?

To convert degrees to radians, multiply the number of degrees by π/180. This conversion is essential in Edexcel International A Levels Mathematics, particularly in Pure Mathematics 1, where problems often require angle measurements in radians for trigonometric functions and calculus applications.

Why are radians preferred over degrees in mathematical calculations?

Radians are preferred in mathematical calculations because they provide a direct relationship between the angle and the arc length, simplifying many mathematical formulas. In calculus, for example, the derivative of trigonometric functions is more straightforward when angles are measured in radians. This is why radians are extensively used in Pure Mathematics 1 for Edexcel International A Levels.

What is the relationship between radians and the unit circle in Pure Mathematics 1?

In Pure Mathematics 1, the unit circle is a fundamental concept where the radius is 1 unit. Radians naturally describe angles on the unit circle, as the length of an arc on the unit circle is numerically equal to the angle in radians. This relationship simplifies the understanding of trigonometric functions and their properties, which is crucial for Edexcel International A Levels Mathematics.

How do radians facilitate the understanding of trigonometric functions in Edexcel International A Levels?

Radians facilitate the understanding of trigonometric functions by providing a natural and consistent way to measure angles. In Edexcel International A Levels, particularly in Pure Mathematics 1, using radians allows for simpler derivatives and integrals of trigonometric functions, making it easier to solve complex mathematical problems involving periodic functions and oscillations.

Why is "Using $s = r\theta$ with $\theta$ in degrees" a common mistake in Radians?

Why it happens: Forgetting the radian requirement. How to avoid it: s = r ONLY works in radians. Convert first: \theta_rad = \theta_° × /180. Or use s = \theta_°360 · 2 r. Source: WMA11/01 examiner reports flag this on every paper

Why is "Confusing segment with sector" a common mistake in Radians?

Why it happens: Similar-sounding terms. How to avoid it: **Sector** = pie-slice (two radii + arc). **Segment** = chord-cut region (chord + arc). To find segment area: subtract the triangle from the sector.

Why is "Applying $\sin\theta \approx \theta$ with $\theta$ in degrees" a common mistake in Radians?

Why it happens: Forgetting the radian requirement. How to avoid it: Small-angle approximations REQUIRE radians. 30° 30 is wrong (it's actually 0.5); (/6) /6 0.524 is decent. Always check that is small AND in radians.

Why is "Giving decimal answer when 'in terms of $\pi$' demanded" a common mistake in Radians?

Why it happens: Habit of going to decimals. How to avoid it: Read the question. 'In terms of ' → leave as /3, 5/6, etc. 'To 3 sf' → give decimal.

Why is "Calculator in degree mode when computing $\sin\theta$ with $\theta$ in radians" a common mistake in Radians?

Why it happens: Mode left from previous trig question. How to avoid it: Test: 1 should give 0.841 in radian mode and 0.0175 in degree mode. Switch to radian mode for all radian work, especially small-angle approximations.

Pure Mathematics 1Edexcel International A Levels Mathematics (XMA01-YMA01)

Radians

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Radians Study Notes — Edexcel IAL Mathematics P1 (WMA11/01, 2018 spec — 2026 onwards)

Radian measure, conversion, arc length, sector area, segment area, and small-angle approximations. Radians are the natural unit for calculus — every later calculus result assumes radians.

What you’ll learn

Mapped to the Cambridge IGCSE XMA01-YMA01 syllabus (2026 onwards).

7.1 — Work with radian measure, including use of arc length and sector area formulae.
7.2 — Understand and use the standard small angle approximations for $\sin\theta$ , $\cos\theta$ , and $\tan\theta$ .
7.3 — Use radians in problems involving circular sectors and segments.

What is a radian?

An angle defined by arc length, not by degrees. Cleaner for calculus.

Definition. One radian is the angle subtended at the centre of a circle by an arc EQUAL IN LENGTH to the radius.

$\theta = \dfrac{s}{r}.$

If $s = r$ , then $\theta = 1$ rad. If $s$ is the full circumference $2\pi r$ , then $\theta = 2\pi$ rad — a full revolution.

The conversion.

$2\pi \text{ rad} = 360° \quad \Longleftrightarrow \quad \pi \text{ rad} = 180°.$

To convert:

Degrees → radians: multiply by $\pi/180$ .
Radians → degrees: multiply by $180/\pi$ .

Quick reference.

Degrees	Radians
$30°$	$\pi/6$
$45°$	$\pi/4$
$60°$	$\pi/3$
$90°$	$\pi/2$
$120°$	$2\pi/3$
$135°$	$3\pi/4$
$150°$	$5\pi/6$
$180°$	$\pi$
$270°$	$3\pi/2$
$360°$	$2\pi$

Why radians? Calculus formulas (e.g. $\frac{d}{dx}\sin x = \cos x$ ) only hold when $x$ is in radians. In degrees you'd get $\frac{d}{dx}\sin x° = \frac{\pi}{180}\cos x°$ , which is ugly. Radian is the 'natural' unit.

Edexcel tip. When 'leave your answer in terms of $\pi$ ' is stated, KEEP $\pi$ symbolic. Don't write $\pi/3 \approx 1.047$ unless decimal form is explicitly asked.

For a sector of radius r and central angle θ (radians), arc length s = rθ and sector area A = ½r²θ.

$\theta = s/r$ (radian definition).
$\pi$ rad $= 180°$ .
Multiply by $\pi/180$ to go to radians.
Multiply by $180/\pi$ to go to degrees.
Memorise the table of standard angles.

Arc length and sector area — formula booklet results

Both formulas in the booklet. Both need $\theta$ in RADIANS.

Arc length. $s = r\theta$ .

The arc subtended by angle $\theta$ in a circle of radius $r$ . Derivation: the full circumference is $2\pi r$ , corresponding to $2\pi$ rad. By proportionality, an arc of $\theta$ rad has length $\dfrac{\theta}{2\pi} \cdot 2\pi r = r\theta$ .

Example. Radius 12, angle $0.8$ rad:

$s = 12 \times 0.8 = 9.6.$

Sector area. $A = \tfrac{1}{2} r^{2} \theta$ .

The 'pie slice' area. Derivation: full circle area is $\pi r^{2}$ , corresponding to $2\pi$ rad. By proportionality, a sector of $\theta$ rad has area $\dfrac{\theta}{2\pi} \cdot \pi r^{2} = \tfrac{1}{2}r^{2}\theta$ .

Example. Radius 5, angle $2\pi/3$ :

$A = \tfrac{1}{2}(25)(2\pi/3) = \dfrac{25\pi}{3} \approx 26.2.$

Combined formula for perimeter of sector.

Perimeter = two radii + arc = $2r + r\theta = r(2 + \theta)$ .

Both formulas REQUIRE radians. If you have degrees:

Convert first: $\theta_{\text{rad}} = \theta_{°} \times \pi/180$ .
Or use the degree versions: $s = \dfrac{\theta_{°}}{360} \cdot 2\pi r$ , $A = \dfrac{\theta_{°}}{360} \cdot \pi r^{2}$ .

Edexcel tip. Both formulas $s = r\theta$ and $A = \tfrac{1}{2}r^{2}\theta$ are in the formula booklet — no need to memorise — but you MUST remember they need radians.

$s = r\theta$ — arc length, radians.
$A = \tfrac{1}{2}r^{2}\theta$ — sector area, radians.
Perimeter of sector = $r(2 + \theta)$ .
Both formulas in the IAL formula booklet.

Segment area — sector minus triangle

Segment = sector − isoceles triangle with two-radii sides.

Segment. The region between a chord and the corresponding arc.

Formula.

$A_{\text{seg}} = A_{\text{sector}} - A_{\triangle} = \tfrac{1}{2}r^{2}\theta - \tfrac{1}{2}r^{2}\sin\theta = \tfrac{1}{2}r^{2}(\theta - \sin\theta).$

The triangle's area is $\tfrac{1}{2}ab\sin C = \tfrac{1}{2}r \cdot r \cdot \sin\theta$ , since two sides of the triangle are both $r$ (radii) and the included angle is $\theta$ .

Worked example. Chord subtends $\pi/3$ rad at the centre of a circle of radius 10. Find the minor segment area.

Sector: $A_{\text{sector}} = \tfrac{1}{2}(100)(\pi/3) = 50\pi/3 \approx 52.36$ .
Triangle: $A_{\triangle} = \tfrac{1}{2}(100)\sin(\pi/3) = 50 \cdot (\sqrt{3}/2) = 25\sqrt{3} \approx 43.30$ .
Segment: $50\pi/3 - 25\sqrt{3} \approx 9.06$ cm².

Minor vs major segment. The MINOR segment is the smaller piece (between the chord and the shorter arc). The MAJOR is the larger. If the question asks for the major, compute it as (full circle area) − (minor segment area).

Visualisation.

     ___
    /   \
   |     |     <- minor segment (the small "bite")
    \___/
     \ /
      X
       \
        \    <- triangle (isoceles, two radii)
         \

Edexcel tip. Drawing a clear sketch of the sector and segment is worth a method mark on most exam papers. Label $r$ and $\theta$ explicitly.

Segment = sector − isoceles triangle.
$A_{\text{seg}} = \tfrac{1}{2}r^{2}(\theta - \sin\theta)$ .
Minor (smaller) vs major (larger).
Always draw a sketch.

Small-angle approximations

For small $\theta$ in RADIANS: $\sin\theta \approx \theta$ , $\cos\theta \approx 1 - \theta^{2}/2$ , $\tan\theta \approx \theta$ .

Approximations (in the formula booklet):

$\sin\theta \approx \theta, \quad \cos\theta \approx 1 - \tfrac{1}{2}\theta^{2}, \quad \tan\theta \approx \theta.$

These hold when $\theta$ is small (in radians, $|\theta| \ll 1$ ). They come from the first few terms of the Taylor series:

$\sin\theta = \theta - \tfrac{\theta^{3}}{6} + \tfrac{\theta^{5}}{120} - \ldots; \qquad \cos\theta = 1 - \tfrac{\theta^{2}}{2} + \tfrac{\theta^{4}}{24} - \ldots.$

Accuracy.

$\theta$ (rad)	$\sin\theta$	Approx $\theta$	Error
$0.05$	$0.04998$	$0.05$	$0.0004\%$
$0.1$	$0.09983$	$0.1$	$0.17\%$
$0.3$	$0.29552$	$0.3$	$1.5\%$
$0.5$	$0.47943$	$0.5$	$4.3\%$

By $\theta = 0.5$ rad ( $\approx 28.6°$ ) the approximation $\sin\theta \approx \theta$ has 4.3% error — bad.

Working with substitutions. If $\theta$ is small, so is any multiple of $\theta$ :

$\sin(2\theta) \approx 2\theta; \quad \cos(2\theta) \approx 1 - \tfrac{(2\theta)^{2}}{2} = 1 - 2\theta^{2}.$

Worked example. Approximate $\dfrac{1 - \cos 2\theta}{\theta \sin\theta}$ for small $\theta$ .

Numerator: $1 - \cos 2\theta \approx 1 - (1 - 2\theta^{2}) = 2\theta^{2}$ .
Denominator: $\theta \sin\theta \approx \theta \cdot \theta = \theta^{2}$ .
Quotient: $2\theta^{2}/\theta^{2} = 2$ .

WHY ONLY RADIANS? The approximation comes from the Taylor series, which is derived using calculus, which uses radians. Plugging in degrees breaks the proportionality — $\sin 1° \approx 0.0175$ , NOT $1$ .

$\sin\theta \approx \theta$ (radians only).
$\cos\theta \approx 1 - \theta^{2}/2$ .
$\tan\theta \approx \theta$ .
Substitute carefully when $\theta$ is replaced by $2\theta$ , $3\theta$ , etc.
Works for $|\theta|$ less than $\sim 0.2$ rad.

How it’s examined

Radians appears every WMA11/01 paper — typically a 6-10 mark question combining arc length, sector area, and segment area, often in a real-world context (e.g. a clock face, a pendulum, a windscreen wiper). Small-angle approximations appear in 4-6 mark questions in later P1 sittings; the spec rates them as worth examining annually. Mark scheme awards a method mark for identifying the segment-as-sector-minus-triangle approach. Calculator mode errors are the #1 cause of dropped marks. A* candidates: leave answers in exact form ( $\pi$ , $\sqrt{}$ ) where possible; convert to decimal only when explicitly demanded.

Sources: Pearson Edexcel International Advanced Level Mathematics specification (2018 — current for 2026 onwards); Edexcel IAL Mathematical Formulae and Statistical Tables (2018); WMA11/01 Examiner Reports — January 2023, June 2023, January 2024, June 2024. Last reviewed 2026-05-12.

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Step-by-step worked examples — Radians

Step-by-step solutions to past-paper-style questions on radians, written exactly the way a tutor would explain them at the board.

1Convert between degrees and radians (3 marks, P1)
Core• Style: WMA11/01 January 2024 Q3• radians, conversion
Question
(a) Convert $150°$ to radians, giving your answer in terms of $\pi$ . (b) Convert $\dfrac{3\pi}{8}$ radians to degrees.
Step-by-step solution
1. Step 1
  (a) Use $180° = \pi$ rad. So $150° = \dfrac{150}{180}\pi = \dfrac{5\pi}{6}$ rad.
2. Step 2
  (b) $\pi$ rad $= 180°$ , so $\dfrac{3\pi}{8}$ rad $= \dfrac{3}{8}(180°) = 67.5°$ .
Answer
(a) $\dfrac{5\pi}{6}$ rad. (b) $67.5°$ .
Examiner tip
B1 for each conversion. Method: multiply by $\pi/180$ to go to radians; by $180/\pi$ to go to degrees. When 'in terms of $\pi$ ' is asked, DON'T give a decimal.
2Arc length (3 marks, P1)
Core• Style: WMA11/01 June 2024 Q3• arc length
Question
A circular sector has radius 12 cm and angle $0.8$ rad. Find the arc length.
Step-by-step solution
1. Step 1
  Use $s = r\theta$ (with $\theta$ in radians):
  $s = 12 \times 0.8 = 9.6 \text{ cm}$
Answer
$s = 9.6$ cm.
Examiner tip
M1 for formula and substitution; A1 for the answer with units. The formula $s = r\theta$ ONLY works in RADIANS. If the angle is in degrees, convert first or use $s = \dfrac{\theta}{360}\cdot 2\pi r$ instead.
3Sector area (3 marks, P1)
Core• Style: WMA11/01 January 2023 Q6• sector area
Question
Find the area of a sector with radius 5 cm and angle $\dfrac{2\pi}{3}$ rad.
Step-by-step solution
1. Step 1
  Use $A = \tfrac{1}{2}r^{2}\theta$ :
  $A = \tfrac{1}{2}(5)^{2}(2\pi/3) = \tfrac{1}{2}(25)(2\pi/3) = \dfrac{25\pi}{3}$
2. Step 2
  $\dfrac{25\pi}{3} \approx 26.18$ cm².
Answer
$\dfrac{25\pi}{3} \approx 26.2$ cm² (3 sf).
Examiner tip
M1 for substitution; A1 for exact form; A1 for decimal to 3 sf. As with arc length, the formula requires RADIANS — never plug in degrees directly. Mark scheme awards either exact or 3 sf form unless one is explicitly demanded.
4Segment area (6 marks, P1)
Extended• Style: WMA11/01 June 2023 Q9• segment, area
Question
A chord $PQ$ subtends an angle of $\dfrac{\pi}{3}$ rad at the centre $O$ of a circle of radius 10 cm. Find the area of the minor segment cut off by the chord, to 3 sf.
Step-by-step solution
1. Step 1
  Segment area = Sector area − Triangle area.
2. Step 2
  Sector area: $A_{\text{sector}} = \tfrac{1}{2}r^{2}\theta = \tfrac{1}{2}(100)(\pi/3) = \dfrac{50\pi}{3}$ .
3. Step 3
  Triangle area (isoceles, two sides $r$ , included angle $\theta$ ): $A_{\triangle} = \tfrac{1}{2}r^{2}\sin\theta = \tfrac{1}{2}(100)\sin(\pi/3) = 50 \cdot \dfrac{\sqrt{3}}{2} = 25\sqrt{3}$ .
4. Step 4
  Segment: $A = \dfrac{50\pi}{3} - 25\sqrt{3} \approx 52.36 - 43.30 = 9.06$ cm².
Answer
$\dfrac{50\pi}{3} - 25\sqrt{3} \approx 9.06$ cm² (3 sf).
Examiner tip
M1 for the sector-minus-triangle method; A1 for sector area; A1 for triangle area; M1 for subtracting; A1 for exact; A1 for 3 sf. Both formulas use radians. Common mistake: confusing 'segment' with 'sector' — segment is the curved region BETWEEN the chord and the arc; sector includes the triangle.
5Small-angle approximations (5 marks, P1)
Extended• Style: WMA11/01 January 2024 Q8• small angle
Question
For small values of $\theta$ (in radians), approximate the expression $\dfrac{1 - \cos 2\theta}{\theta \sin\theta}$ .
Step-by-step solution
1. Step 1
  Use small-angle approximations: $\sin\theta \approx \theta$ ; $\cos\theta \approx 1 - \tfrac{1}{2}\theta^{2}$ , so $\cos 2\theta \approx 1 - \tfrac{1}{2}(2\theta)^{2} = 1 - 2\theta^{2}$ .
2. Step 2
  Numerator: $1 - \cos 2\theta \approx 1 - (1 - 2\theta^{2}) = 2\theta^{2}$ .
3. Step 3
  Denominator: $\theta \sin\theta \approx \theta \cdot \theta = \theta^{2}$ .
4. Step 4
  Quotient:
  $\dfrac{1 - \cos 2\theta}{\theta \sin\theta} \approx \dfrac{2\theta^{2}}{\theta^{2}} = 2$
Answer
$\approx 2$
Examiner tip
M1 for using $\sin\theta \approx \theta$ ; M1 for using $\cos\theta \approx 1 - \theta^{2}/2$ (with the $2\theta$ correctly substituted); A1 for both approximations; M1 for the ratio; A1 for the answer $2$ . Small-angle approximations REQUIRE radians. Approximations are listed in the IAL formula booklet.

Key Formulae — Radians

The formulae you need to memorise for radians on the Pearson Edexcel IAL XMA01 / YMA01 paper, with every variable defined in plain English and a note on when to use it.

Radian-degree conversion
$180° = \pi \text{ rad}; \quad 1° = \dfrac{\pi}{180} \text{ rad}; \quad 1 \text{ rad} = \dfrac{180°}{\pi}$
$—$
no variables — it's a unit conversion
When to use
Whenever you need to convert. NOT in the formula booklet — memorise.
Example
$60° = \pi/3$ rad. $\pi/4$ rad $= 45°$ .
Arc length
$s = r\theta$
$s$
arc length
$r$
radius
$θ$
angle subtended at centre, in RADIANS
When to use
Length of a circular arc. IS in the IAL formula booklet. ONLY works with radians.
Example
Radius 12, angle $0.8$ rad: $s = 12 \cdot 0.8 = 9.6$ .
Sector area
$A = \tfrac{1}{2}r^{2}\theta$
$A$
sector area
$r$
radius
$θ$
angle in RADIANS
When to use
Area of a 'pie slice'. IS in the formula booklet.
Example
$r = 5$ , $\theta = 2\pi/3$ : $A = \tfrac{1}{2}(25)(2\pi/3) = 25\pi/3$ .
Segment area
$A_{\text{seg}} = \tfrac{1}{2}r^{2}(\theta - \sin\theta)$
$A_seg$
minor segment area
$r$
radius
$θ$
angle in RADIANS
When to use
Area of the region between a chord and the corresponding arc. Derived from $A_{\text{sector}} - A_{\triangle}$ , with $A_{\triangle} = \tfrac{1}{2}r^{2}\sin\theta$ . The combined formula IS sometimes in extended formula sheets, but the individual pieces are always available.
Example
$r = 10$ , $\theta = \pi/3$ : $A = 50(\pi/3 - \sin(\pi/3)) = 50\pi/3 - 25\sqrt{3}$ .
Small-angle approximations
$\sin\theta \approx \theta, \quad \cos\theta \approx 1 - \tfrac{1}{2}\theta^{2}, \quad \tan\theta \approx \theta$
$θ$
angle in RADIANS, small (|θ| ≪ 1)
When to use
When $\theta$ is small enough that higher-order terms can be ignored. IS in the IAL formula booklet.
Example
$\sin 0.1 \approx 0.1$ (true value: $0.0998$ , error $\sim 0.2\%$ ).

Key Definitions and Keywords — Radians

Definitions to memorise and the exact keywords mark schemes credit for radians answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IAL XMA01 / YMA01 sitting.

Radian
Examiner keyword
The angle subtended at the centre of a circle by an arc EQUAL IN LENGTH to the radius. Dimensionless ratio: $\theta = s/r$ .
Example
A full revolution is $2\pi$ radians, equivalent to $360°$ .
Arc
Examiner keyword
A portion of the circumference of a circle.
Sector
Examiner keyword
A 'pie-slice' region bounded by two radii and an arc.
Segment
Examiner keyword
The region between a chord and the arc it cuts off. Minor segment (smaller) or major segment (larger).
Chord
A straight line joining two points on a circle.

Common Mistakes and Misconceptions — Radians

The traps other students keep falling into on radians questions — taken from recent Pearson Edexcel IAL XMA01 / YMA01 examiner reports and mark schemes — and how to avoid them.

✕Using $s = r\theta$ with $\theta$ in degrees
WMA11/01 examiner reports flag this on every paper
Why it happens
Forgetting the radian requirement.
How to avoid it
$s = r\theta$ ONLY works in radians. Convert first: $\theta_{\text{rad}} = \theta_{°} \times \pi/180$ . Or use $s = \dfrac{\theta_{°}}{360} \cdot 2\pi r$ .
✕Confusing segment with sector
Why it happens
Similar-sounding terms.
How to avoid it
Sector = pie-slice (two radii + arc). Segment = chord-cut region (chord + arc). To find segment area: subtract the triangle from the sector.
✕Applying $\sin\theta \approx \theta$ with $\theta$ in degrees
Why it happens
Forgetting the radian requirement.
How to avoid it
Small-angle approximations REQUIRE radians. $\sin 30° \approx 30$ is wrong (it's actually $0.5$ ); $\sin (\pi/6) \approx \pi/6 \approx 0.524$ is decent. Always check that $\theta$ is small AND in radians.
✕Giving decimal answer when 'in terms of $\pi$ ' demanded
Why it happens
Habit of going to decimals.
How to avoid it
Read the question. 'In terms of $\pi$ ' → leave as $\pi/3$ , $5\pi/6$ , etc. 'To 3 sf' → give decimal.
✕Calculator in degree mode when computing $\sin\theta$ with $\theta$ in radians
Why it happens
Mode left from previous trig question.
How to avoid it
Test: $\sin 1$ should give $0.841$ in radian mode and $0.0175$ in degree mode. Switch to radian mode for all radian work, especially small-angle approximations.

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Radians

Short Study Notes in the form of Slides

Short Study Notes — Radians

Detailed Notes

What you’ll learn

How it’s examined

1Convert between degrees and radians (3 marks, P1)

2Arc length (3 marks, P1)

3Sector area (3 marks, P1)

4Segment area (6 marks, P1)

5Small-angle approximations (5 marks, P1)

Radian-degree conversion

Arc length

Sector area

Segment area

Small-angle approximations

Radian

Arc

Sector

Segment

Chord

✕Using s=rθs = r\thetas=rθ with θ\thetaθ in degrees

✕Confusing segment with sector

✕Applying sin⁡θ≈θ\sin\theta \approx \thetasinθ≈θ with θ\thetaθ in degrees

✕Giving decimal answer when 'in terms of π\piπ' demanded

✕Calculator in degree mode when computing sin⁡θ\sin\thetasinθ with θ\thetaθ in radians

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Radians — frequently asked questions

✕Using $s = r\theta$ with $\theta$ in degrees

✕Applying $\sin\theta \approx \theta$ with $\theta$ in degrees

✕Giving decimal answer when 'in terms of $\pi$ ' demanded

✕Calculator in degree mode when computing $\sin\theta$ with $\theta$ in radians